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10 Some Regression models for AF measures

From a policy perspective, in addition to measuring poverty we must perform some vital analyses regarding the transmission mechanisms between policies and poverty measures. Issues we may wish to explore with a regression model include the determinants of poverty at the household level in the form of poverty profiles or the elasticity of poverty to economic growth, while controlling for other determinants. We may also be interested in understanding how macro variables such as average income, public expenditure, decentralization, information technology, and so on relate to multidimensional poverty levels or changes across groups or regions—and across time. Through regression analysis, we can partially study these transmission mechanisms by looking at the determinants of multidimensional poverty. In a regression model, we can account for the effect or the ‘size’ of determinants of multidimensional poverty, which would not be possible with a purely descriptive analysis.

Such analyses are routinely performed for income poverty using what we will term ‘micro’ or ‘macro’ regressions. As is explained below, the term ‘micro’ refers to analyses in which the unit of analysis is a person or household; the term ‘macro’ refers to analyses in which the unit of analysis is a subgroup, such as a district, a state, a province, or a country. This section provides the reader with a general modelling framework for analysing the determinants of Alkire–Foster poverty measures, at both micro and macro levels of analyses.

In general in micro regressions, the focal variable to be modelled may be a binary variable denoting a person’s status as poor (or non-poor) or a variable denoting the deprivation score assigned to the poor. In macro regressions, the focal variable to model is a subgroup poverty measure like the poverty headcount ratio or any other Foster–Greer–Thorbecke (FGT) poverty measure. As with regressions that model the monetary headcount ratio or the poverty gap, macro regressions with -dependent variables must respect their nature as cardinally meaningful values ranging from zero to one. In these cases, a classic linear regression is not the appropriate model. The common assumptions of the classic linear regression fall short because the range of the dependent variable is bounded and may not be continuous or follow a normal distribution that is often assumed in linear regression models.

Generalized linear models (GLMs), by contrast, are preferred as the data analytic technique because they account for the bounded and discrete nature of the AF-type dependent variables. GLMs extend classic linear regression to a family of regression models where the dependent variable may be normally distributed or may follow a distribution within the exponential family—such as the gamma distribution, bernoulli distribution, or binomial distribution. GLMs encompass models for quantitative and qualitative dependent variables, such as linear regression models, logit and probit models, and models for fractional data. Hence they offer a general framework for our analysis of functional relationships.[1]

This section presents the GLM as an overall framework to study micro and macro determinants of multidimensional poverty. Within this framework we are able to account for the bounded nature of the Adjusted Headcount Ratio and the incidence while modeling their determinants. We are also able to model these determinants for the probability of being multidimensionally poor.

This Chapter is structured as follows. We begin by differentiating micro and macro regression analyses. For this purpose, we review the measure of the AF class, its consistent partial indices, and the type of variables they represent in a regression framework. We then present the general structure and possible applications of the GLMs to AF measures. We begin with an exposition of linear regression models and how these extend to models for binary dependent variables—logit and probit—and fractional[2] data. We assume readers have some background in applied statistics and key elements of estimation and inference. Our exposition deals with cross-sectional data but could be easily extended to panel data.[3] Before we begin we should point out that the notation used in this chapter is self-contained. Some notation may duplicate that used in other sections or chapters for different purposes. When the notation is linked to discussions in other sections or chapters, it will be specified accordingly.

10.1 Micro and Macro Regressions

The AF measures can be used to analyse poverty determinants[4] for a household or person (henceforth we use the term ‘household’) and for a population subgroup. We could study determinants of household or subgroup poverty in a ‘micro’ and a ‘macro’ context. In what follows, the term ‘micro’ refers to regressions where the unit of analysis is the person or household. The term ‘macro’ refers to regressions where the unit of analysis is some spatial or social aggregate, such as a district, state, province, ethnic group, or country. Micro regressions are useful for describing the distinctive features of multidimensional poverty profiles across households (in a given country) or to understand the determinants of poverty. Macro regressions, on the other hand, are useful for studying the determinants of poverty at the province, district, state, or country levels. Both types of regressions use specific components of the AF measures. In the case of micro regressions, the focal variable is the (household) censored deprivation score. From the exposition of Chapter 5, we know that if the deprivation score of a household is equal to or greater than the multidimensional poverty cutoff (), the household is identified as multidimensionally poor. This poverty status of a household is represented by a binary variable (indicator function) that takes the value of one if the household is identified as multidimensionally poor and zero otherwise.

A natural question that arises is how to analyse the ‘causes’ (in the sense of determinants) that underlie the (multidimensional) poverty status of a household. An intuitive way would be to model the probability of a household becoming multidimensionally poor or falling into multidimensional poverty. A crucial point should be noted here, which may be more particular to multidimensional notions of poverty than their monetary counterparts: when modelling the probability of a household being in monetary poverty, various health- and education-related variables, which are not embedded in the monetary poverty measures, are used as exogenous variables.[5] In a multidimensional case, these exogenous variables may be used directly to construct the poverty measure and so the probability models at the household level are subject to a potential endogeneity issue. For example, if among the explanatory variables we include an asset variable like car ownership, and if that indicator was also included among the ‘assets’ indicator that appears in the multidimensional poverty measure, there will be an endogeneity issue in the model. A typical approach to deal with endogeneity is to use an instrumental variable, but often it is very difficult to find a valid instrument.[6] An alternative approach would be to restrain the set of explanatory variables of the household regression model to non-indicator measurement variables[7]—like certain demographic variables—or additional socioeconomic characteristics of the household. From such a perspective one would be interested in examining household poverty profiles. Sample research questions would be: are female-headed households more likely to be multidimensionally poor? Are larger households more prone to be multidimensionally poor? How does the probability of being multidimensionally poor vary by household size and composition, caste, or ethnicity?

In the case of cross-sectional macro regressions, the focal variables are the measures at the province, district, state, or country levels, or some other population sub-group or aggregate which leads to a proper sample size.[8] If the focus is on the Adjusted Headcount Ratio , the focal variables in a macro regression could comprise or could use the intensity and incidence of multidimensional poverty. However, from Chapter 5 we know that and are partial indices that do not enjoy the same properties as the measure. In this Chapter we do not further consider regression models for . Although is also a partial index, which violates dimensional monotonicity, we do discuss its analysis, given the prominence of existing studies using the unidimensional poverty headcount ratio.

As already noted, and are bounded between zero and one. In statistical terms, and are fractional (proportion) variables that lie in the unit interval. Their restricted range of variation limits the use of the linear regression model because these models assume continuous variables comprised between and +. A natural model to be considered is one that reflects the fractional nature of any of these two indices (see section 10.4).

10.2 Generalized Linear Models

Our exposition of GLMs draws on Nelder and Wedderburn (1972), McCullagh and Nelder (1989) and Firth (1991). We treat GLMs in an applied manner covering the basic structure of the models, estimation, and model fitting. We do not provide a detailed exposition of the method itself. Readers interested in a complete statistical treatment of GLMs can refer to McCullagh and Nelder (1989) or to Dobson (2001). The former presents an excellent and comprehensive statistical overview of GLMs, but assumes an advanced statistics background on the part of the reader. The latter presents a briefer and more synthetic exposition of GLMs at a moderate level of statistical complexity.

Generalized linear models are an extension of classic linear models. The linear regression model has found widespread application in the social sciences mainly due to its simple linear formulation, easy interpretation, and estimation. In monetary poverty analysis, linear regression analysis has been used to study the determinants of household consumption expenditures or to model the growth elasticity of per capita income or income poverty aggregates like the headcount ratio or the poverty gap index.[9] Linear regressions are also used to model changes in (i) the income share of the poorest quintile (Dollar and Kray 2004); (ii) adjusted GDP incomes (Foster and Szekely 2008); (iii) the poverty rate (Ravallion 2001); and (iv) the growth rates of real per capita GDP (Barro 2003).

10.2.1 Classic Linear Regression

We begin with a brief review of the classic linear regression model and its notation and build on this to present the more generic case of GLMs. The classic linear regression model (LRM) assumes that the endogenous or dependent variable () (hitherto referred to as ‘endogenous’) is a linear function of a set of exogenous[10] variables (). The LRM assumes that the endogenous variable is continuous and distributed with constant variance. In addition the LRM may also assume that the endogenous variable is normally distributed. However this assumption is not needed for estimating the model but only to obtain the exact distribution of the parameters in the model. In the case of large samples one may not need to assume normality in a LRM as inference on parameters is based on asymptotic theory (c.f Amemiya, 1985). These assumptions may be inappropriate if the endogenous variable is discrete (binary or categorical)—or continuous but non-normal.[11] GLMs overcome these limitations. They extend classic linear regression to a family of models with non-normal endogenous variables. In what follows, random variables are denoted in uppercase and observations in lowercase; vectors are represented with lowercase bold and matrices with uppercase bold.

Consider a sample of observations of a scalar dependent variable () and a set of K exogenous variables (). This data is specified as , where is a column vector. Each observation is assumed to be a realization of a random variable independently distributed with mean . The classic regression model with additive errors for the observation can be written as



where denotes the conditional expectation[12] of the random variable given , and is a disturbance or random error. From equation (10.1) we see that the dependent variable is decomposed into two components: a systematic or deterministic component given the exogenous variables and an error component. The deterministic component is the conditional expectation , while the error component, attributed to random variation, is .

Equation (10.1) is a general representation of regression analysis. It attempts to explain the variation in the dependent variable through the conditional expectation without imposing any functional form on it. If we specify a linear functional form of the conditional expectation we obtain the classic linear regression model. Then, the systematic part of the model may be written



where is the value of the exogenous variable for observation . To show the relation between a linear regression model and a generalized linear model it will become convenient to denote the right-hand side of equation (10.1) by , referred to as the predictor in the generalized linear model. Thus we can write



and the systematic part can be expressed as



Equations (10.1) to(10.4) lead to the familiar linear regression model:



where , ,…, are parameters whose values are unknown and need to be estimated from the data.[13] Note that in the linear regression model of equation (10.6), the conditional expectation is equal to the linear predictor:




The LRM additionally assumes that the errors are independent, with zero mean, constant variance and follow a Gaussian or normal distribution.[14] Often the assumptions on are conditional on the exogenous variables, as these are possibly stochastic or random. Then, the errors have zero mean and homoscedastic or identical variance conditional on the exogenous variables, that is, . Due to the relationship between and , the dependent variable is also normally distributed with constant variance. In other words, in a LRM, the distribution of the dependent variable is derived from the distribution of the disturbance. As explained in section 10.2.2, in a GLM the distribution of the dependent variable is specified directly.

10.2.2 The Generalization

The GLM family of models involves predicting a function of the conditional mean of a dependent variable as a linear combination of a set of explanatory variables. Classic linear regression is a specific case of a GLM in which the conditional expectation of the dependent variable is modelled by the identity function. GLMs extend the domain of applicability of classic linear regression to contexts where the dependent variable is not continuous or normally distributed. GLMs also permit us to model continuous dependent variables that have positively skewed distributions.

Generalized linear models relax the assumption of additive error in equation (10.1). The random component is now attributed to the dependent variable itself. Thus, for GLMs we need to specify the conditional distribution of the dependent variable given the values of the explanatory variables, denoted as . These distributions often belong to the linear exponential family, such as the Gaussian, binomial, poisson, and gamma, among others—although recently have been extended to non-exponential families (McCullagh and Nelder 1989).

A generalized linear model is one that takes the form:



where the systematic part or linear predictor () is now a function () of the conditional expectation of the dependent variable ; is a one-to-one differentiable function referred to as the link function, and is referred to as the linear predictor. The link function transforms the conditional expectation of the dependent variable to the linear predictor, which is a linear function of the explanatory variables that could be of any nature. This allows the linear predictor to include continuous or categorical variables, a combination of both, or interactions—as well as transformations of continuous variables. Note that when the link function is the identity function, we have an LRM.

In most applications, as in the regression analysis with AF measures, the primary interest is the conditional mean . This could be easily retrieved from equation (10.7) by inverting the link function; hence we can write




where is the inverse link also called the mean function. Equations (10.7) and (10.8) provide two alternative specifications for a GLM, either as a linear model for the transformed conditional expectation of the dependent variable—given by (10.7)—or as a non-linear model for the conditional mean—given by (10.8).

A GLM is thus composed of three components: (i) a random component resulting from the specification of the conditional distribution of the dependent variable given the values of the explanatory variables (this is implicit and cannot be seen directly); (ii) a linear predictor , and iii) a link function (cf. Fox 2008: ch.15).

The distribution of the dependent variable and the choice of the link function are intimately related and depend on the type of variable under study. The form of a proper link function is determined to some extent[15] by the range of the dependent variable and consequently by the range of variation of its conditional mean.

In the case of AF poverty measures, we may consider two types of dependent variables with a different range of variation and distribution. The first type is a binary indicator identifying multidimensionally poor households. This variable takes the value of one if the household is identified as multidimensionally poor and zero otherwise. The Bernoulli distribution is suitable to describe this kind of variable. A typical model in this case is the probit or logit model. As we will see, in a GLM this is equivalent to choosing a logit link. The second type of dependent variable that we could study in the AF approach is a proportion. The Adjusted Headcount Ratio and the incidence are fractions or proportions that take values in the unit interval. The binomial distribution may be suitable as a model for these proportions.

In each of these cases, the link function should map the range of variation of the dependent variable— for the binary indicator and for the proportion—to the whole real line . The scale is chosen in such a way that the fitted values respect the range of variation of the dependent variable. Columns one to five in Table 10.1 present the two types of dependent variables with AF measures that we study in this section, along with their range of variation, type of model, level of analysis, and random variation described by the conditional distribution. The link and mean functions are explained in the examples in sections 10.3 and 10.4. Before presenting the examples, we briefly explain the estimation and goodness of fit of GLMs.

Table 10.1 Generalized Linear Regression Models with AF Measures

Dependent variable AF measure:

Range of

Regression Model


Conditional Distribution )


Mean function












Note: and are the cumulative distribution functions of the standard-normal and logistic distributions, respectively. For the binary model, the conditional mean is the conditional probability .


10.2.3 Estimation and Goodness of Fit

Once we have selected the particular models of our study, we need to estimate the parameters and measure their precision. For this purpose we maximize the likelihood or log likelihood[16] of the parameters of our data denoted by .[17] The likelihood function of a parameter is the probability distribution of the parameter given .

To assess goodness of fit of the possible estimates we use the scaled deviance. This statistic is formed from the logarithm of a ratio of likelihoods and measures the discrepancy, or goodness of fit, between the observed data and the fitted values generated by the model. To assess the discrepancy we use as a baseline the full or ‘saturated’ model. Given observations, the full model has parameters, one per observation. This model fits the data perfectly but is uninformative because it simply reproduces the data without any parsimony. Nonetheless it is useful for assessing discrepancy vis-à-vis a more parsimonious model that uses K parameters. Hence in the saturated model the estimated conditional mean = and the scaled deviance is zero. For intermediate models, say with K parameters, the scaled deviance is positive.

The scaled deviance statistic




is twice the difference between which is the maximum log likelihood of a saturated model or exact fit, and the log likelihood of the current or reduced model.

The goodness of fit is assessed by a significance test of the null hypothesis that the current model holds against the alternative given by the saturated or full model. Under the null hypothesis, is approximately distributed as a random variable where the number of degrees of freedom equals the difference in the number of regression parameters in the full and the reduced models. However, an appropriate assessment of the goodness of fit is based on the conditional distribution of given . If is not significant, it suggests that the additional parameters in the full model are unnecessary and that a more parsimonious model with fewer parameters may be sufficient.

The scaled deviance statistic is also useful for model selection. Due to its additive property, the discrepancy between nested sets of models can be compared if maximum likelihood estimates are used. Suppose we are interested in comparing two models, A and B, that represent two different choices of explanatory variables, and , that are nested. Intuitively this means that all explanatory variables included in model A are also present in model B, a more complex or less parsimonious model. The improvement in fit may be assessed by a significance test of the null hypothesis that model A holds against the alternative given by model B. If the value of the scaled deviance statistic is found to be significant, there is an improvement in the fit of model B vis-à-vis model A, although a general conclusion on model selection should also consider the added complexity of model B.

10.3 Micro Regression Models with AF Measures

In the case of micro regression analysis, the focal variable is the (household) censored deprivation score . This score reflects the joint deprivations characterizing a household identified as multidimensionally poor. From a policy perspective a natural question that arises consequently is how to understand the ‘causes’ that underlie the (multidimensional) poverty status of a household. The simplest model for this purpose is a probability model, which we illustrate in this section; although one could also consider modelling the vector directly. We are thus interested in assessing the probability of a household being multidimensionally poor. Within the AF framework this is equivalent to comparing the deprivation score of a household with the multidimensional poverty cutoff (). If is above the multidimensional poverty cutoff (), the household is identified as multidimensionally poor. This is represented by a binary random variable () that takes the value of one if the household is identified as multidimensionally poor and zero otherwise, as follows:



The outcomes of this binary variable occur with probability which is a conditional probability on the explanatory variables. For a (sampled) household identified as multidimensionally poor this is represented as




and thus the conditional mean equals the probability as follows:




For a binary model the conditional distribution of the dependent variable, or random component in a GLM, is given by a Bernoulli distribution (Table 10.1). Thus the probability function of is




To ensure that the conditional mean given by the conditional probability stays between zero and one, a GLM commonly considers two alternative link functions (). These are given by the quantile functions of the standard normal distribution function and the logistic distribution function . The former is referred to as the probit link function and the latter as the logit link function. The probit link function does not have a direct interpretation, while the logit is directly interpretable as we discuss below.[19]

The logit of is the natural logarithm of the odds that the binary variable takes a value of one rather than zero. In our context, this gives the relative chances of being multidimensionally poor. If the odds are ‘even’—that is, equal to one—the corresponding probability ( of falling into either category, poor or non-poor, is 0.5, and the logit is zero. The logit model is a linear, additive model for the logarithm of odds as in equation (10.14), but it is also a multiplicative model for the odds as in equation (10.15):





The conditional probability is then



The partial regression coefficients are interpreted as marginal changes of the logit, or as multiplicative effects on the odds. Thus, the coefficient indicates the change in the logit due to a one-unit increase in , and is the multiplicative effect on the odds of increasing by one, while holding constant the other explanatory variables. For example, if the first explanatory variable increases by one unit, the odds ratio in equation (10.15) associated with this increase is , and . For this reason, is known as the odds ratio associated with a one-unit increase in . To see the percentage change in the odds, we need to consider the sign of the estimated parameter. If is negative, the change in denotes a decrease in the odds; this decrease is obtained as . Likewise if is positive, the change in indicates an increase in the odds. In this case, the increase is obtained as

10.3.1 A Micro-Regression Example

To illustrate the type of micro regression models that have been discussed, we use a subsample of the Indonesian Family Life Survey (IFLS) dataset. This is a dataset analysed by Ballon and Apablaza (2012) to assess multidimensional poverty in Indonesia during the period 1993–2007. The IFLS is a large-scale longitudinal survey of the socioeconomic, demographic, and health conditions of individuals, households, families, and communities in Indonesia. The sample is representative of about 83% of the population and contains over 30,000 individuals living in thirteen of the twenty-seven provinces in the country. Ballon and Apablaza (2012) measure multidimensional poverty at the household level in five equally weighted dimensions: education, housing, basic services, health issues, and material resources. For this illustration we retain a poverty cutoff of 33%. Thus a household is identified as multidimensionally poor if the sum of the weighted deprivations is greater than 33%. That is, takes the value of one if 33% and zero otherwise. Within the GLM framework this binary dependent variable is estimated by specifying a Bernoulli distribution and a logit link function. This is equivalent to a logit regression.

The household poverty profile that we specify regresses the log of the odds of being multidimensionally poor (using on the demographic and socioeconomic characteristics of the household head. For this illustration we use data for West Java in 2007. West Java is a province of Indonesia located in the western part of the island of Java. It is the most populous and most densely populated of Indonesia’s provinces, which is why we selected it. The explanatory variables included in this illustration are non-indicator measurement variables and comprise:

·         Education of the household head, defined as the number of years of education (not necessarily completed);

·         The presence of a female household head, represented by a dummy variable taking a value of one if the household head is a female and zero if male;

·         Household size, defined by the number of household members;

·         The area in which the household resides, represented by a dummy variable taking a value of one if the household resides in the urban areas of West Java and zero otherwise;

·         Muslim religion, represented by a dummy variable taking a value of one if the household’s main religion is Muslim and zero if not.

Table 10.2 Logistic Regression Model of Multidimensional Poverty in West Java

table10_2Table 10.2 reports the logistic regression results of this poverty profile for West Java in 2007. Columns two to five report the estimated regression parameters along with their standard errors, t ratios, and significance levels at 5%[20]. Apart from being Muslim, all other determinants are significant at the 5% level and show the expected signs. For a given household, the log of the odds of being multidimensionally poor decreases with the education of the household head and with an urban location and increases with the presence of a female household head and with household size. The odds ratio for years of education of the household head indicates that an increase of one year of education decreases the odds of being multidimensionally poor by 49%, ceteris paribus, whereas having a female household head increases the odds of being multidimensionally poor by 28%, ceteris paribus.[21] Similarly, the odds of a household of being multidimensionally poor decrease by 57% for households living in urban areas, ceteris paribus, and increase by 10% for each additional household member. Figure 10.1 shows the odds model for urban and rural areas as a function of the education of the household head, holding constant the gender status of the household head (female), assuming five household members (average), and being Muslim. The logistic curves show a decrease in the probability of a household being multidimensionally poor as the education of the household head improves. These probabilities are lower for households living in urban areas compared to rural ones.

Figure 10.1 Logistic Regression Curve—West Java



As religion turns out to be statistically insignificant, we could consider an alternative poverty profile without religion as an explanatory variable (model B). To test whether this restrained model (without religion) is as good as the current model (model A), we compare the deviance statistics of both models.[22] Formally we test the following hypothesis:

Model A is as good as model B

Model A fits better than model B.

To reject the null hypothesis we compare with the corresponding chi-square statistic with degrees of freedom. These degrees of freedom correspond to the difference in the number of parameters in model A and model B. A non-rejection of the null hypothesis indicates that both models are statistically equivalent and thus the most parsimonious model, which has the smaller number of explanatory variables, should be chosen—which is B in this context. A rejection of the null indicates a statistical justification for model A. In our case the comparison of the two nested models, A and B, gives a scaled deviance statistic of 0.05. We compare this value with the corresponding chi-square statistic of one degree of freedom and a 5% type I error rate; this gives a value of 3.84. As is smaller than 3.84, we cannot reject the null hypothesis; so we choose the more parsimonious model B and drop religion as an explanatory variable.

10.4 Macro Regression Modelsfor M0 and H

We now turn to the econometric modelling for the Adjusted Headcount Ratio and the incidence of multidimensional poverty as endogenous or dependent variables. As and are bounded between zero and one, an econometric model for these endogenous variables must account for the shape of their distribution. and are fractional (proportional) variables bounded between zero and one with the possibility of observing values at the boundaries. This restricted range of variation also applies for the conditional mean, which is the focus of our analysis. Thus specifying a linear model, which assumes that the endogenous variable and its mean take any value in the real line, and estimating it by ordinary least squares is not the right strategy, as this ignores the shape of the distribution of these dependent variables. Clearly if the interest of the research question is not in modelling the conditional mean of the proportion but rather in modelling the absolute change (between two time periods) of or , which can take any value, standard linear regression models may apply. In what follows we describe the statistical strategy for modelling the conditional mean of or as a function of a set of explanatory variables.

Various approaches have been used in the literature to model a fraction or proportion. We can differentiate between two types of approaches—often referred to as one-step or two-step approaches. These differ in the treatment of the boundary values of the fractional dependent variable. In a one-step approach, one considers a single model for the entire distribution of the values of the proportion, where both the limiting observations and those falling inside the unit interval are modelled together. In a two-step approach, the observations at the boundaries are modelled separately from those falling inside the unit interval. In other words, in a two-step approach one considers a two-part model where the boundary observations are modelled as a multinomial model and remaining observations as a fractional one-step regression model (Wagner 2001; Ramalho, Ramalho and Murteira 2011). The decision whether a one- or a two-part model is appropriate is often based on theoretical economic arguments. Wagner (2001) illustrates this point. He models the export/sales ratio of a firm and argues that firms choose the profit-maximizing volume of exports, which can be zero, positive, or one. Thus the boundary values of zero or one may be interpreted as the result of a utility-maximizing mechanism. Following this theoretical economic argument he specifies a one-step fractional model for the exports/sales ratio. In the absence of an a priori criteria for the selection of either a one- or two-part model, Ramalho et al. (2011) propose a testing methodology that can be used for choosing between one-part and two-part models. In the case of or we consider that non-poverty and full poverty, the boundary values, as well as the positive values, are characterized by the same theoretical mechanism. This is thus represented by a one-part model. For further references on alternative estimation approaches for one-part models, see Wagner (2001) and Ramalho et al. (2011).

10.4.1 Modelling M0 or H

To model or we follow the modelling approach proposed by Papke and Wooldridge (1996). For this purpose we denote the Adjusted Headcount Ratio or the incidence by . For a given spatial aggregate, say a country, the Adjusted Headcount Ratio or the incidence is . Papke and Wooldridge (PW hereafter) propose a particular quasi-likelihood method to estimate a proportion. The method follows Gourieroux, Monfort, and Trognon (1984) and McCullagh and Nelder (1989) and is based on the Bernoulli log-likelihood function which is given by



where is a known nonlinear function satisfying . In the context of a GLM, is the mean function defined in equation (10.8) as the inverse link function. PW suggest as possible specifications for any cumulative distribution function, with the two most typical examples being the logistic function and the standard normal cumulative density function as described in Table 10.1.

The quasi-maximum likelihood estimator (QML) obtained from equation (10.17) is consistent and asymptotically normal, provided that the conditional mean is correctly specified. This follows the QML theory where consistency and asymptotic normality characterize all QML estimators belonging to the linear exponential family of distributions, which is the case of the Bernoulli distribution of equation (10.17).

10.4.2 Econometric issues for an empirical model of M0 or H

We would like to conclude with a few recommendations for performing a macro regression with or as explained variables. First, we suggest testing for linearity before specifying a non-linear functional form. For this purpose one can apply the Ramsey RESET[23] test of functional misspecification. The test consists of evaluating the presence of nonlinear patterns in the residuals that could be explained by higher-order polynomials of the dependent variable. Second, we recommend testing for possible endogeneity using a two-stage or instrumental variable (IV) estimation. In regressions of the type of the macrodeterminants of or it is very likely that there will be a correlation between one or more of the explanatory variables and the error term. Let us suppose we regress the Adjusted Headcount Ratio on the logarithm of the per capita gross national income in PPP of the same year for a group of countries. This is the GNI converted to international dollars using purchasing power parity rates. This gives a contemporaneous model for the semi-elasticity between growth and poverty. In this very simple model, it is highly likely that the GNI would be correlated with the disturbance of the equation, which consists of unobserved variables affecting the poverty rate. This violates a necessary condition for the consistency of standard linear estimators. To deal with endogeneity, often one replaces the endogenous explanatory variable with a proxy assumed to be correlated with the endogenous explanatory variable but uncorrelated with the error term.

Third, one is also very likely to find measurement errors among the explanatory variables in a model for or . This issue can also be treated with the IV method by replacing the measured-with-error variable with a proxy. To minimize the loss of efficiency that may result from an IV estimation, one can complement the estimation results using the Generalized Method of Moments. Lastly, we would like to point out that although this Chapter has focused on the modelling of levels of poverty (rates of poverty: , ), it is at once straightforward and necessary to analyse changes in poverty. It suffices to estimate the model in levels and then compute the marginal effects of the expected poverty rate with respect to the explanatory variables included in the model.


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Papke, L. E. and Wooldridge, J. M. (1996). ‘Econometric Methods for Fractional Response Variables with and Application to 401(k) Plan Participation Rates’. Journal of Applied Econometrics, 11(6): 619–632.

Rabe-Hesketh, S. and Skrondal, A. (2012). Multilevel and Longitudinal Modeling Using Stata. Volume I: Continuous Responses. (3rd ed.). Stata Press.

Ramalho, E. A., et al. (2011): Ramalho, E. A., Ramalho, J. J., and Murteira, J. M. (2011). ‘Alternative Estimating and Testing Empirical Strategies for Fractional Regression Models’. Journal of Economic Surveys, 25(1): 19–68.

Ravallion, M. (2001). ‘Growth, Inequality and Poverty: Looking beyond Averages’. World Development, 29(11): 1803–1815.

Roelen, K. and Notten, G. (2011). The Breadth of Child Poverty in Europe: An Investigation into Overlap and Accumulation of Deprivations. UNICEF Innocenti Research Centre.

Skrondal, A. and Rabe-Hesketh, S. (2004). Generalized Latent Variable Modeling: Multilevel, Longitudinal, and Structural Equation Models. CRC Press.

Stock, J. H., Wright, J. H., and Yogo, M. (2002). ‘A Survey of Weak Instruments and Weak Identification in Generalized Method of Moments’. Journal of Business & Economic Statistics, 20(4): 518–529.

Wagner, J. (2001). ‘A Note on the Firm Size–Export Relationship’. Small Business Economics, 17(4): 229–237.

[1] Cf. Nelder and Wedderburn (1972) and McCullagh and Nelder (1989).

[2] Also referred to as models for proportions.

[3] Skrondal and Rabe-Hesketh (2004), Rabe-Hesketh and Skrondal (2012) address this extension.

[4] The term determinants shall be understood in a ‘weak’ sense and refers to ‘proximate’ causes of poverty as defined in Haughton and Khandker (2009: 147).

[5] Also called independent, exogenous, or explanatory variables. We prefer the terms ‘exogenous’ or ‘explanatory’ to refer to the right-hand-side variables of a regression. In this section we use both terms interchangeably.

[6] See, for example, Bound, Jaeger, and Baker (1995) and Stock, Wright, and Yogo (2002).

[7] These are variables with explanatory power that were not used when constructing the poverty measure. These variables are expected to be uncorrelated with the error term of the model.

[8] Small-sample econometric and statistical techniques could be envisaged in the case of aggregates with very few categories.

[9] See, for example, De Janvry and Sadoulet (2010) and Roelenand Notten (2011).

[10] In the statistical literature is referred to as a regressor or covariate that is exogenous when the assumptions on the disturbance term are conditional on the covariates. In our exposition, all assumptions on the disturbance term or the dependent variable are conditional on the regressors so we use the term ‘exogenous’ instead of the generic term regressor. By ‘exogenous’ we mean non-stochastic or conditionally stochastic right-hand-side variables.

[11] An example of a non-normal continuous variable is income (consumption expenditures). The distribution of income is skewed (to the right), takes on only positive values, and is often heteroscedastic.

[12] Or conditional mean. We use both terms interchangeably.

[13] An equivalent expression of the LRM is a matrix representation of the form
, where is an vector of observations; is an vector of disturbances; is a matrix of explanatory variables, where each row refers to a different observation each column to a different explanatory variable; and is a vector of parameters. However for the expositional purposes of this Chapter we do not use the matrix representation but rather the one specified in equation (10.5).

[14] To denote a random variable as normally distributed we follow the statistical convention and denote it as .

[15] The range of variation of the dependent variable is a mild requirement for the choice of a proper link function. As noted by Firth (1991) this mild requirement is complemented by multiple criteria where the choice of a proper link function is made on the grounds of its fit to the data, the ease of interpretation of parameters in the linear predictor, and the existence of simple sufficient statistics.

[16] The parameters in a GLM are estimated by a numerical algorithm, namely, iterative weighted least squares (IWLS). For models with the links considered in this section, the IWLS algorithm is equivalent to the Newton–Raphson method and also coincides with Fisher scoring (McCullagh and Nelder 1989).

[17] Note we drop the subscript as the log likelihood depends on the full sample. For ease of exposition we also write as .

[18] Note and are the cumulative distribution functions of the standard-normal and logistic distributions, respectively.

[19] Alternative link functions include the log-log and the complementary log-log links; however, these two are not symmetric around the median.

[20] Note we can also report marginal effects if the interest is to see the effect of an explanatory variable on the change of the probability.

[21] All estimated parameters exhibiting a negative sign denote a decrease in the odds; this is obtained as (1-odds ratio)×100. Likewise, estimated parameters with a positive sign denote an increase in the odds; this is obtained as (odds ratio-1)×100. For the effect of education we have (1-0.51)×100, and for the effect of gender we have (1.28-1)×100%.

[22] The deviance statistic: .

[23] RESET stands for Regression Equation Specification Error Test.

9 Distribution and Dynamics

This chapter provides techniques required to measure and analyse inequality among the poor (section 9.1), to describe changes over time using repeated cross-sectional data (section 9.2), to understand changes across dynamic subgroups (section 9.3) and to measure chronic multidimensional poverty (section 9.4). Each of these sections extends the methodological toolkit beyond the partial indices presented in Chapter 5, to address common empirical problems such as poverty comparisons, and illustrate these with examples. We build upon and do not repeat material presented in earlier chapters, and as in other chapters, confine attention to issues that are distinctive in multidimensional poverty measures.

9.1     Inequality among the Poor

Given the long-standing interest in inequality among the poor, we first enquire whether  can be extended to reflect inequality among the poor. To make a long story short, it can easily do so. But the problem is that the resulting measure loses the property of dimensional breakdown that provides critical information for policy. So, taking a step back, we consider key properties a measure should have in order to reflect inequality among the poor and be analysed in tandem with  Our chosen measure uses the distribution of censored deprivation scores to compute a form of variance across the multidimensionally poor. We also illustrate interesting related applications of this measure: for example to assess horizontal disparities across groups.

Chapter 5 showed that the Adjusted Headcount Ratio  can be expressed as a product of the incidence of poverty () and the intensity of poverty () among the poor. Thus,  captures two very important components of poverty—incidence and intensity. But it remains silent on a third important component: inequality across the poor. Now, the ultimate objective is to eradicate poverty—not merely reduce inequality among the poor. However, the consideration of inequality is important because the same average intensity can hide widely varying levels of inequality among the poor. For this reason, following the seminal article by Sen (1976), numerous efforts were made to incorporate inequality into unidimensional and latterly multidimensional poverty measures.[1]

This section explores how inequality among the poor can be examined when poverty analyses are conducted using the  measure (Alkire and Foster 2013, Seth and Alkire 2014a, b).[2]

9.1.1   Integrating Inequality into Poverty Measures

Section 5.7.2 already presented one way of bringing inequality into multidimensional poverty measures. This was achieved by using  or some other gap measure applied to cardinal data, where the exponent on the normalized gap is strictly greater than one. Such an approach is linked to Kolm (1977) and generalizes the notion of a progressive transfer (or more broadly a Lorenz comparison) to the multidimensional setting by applying the same bistochastic matrix to every variable to smooth out the distribution of each variable (the powered normalized gap) while preserving its mean.[3] Poverty measures that are sensitive to inequality fall (or at least do not rise) in this case.

A second form of multidimensional inequality is linked to the work of Atkinson and Bourguignon (1982) and relies on patterns of achievements across dimensions. Imagine a case where one poor person initially has more of everything than another poor person and the two persons switch achievements for a single dimension in which both are deprived. This can be interpreted as a progressive transfer that preserves the marginal distribution of each variable and lowers inequality by relaxing the positive association across variables under the assumption that the dimensions are substitutes. The resulting transfer principle specifies conditions under which this alternative form of progressive transfer among the poor should lower poverty, or at least not raise it. The transfer properties are motivated by the idea that poverty should be sensitive to the level of inequality among the poor, with greater inequality being associated with a higher (or at least not lower) level of poverty.[4] Alkire and Foster (2011a) observe that the AF class of measures can be easily adjusted to respect the strict version of the second kind of transfer (the strong deprivation rearrangement property as discussed in section 5.2.5) involving a change in association between dimensions by replacing the deprivation count or score  with a related individual poverty function  for some , and averaging across persons.[5]

Many multidimensional poverty measures that employ cardinal data, including  satisfy one or both of these transfer principles.[6] Alkire and Foster (2013) formulate a strict version of distribution sensitivity — ‘dimensional transfer’ (defined in 5.2.5)—which is applicable to poverty measures such as  that use ordinal data. This property follows the Atkinson–Bourguignon type of distribution sensitivity, in which greater inequality among the poor strictly raises poverty. Alkire and Foster (2013) also prove a general result establishing that ‘the highly desirable and practical properties of subgroup decomposability, dimensional breakdown, and symmetry prevent a poverty measure from satisfying the dimensional transfer property’. In other words,  does not reflect inequality among the poor, and, furthermore, no measure that satisfies dimensional breakdown and symmetry will be found that does satisfy dimensional transfer.

Given that it is necessary to choose between measures that satisfy dimensional transfer and those that can be broken down by dimension, and given that both properties are arguably important, how should empirical studies proceed? The first option is to employ the class of measures that respect dimensional breakdown and to supplement these with associated inequality measures. The second is to employ poverty measures that are inequality-sensitive but cannot be broken down by dimension, and to supplement them with separate dimensional analyses.

9.1.2  Analyzing Inequality Separately: A Descriptive Tool

While both should be explored, this book favours the first route in applied work for several reasons. Dimensional breakdown enriches the informational content of poverty measures for policy, enabling them to be used to tailor policies to the composition of poverty, to monitor changes by dimension, and to make comparisons across time and space. Poverty reduction in measures respecting dimensional breakdown can be accounted for in terms of changes in deprivations among the poor and analysed by region and dimension. This creates positive feedback loops that reward effective policies. Also, the inequality-adjusted poverty measures may lack the intuitive appeal of the  measure. Some of the inequality-adjusted measures (Chakravarty and D’Ambrosio, 2006, Rippin, 2012) are broken down into different components separately capturing incidence, intensity, and inequality, but without clarifying the relative weights attached to these components.

Whether or not an inequality measure is not computed,  measures can be supplemented by direct descriptions of inequality among the poor. A first descriptive but powerfully informative tool is to report subsets of poor people which have mutually exclusive and collectively exhaustive graded bands of deprivation scores. This is possible by effectively ordering all  poor persons according to the value of their deprivation score  and dividing them into groups. If the poverty cutoff is 30%, the analysis might then report the percentage of poor people whose deprivation scores fall in the band of 30–39.9% of deprivations, 40–49.9%, and so on to 100%. The percentage of people who experience different intensity gradients of poverty across regions and time can be compared to see how inequality among the poor is evolving.[7] Figure 9.1 presents an example of two countries—Madagascar and Rwanda—which have similar multidimensional headcount ratios () and MPIs. However, the distributions of intensities across the poor are quite different. Also, data permitting, these intensity groups can be decomposed by population subgroups such as region or ethnicity. The comparisons can be enriched by applying a dimensional breakdown to examine the dimensional composition of poverty experienced by those having different ranges of deprivation scores.

Figure 9.1 Distribution of Intensities among the Poor in Madagascar and Rwanda


9.1.3  Using a Separate Inequality Measure

Another tool is to supplement  with a measure of inequality among the poor. Using the distribution of (censored) deprivation scores across the poor or some transformation of these, it is actually elementary to create an inequality measure, much in the same way that traditional inequality measures such as Atkinson, Theil, or Gini are constructed.  Such measures will offer a window onto one type of multidimensional inequality—one that is oriented to the breadth of deprivations people experience. This approach is quite different from other constructions of multidimensional inequality, but it is useful, particularly when data are ordinal. Building on Chakravarty (2001), Seth and Alkire (2014a,b) propose such an inequality measure that is founded on certain properties. Note that these are properties of inequality measures, and are defined differently from those presented in Chapter 2 (despite similar names), but introduced intuitively below. Let us briefly discuss these properties before introducing the measure.  Properties

The first property, translation invariance, requires inequality not to change if the deprivation score of every poor person increases by the same amount. Implicitly, we assume that the measure reflects absolute inequality. Seth and Alkire (2014) argue that measures reflecting absolute inequality are more appropriate when each deprivation is judged to be of intrinsic importance. In addition, the use of the absolute inequality measure ensures that inequality remains the same whether poverty is measured by counting the number of deprivations or by counting the number of attainments. The use of the relative inequality measure is more common in the case of income inequality, where it is often assumed that as long as people’s relative incomes remain unchanged, inequality should not change. However, it is difficult to argue that inequality between two poor persons who are deprived in one and two dimensions respectively is the same as the inequality between two poor persons who are deprived in five and ten dimensions, respectively, if these deprivations referred to, for example, serious human rights violations. Any relative inequality measure, such as the Generalized Entropy measures (which include the Squared Coefficient of Variation associated with the FGT2 index) or Gini Coefficient, would evaluate these two situations as having identical inequality across the poor. Moreover, a relative inequality measure may provide counterintuitive conclusion while assessing inequality within a counting approach framework. In fact, no non-constant inequality measure exists that is simultaneously invariant to absolute as well as relative changes in a distribution.

The second property requires that the inequality measure should be additively decomposable so that overall inequality in any society can be broken down into within-group and between-group components. This can be quite useful for policy (Stewart 2010). We have shown in Chapter 5 that the additive structure of the indices in the AF class allows the overall poverty figure to be decomposed across various population subgroups. A country or a region with same level of overall poverty may have very different poverty levels across different subgroups, or a country may have the same level of poverty across two time periods, but the distribution of poverty across different subgroups may change over time. Furthermore, within each population subgroup, there may be different distributions of deprivation scores across poor persons living within that subgroup, thus reflecting various levels of within-group inequality.

The third property, within-group mean independence, requires that overall within-group inequality should be expressed as a weighted average of the subgroup inequalities, where the weight attached to a subgroup is equal to the population share of that subgroup. This assumption makes the interpretation and analysis of the inequality measure more intuitive.

Four additional properties are commonly satisfied when constructing any inequality measure. The anonymity property requires that a permutation of deprivation scores should not alter inequality. According to the replication invariance property, a mere replication of population leaves the inequality measure unaltered. The normalization property requires that the inequality measure should be equal to zero when the deprivation scores are equal for all. The transfer property requires that a progressive dimensional rearrangement among the poor should decrease inequality.  A Decomposable Measure

The proposed inequality measure, which is the only one to satisfy those properties, takes the general form



where  is a vector with  elements. Relevant applications using our familiar notation will be provided in equations (9.6) and (9.7) below, but we first present the general form and notation. As we will show, in relevant applications an element  may be the deprivation score of a person  or  or the average poverty level of a region. The size of the vector  for an entire population would be  and for the poor it would be  The functional form in equation (9.1) is a positive multiple () of the variance. The measure reflects the average squared difference between person ’s deprivation score and the mean of the deprivation scores in . The value of parameter  can be chosen in such a way that it normalizes the inequality measure to lie between 0 and 1.

The overall inequality in  may be decomposed into two components: total within-group inequality and between-group inequality. Following the notation in Chapter 2, suppose there are  population subgroups. The deprivation score vector of subgroup  is denoted by  with  elements. The decomposition expression is given as follows:


                            Total within-group                    Total between-group


where  is the population share of subgroup  in the overall population and  is the mean of all elements in  for all .

The between-group inequality component  in (9.2) can be computed as



where  is the mean of all elements in .

The within-group inequality component of subgroup  can be computed using (9.1) as



and thus the total within-group inequality component in (9.2) can be computed as


(9.5)  Two Important Applications

There are different relevant applications of this inequality framework to multidimensional poverty analyses based on . The first central case is to assess inequality among the poor. To do so we suppose that the deprivation scores are ordered in a descending order and the first  persons are identified as poor. The elements are taken from the censored deprivation score vector,  We choose vector  such that it contains only the deprivation scores of the poor (). The average of all elements in  then is the intensity of poverty which for  persons is . We can then denote the inequality measure that reflects inequality in multiple deprivations only among the poor by , which can be expressed as




The  measure effectively summarizes the information underlying Figure 9.1. It goes well beyond that figure because each individual deprivation score is used, which effectively creates a much finer gradation of intensity than that figure portrays. Furthermore, it can be decomposed by subgroup, to permit comparisons of within-subgroup inequalities among the poor. It can also be used over time to show how inequality among the poor changed.

Our second central case considers inequalities in poverty levels across population subgroups. It is motivated by studies of horizontal inequalities that find group-based inequalities to predict tension and in some cases conflict (Stewart 2010). Essentially, the measure reflects population-weighted disparities in poverty levels across population subgroups.

Suppose the censored deprivation score vector of subgroup  is denoted by  with  elements. If instead of only considering the deprivation scores of the poor, we now sum across the whole population so (), then we realize that  or the average of all elements in  is actually the  of subgroup , which for simplicity we denote by . The between-group component of  shows the disparity in the national Adjusted Headcount Ratio  across subgroups and is written using (9.3) as



Thus, equation (9.7) captures the disparity in s across  population subgroups, which can be used to detect patterns in horizontal disparities over time. Naturally, the number and population share of the subgroups must be considered in such comparisons.

While studying disparity in MPIs across sub-national regions, Alkire, Roche, and Seth (2011) found that the national MPIs masked a large amount of sub-national disparity within countries, and Alkire and Seth (2013) and Alkire, Roche, and Vaz (2014) found considerable disparity in poverty trends across subnational groups. In some countries, the overall situation of the poor improved, but not all subgroups shared the equal fruit of success in poverty reduction and indeed poverty levels may have stagnated or risen in some groups. Therefore, it is also important to look at inequality or disparity in poverty across population subgroups. This separate inequality measure, elaborated in Seth and Alkire (2014), provides such framework.  An Illustration

Table 9.1 presents two pair-wise comparisons. For the inequality measure, we choose  4 because the deprivation scores are bounded between 0 and 1; hence the maximum possible variance is 0.25.  4 ensures that the inequality measure lies between 0 and 1. The first pair of countries, India and Yemen, have exactly the same levels of MPI. The multidimensional headcount ratios and the intensities of poverty are also similar. However, the inequality among the poor (computed using equation (9.1)) is much higher in Yemen than in India. We also measure disparity across sub-national regions. Yemen has twenty-one sub-national regions whereas, India has twenty-nine sub-national regions. We find that, like the national MPIs, the disparities across subnational MPIs—computed using equation 9.7)—are similar. This means that the inequality in Yemen is not primarily due to regional disparities in poverty levels, but may be affected by non-geographic divides such as cultural or rural–urban.

A contrasting finding for regional disparity is obtained across Togo and Bangladesh. As before, the MPIs, headcount ratios, and intensities are quite similar across two countries—but with two differences. The inequality among the poor is very similar, but the regional disparities are stark. Even though both countries have similar number of sub-national regions, the level of sub-national disparity is much higher in Togo than that in Bangladesh.

Table 9.1: Countries with Similar Levels of MPI but Different Levels of Inequality among the Poor and Different Levels of Disparity across Regional MPIs


9.2 Descriptive Analysis of Changes over Time

A strong motivation for computing multidimensional poverty is to track and analyse changes over time. Most data available to study changes over time are repeated cross-sectional data, which compare the characteristics of representative samples drawn at different periods with sampling errors, but do not track specific individuals across time. This section describes how to compare  and its associated partial indices over time with repeated cross-sectional data. It offers a standard methodology of computing such changes, and an array of small examples. This section does not treat the data issues underlying poverty comparisons, and readers are expected to know standard techniques that are required for such rigorous empirical comparisons. For example, the definition of indicators, cutoffs, weights, etc. must be strictly harmonized for meaningful comparisons across time, which always requires close verification of survey questions and response structures, and may require amending or dropping indicators. The sample designs of the surveys must be such that they can be meaningfully compared, and basic issues like the representativeness and structure of the data must be thoroughly understood and respected. We presume this background in what follows. This section focuses on changes across two time periods; naturally the comparisons can be easily extended across more than two time periods.

9.2.1    Changes in M0, H and A across Two Time Periods

The basic component of poverty comparisons is the absolute pace of change across periods.[10] The absolute rate of change is the difference in levels between two periods. Changes (increases or decreases) in poverty across two time periods can also be reported as a relative rate. The relative rate of change is the difference in levels across two periods as a percentage of the initial period.

For example, if the  has gone down from 0.5 to 0.4 between two consecutive years, then the absolute rate of change is (0.5 – 0.4) = 0.1. It tells us how much level of poverty () has changed: 10% of the total possible set of deprivations that poor people in that society could have experienced has been eradicated; 40% remains. The relative rate of change is (0.5 – 0.4)/0.5 = 20%, which tells us that  has gone down by 20% with respect to the initial level. While absolute changes are in some sense prior, because they are easy to understand and compare, both absolute and relative rates may be important to report and analyse. The value-added of the relative changes is evident in relatively low-poverty regions.  A region or country with a high initial level of poverty may be able to reduce poverty in absolute terms much more than one having a low initial level of poverty. It is however possible that although a region or country with low initial poverty levels did not show a large absolute reduction, the reduction was large relative to its initial level and thus it should not be discounted for its slower absolute reduction.[11] The analysis of both absolute and relative changes gives a clear sense of overall progress.

In expressing changes across two periods, we denote the initial period by  and the final period by . This section mostly presents the expressions for , but they are equally applicable to its partial indices: incidence (), intensity (), censored headcount ratios (), and uncensored headcount ratios (). The achievement matrices for period  and  are denoted by  and , respectively. As presented in Chapter 5,  and its partial indices depend on a set of parameters: deprivation cutoff vector , weight vector  and poverty cutoff . For simplicity of notation though, we present  and its partial indices only as a function of the achievement matrix. For strict intertemporal comparability, it is important that the same set of parameters be used across two periods.

The absolute rate of change () is simply the difference in Adjusted Headcount Ratios between two periods and is computed as



Similarly, for  and :






The relative rate of change () is the difference in Adjusted Headcount Ratios as a percentage of the initial poverty level and is computed forand  (only  shown) as



If one is interested in comparing changes over time for the same reference period, the expressions (5.7) and (9.11) are appropriate. However, in cross-country exercises, one may often be interested in comparing the rates of poverty reduction across countries that have different periods of references. For example the reference period of one country may be five years; whereas the reference period for another country is three years. It is evident in Table 9.2 that the reference period of Nepal is five years (2006–2011); whereas that of Peru is only three years (2005-–2008). In such cases, it is essential to annualize the change in order to preserve strict comparison.

The annualized absolute rate of change () is the difference in Adjusted Headcount Ratios between two periods divided by the difference in the two time periods () and is computed for  as



The annualized relative rate of change () is the compound rate of reduction in  per year between the initial and the final periods, and is computed for  as




As formula (9.8) has been used to compute the changes in  and  using formulae (9.9) and (9.10), formulae (9.11) to (9.13) can be used to compute and report annualized changes in the other partial indices, namely , or

9.2.2  An Example: Analyzing Changes in Global MPI for Four Countries

Table 9.2 presents both the annualized absolute and annualized relative rates of change in Global MPI, as outlined in Chapter 5, and its two partial indices— and —for four countries: Nepal, Peru, Rwanda, and Senegal, drawing from Alkire, Roche and Vaz (2014). Taking the survey design into account, we also present the standard errors (in parentheses) and the levels of statistical significance of the rates of reduction, as described in the Appendix of Chapter 8. The figures in the first four columns present the values and standard errors for ,  and  in both time periods. The results show that Peru had the lowest MPI with 0.085 in the initial year, while Rwanda had the highest with 0.460.

Under the heading ‘annualized change’, Table 9.2 provides the annualized absolute and annualized relative reduction for ,  and , which are computed using equations (9.12) and (9.13). It shows, for example, that Nepal with a much lower initial poverty level than Rwanda, has experienced a greater absolute annualized poverty reduction of ‒0.027. In relative terms Nepal outperformed Rwanda. Peru had a low initial poverty level, and reduced it in absolute terms by only ‒0.006 per year, which means that the share of all possible deprivations among poor people that were removed was only less than one-fourth that of Nepal or Rwanda. But relative to its initial level of poverty, its progress was second only to Nepal. It is thus important to report both absolute and relative changes and to understand their interpretation. The same results for  and  are provided in the Panel II and III of the table. We see that Nepal reduced the percentage of people who were poor by 4.1 percentage points per year—for example, if the first year 64.7% of people were poor, the next year it would be 60.6%. Peru cut the poverty incidence by 1.3 percentage points per year. Relative to their starting levels, they had similar relative rates of reduction of the headcount ratio. Note that when estimates are reported in percentages, the absolute changes are reported in ‘percentage points’ and not in ‘percentages’. Thus, Nepal’s reduction in  from 64.7% to 44.2% is equivalent to an annualized absolute reduction of 4.1 percentage points and an annualized relative reduction of 6.3%.

Table 9.2: Reduction in Multidimensional Poverty Index, Headcount Ratio and Intensity of Poverty in Nepal, Peru, Rwanda and Senegal


The third column provides the results for the hypothesis tests which assess if the reduction between both years is statistically significant.[12] The reductions in  in Nepal and Rwanda are significant at =0.01, but the same in Peru is only significant at =0.10. Interestingly, the reduction in intensity of poverty in Peru is significant at =0.05. The case of Senegal is different in that the small reduction in  is not even significantly different at =0.10, preventing the null hypothesis that poverty level in both years remained unchanged from being rejected.

9.2.3  Population Growth and Change in the Number of Multidimensionally Poor

Besides comparing the rate of reduction in ,  and  as in Table 9.2, one should also examine whether the number of poor people is decreasing over time. It may be possible that the population growth is large enough to offset the rate of poverty reduction. Table 9.3 uses the same four countries as Table 9.2 but adds demographic information. Nepal had an annual population growth of 1.2% between 2006 and 2011, moving from 25.6 to 27.2 million people, and reduced the headcount ratio from 64.7% to 44.2%. This means that Nepal reduced the absolute number of poor by 4.6 million between 2006 and 2011.

Table 9.3 Changes in the Number of Poor Accounting for Population Growth

table9_3In order to reduce the absolute number of poor people, the rate of reduction in the headcount ratio needs to be faster than the population growth. The largest reduction in the number of multidimensionally poor has taken place in Nepal. A moderate reduction in the number of poor has taken place in Peru and Rwanda. In contrast, there has been an increase in the total number of multidimensional poor in Senegal, from 8 million to over 9 million between 2005 and 2011.

9.2.4  Dimensional Changes (Uncensored and Censored Headcount Ratios)

The reductions in , , or  can be broken down to reveal which dimensions have been responsible for the change in poverty. This can be seen by looking at changes in the uncensored headcount ratios () and censored headcount ratios () described in section 5.5.3. We present the uncensored and censored headcount ratios of MPI indicators for Nepal in Table 9.4 for years 2006 and 2011 and analyse their changes over time. For definitions of indicators and their deprivation cutoffs, see section 5.6. Panel I gives levels and changes in uncensored headcount ratios, i.e. the percentage of people that are deprived in each indicator irrespective of deprivations in other indicators. Panel II provides levels and changes in the censored headcount ratios, i.e. the percentage of people that are multidimensionally poor and simultaneously deprived in each indicator. By definition, the uncensored headcount ratio of an indicator is equal to or higher than the censored headcount of that indicator. The standard errors are reported in the parenthesis.

Table 9.4: Uncensored and Censored Headcount Ratios of the Global MPI, Nepal 2006–2011

table9_4As we can see in the table, Nepal made statistically significant reductions in all indicators in terms of both uncensored and censored headcount ratios. The larger reductions in censored headcount are observed in electricity, assets, cooking fuel, flooring, and sanitation; all censored headcount ratios have decreased by more than 3 percentage points. Nutrition, mortality, schooling and attendance follow with annual reductions of 3, 2.3, 1.8, and 1.5 percentage points, respectively.

The changes in censored headcount ratios depict changes in deprivations among the poor. Recall that the overall  is the weighted sum of censored headcount ratios of the indicators as presented in equation (5.13) and the contribution of each indicator to the  can be computed by equation (5.14). Because of this relationship, the absolute rate of reduction in  in equation (9.8) and the annualized absolute rate of reduction in  in equation (9.12) can be expressed as weighted averages of absolute rate of reductions in censored headcount ratios and annualized absolute rate of reductions in censored headcount ratios, respectively. When different indicators are assigned different weights, the effects of their changes on the change in  reflect these weights.[13] For example, in the MPI, the nutrition indicator is assigned a three times more weight than electricity. This implies that a one percentage point reduction in nutrition ceteris paribus would lead to an absolute reduction in  that is three times larger than a one percentage point reduction in the electricity indicator.

Recall that it is straightforward to compute the contribution of each indicator to  using its weighted censored headcount ratio as given in equation (5.14). Note that interpreting the real on-the-ground contribution of each indicator to the change in  is not so mechanical. Why? A reduction in the censored headcount ratio of an indicator is not independent of the changes in other indicators. It is possible that the reduction in the censored headcount ratio of a certain indicator  occurred because a poor person became non-deprived in indicator But it is also possible that the reduction occurred because a person who had been deprived in  became non-poor due to reductions in other indicators, even though they remain deprived in . In the second period, their deprivation in  is now censored because they are non-poor (their deprivation score does not exceed ). The comparison between the uncensored and censored headcount distinguishes these situations. For example, we can see from Panel I of Table 9.4 that the reductions in the uncensored headcount ratios of flooring and cooking fuel are lower than the annualized reductions of the censored headcount ratios of the these two indicators. Thus some non-poor people are deprived in these indicators. In intertemporal analysis it is useful to compare the corresponding censored and uncensored headcount ratios to analyse the relation between the dimensional changes among the poor and the society-wide changes in deprivations. Of course in repeated cross-sectional data, this comparison will also be affected by migration and demographic shifts as well as changes in the deprivation profiles of the non-poor.

Panel III of Table 9.5 presents the contribution of the indicators to the  for Nepal in 2006 and in 2011. The contributions of assets, electricity, and attendance have gone down; whereas the contributions of flooring, cooking fuel, sanitation, and schooling have gone up. The contributions of water, nutrition and mortality have not shown large changes. Dimensional analyses is vital and motivating because any real reduction in a dimensional deprivation will certainly reduce . Real reductions are normally those which are visible both in raw and censored headcounts.[14]

9.2.5  Subgroup Decomposition of Change in Poverty

One important property that the adjusted-FGT measures satisfy is population subgroup decomposability, so that the overall  can be expressed as:  where denotes the Adjusted Headcount Ratio and  the population share of subgroup  as in equation (5.14). It is extremely useful to analyse poverty changes by population subgroups, to see if the poorest subgroups reduced poverty faster than less poor subgroups, and to see the dimensional composition of reduction across subgroups (Alkire and Seth 2013b; Alkire and Roche 2013; Alkire, Roche and Vaz 2014). Population shares for each time period must be analysed alongside subgroup trends. For example, let us decompose the Indian population into four caste categories: Scheduled Castes (SC), Scheduled Tribes (ST), Other Backward Classes (OBC), and the General category. As Table 9.5 shows,  as well as  have gone down statistically significantly at the national level and across all four subgroups, which is good news. However, the reduction was slowest among STs who were the poorest as a group in 1999, and their intensity showed no significant decrease. Thus, the poorest subgroup registered slowest progress in terms of reducing poverty.

Table 9.5 Decomposition of ,  and  across Castes in India

table9_5To supplement the above analysis it is useful to explore the contribution of population subgroups to the overall reduction in poverty, which not only depends on the changes in subgroups’ poverty but also on changes in the population composition. This can be seen by presenting the overall change in  between two periods () as



Note that the overall change depends both on the changes in subgroup ’s and the changes in population shares of the subgroups.

9.3     Changes Over Time by Dynamic Subgroups

The overall changes in  and discussed thus far could have been generated in many ways. It might be desirable for policy purposes to monitor how poverty changed. In particular, one may wish to pinpoint the extent to which poverty reduction occurred due to people leaving poverty vs. a reduction of intensity among those who remained poor, and also to know the precise dimensional changes which drove each.

For example, a decrease in the headcount ratio by 10% could have been generated by an exit of 10% of the population who had been poor in the first period. Alternatively, it could have been generated by a 20% decrease in the population who had been poor, accompanied by an influx of 10% of the population who became newly poor. Furthermore, the people who exited poverty could have had high deprivation scores in the first period – that is, been among the poorest – or they could have been only barely poor. The deprivation scores of those entering and leaving poverty will affect the overall change in intensity  as will changes among those who stay poor.  In addition these entries into and exits from poverty could have been precipitated by diferent possible increases or decreases in the dimensional deprivations people experienced in the first period, which will then be reflected in the changes in uncensored and censored headcount ratios.

This section introduces more precisely these dynamics of change. We first show what can be captured with panel data, then show empirical strategies to address this situation with repeated cross-sectional data. Finally we present two approaches related to Shapley decompositions which decompose changes precisely, but rely on some crucial assumptions so their empirical accuracy is questionable.

9.3.1  Exits, Entries, and the Ongoing Poor: A Two-Period Panel

Let us consider a fixed set of population of size  across two periods, and . The achievement matrices of these periods are denoted by  and . The population can be mutually exclusively and collectively exhaustively categorized into four groups that we refer to as dynamic subgroups as follows:


Contains  people who are non-poor in both periods  and


Contains  people who are poor in both periods  and  (ongoing poor)


Contains  people who are poor in period  but exit poverty in period


Contains  people who are not poor in period  but enter poverty in period

We denote the achievement matrices of these four subgroups in period  by , ,  and  for all , . The proportion of multidimensionally poor populationin period  is  and that in period  is . The change in the proportion of poor people between these two periods is =. In other words, the change in the overall multidimensional headcount ratio is the difference between the proportion of poor entering and the proportion of poor exiting poverty. Note that, by construction, no person is poor in , , , and  and thus . This also implies, . On the other hand, all persons in , , , and  are poor and thus . Therefore, the  of each of these four subgroups is equal to its intensity of poverty.

In a fixed population, the overall population and the population share of each dynamic group remains unchanged across two time periods.[15] The change in the overall  can be decomposed using equation (9.14) as



Thus, the right-hand side of equation (9.15) has three components. The first component is due to the change in the intensity of those who remain poor in both periods—the ongoing poor—weighted by the size of this dynamic subgroup. The second component  is due to the change in the intensity of those who exit poverty (weighted by the size of this subgroup) and the third component  is due to the population-weighted change in the intensity of those who enter poverty. Together .

From this point there are many interesting possible avenues for analyses. Each group can be studied separately or in different combinations. For policy, it could be interesting to know who exited poverty, and their intensity in the previous period, to see if the poorest of the poor moved out of poverty. The intensity of those who entered poverty shows whether they dipped into the barely poor group, or catapulted into high-intensity poverty, perhaps due to some shock or crisis or (if the population is not fixed) migration. Intensity changes among the ongoing poor show whether their deprivations are declining, even though they have not yet exited poverty. Dimensional analyses of changes for each dynamic subgroup, which are not covered in this book but are straightforward extensions of this material, are also both illuminating and policy relevant.

In the case of panel data with a fixed population we are able to estimate these precisely. We can thus monitor the extent to which the change in  is due to movement into and out of poverty, and the extent to which it is due to a change in intensity among the ongoing poor population. The example in Box 9.1 may clarify.

Box 9.1 Decomposing the Change inacross Dynamic Subgroups: An Illustration

Consider the following six-person, six-dimension  matrices, in which people enter and exit poverty, and intensity among the poor also increase and decreases.

Let us use a poverty cutoff of 33% or two out of six dimensions. Increases and decreases are depicted in bold. Below we summarize  ,  and  in two periods and their changes across two periods.


So in period two there are four kinds of changes affecting the dynamic subgroups as follows:

1)       :  persons 1 and 2 become non-poor (move out of or exit poverty)

2)       person 6 enters poverty.

3)         two kinds of changes occur

a.        deprivations of ongoing poor persons 3 and 4 reduce by one deprivation each

b.       deprivations of ongoing poor person 5 increases by one deprivation.

The descriptions and the decompositions of  for the changes are in the following table.


What is particularly interesting for policy is that we can notice that, in this example, 11% of the reduction in poverty was due to changes in intensity among the 50% of the population who stayed poor, that poverty was effectively increased 33% by the new entrant, but that this was more than compensated by those who exited poverty (–122%), because they initially had very high intensities. In this dramatic example, the poorest of the poor exited poverty, while the less poor experienced smaller reductions.

9.3.2  Decomposition by Incidence and Intensity for Cross-Sectional Data

The previous section explained the changes for a fixed population over time. To estimate that empirically requires panel data with data on the same persons in both periods which can be used to track their movement in and out of poverty. Yet analyses over time are often based on repeated cross-sectional data having independent samples that are statistically representative of the population under study, but that do not to track each specific observation over time. This section examines the decomposition of changes in  for cross-sectional data.

With cross-sectional data, we cannot distinguish between the three groups identified above, nor can we isolate the intensity of those who move into or out of poverty. Observed values are only available for:  and . Using these, it is categorically impossible to decompose  with the empirical precision that panel data permits.

Nonetheless, if required one can move forward with some simplifications. Instead of three groups ( let us consider just two, which might be referred to (somewhat roughly) as movers and stayers.  We define movers as the  people who reflect the net change in poverty levels across the two periods. Stayers are ongoing poor plus the proportion of previously poor people who were replaced by ‘new poor’, and total those who are poor in period 2 .  In considering only the ‘net’ change in headcount, one effectively permits the larger of  to dominate: if poverty rose nationally, it is the group who entered poverty who dominate; if poverty fell nationally, the group who exited poverty. The subordinate third group is allocated among the ongoing poor and the dominant group. For the remainder of this section we presume that both  and  decreased overall. In this case, . So , and =. As is evident, this simplification is performed because empirical data exist in repeated cross-sections for and .

Example: Suppose that 37% of people are ongoing poor, 3% enter poverty, 13% exit poverty, and 47% remain non-poor. Suppose the overall headcount ratio decreased by 10 percentage points, and the headcount ratio in period 2 is 40%, whereas in period 1 it was 50% (37%+13%). We now primarily consider two numbers: the headcount ratio in period 2 of 40% (interpreted broadly as ongoing poverty) and the change in headcount ratio of 10% (interpreted broadly as moving into/out of poverty). In doing so we are effectively permitting the ‘new poverty entrants’ to be considered as among the group in ongoing poverty in period 2 (37% + 3% = 40%). To balance this, we effectively replace 3% of those who exited poverty (13% – 3% = 10% =), and consider this slightly reduced group to be those who moved out of poverty. If poverty had increased overall, the swaps would be in the other direction.

If poverty has reduced and there has not been a large influx of people into poverty, that is, if  is presumed to be relatively small empirically, then this strategy would be likely to shed light on the relative intensity levels of those who moved out of poverty , and the changes in intensity among those who remained poor . If empirically is expected (from other sources of information) to be large, or if their intensity is expected to differ greatly from the average, this strategy is not advised.[16]

Consider the intensity of the net population who exited poverty – under these simplifying assumptions reflected by the net change in headcount, denoted and the intensity change of the net ongoing poor, whom we will presume to be , denoted . The  can be decomposed according to these two groups. These decompositions can be interpreted as showing percentage of the change in  that can be attributed to those who moved out of poverty, versus the percentage of change which was mainly caused by a decrease in intensity among those who stayed poor. We use the terms movers and stayers to refer to these less precise dynamic subgroups in cross-sectional data analysis.




Cross sectional data does not provide the intensity of either of those who stayed poor or of those who moved out of poverty. One way forward is to estimate these using existing data. First, identify the poor persons having the lowest intensity in the dataset (sampling weights applied), and use the average of these scores for  then solve for  Subsequently, identify the  poor persons having the highest intensity in that dataset, and repeat the procedure.  This will generate upper and lower estimates for  and  in a given dataset, which will provide an idea of the degree of uncertainty that different assumptions introduce. To estimate stricter upper and lower bounds it could be assumed that those moved out of poverty had an intensity score of the value of  (the theoretically minimum possible), and subsequently assume that their intensity was 100% (the theoretically maximum possible).[18]

Table 9.6 provides the empirical estimations for the upper and lower bounds for the same four countries discussed above plus Ethiopia. At the upper bound those who moved out of poverty could have had average intensities ranging from 59% in Peru (the least poor country) to 99% in Ethiopia or 100% in Senegal, according to the datasets. This in itself is interesting, as it shows that Rwanda—which is the poorest country of the four—had ‘movers’ with lower average deprivations than Ethiopa. Those who stayed poor would have had, in this case, small if any increases or decreases in intensity—less than four percentage points. At the lower bound, those who moved out of poverty could have had intensities from 33% in Peru and Senegal to 38% in Nepal, and intensity reductions among the ongoing poor could have ranged from 2% in Senegal to 13% in Nepal. At the upper bound (where we assume the poorest of the poor moved out of poverty), for Nepal, Rwanda and Peru, over 100% of the poverty reduction was due to the movers, because intensity among the ongoing poor would have had to increase (to create the observed ). At the lower bound, where the least poor people moved out of poverty, movers contributed 47–67% to . Senegal did not have a statistically significant reduction in poverty. Ethiopia provides a different example where the upper and lower bound are closer together and reductions in intensity among the ongoing poor would have contributed 31% to 73%.

Table 9.6 Decomposing the Change in  by Dynamic Subgroups

table9_6This empirical investigation shows that, when implemented with the mild assumptions that are required for cross-sectional data, the upper and lower bounds according to each country’s dataset are very wide apart. In reality, the relative contribution, of the ‘movers’ and ‘stayers’ to overall poverty reduction could vary anywhere in this range.

As the example shows, the empirical upper and lower bounds vary greatly across countries. In the case of Ethiopia, movers explain 27% to 69% of the changes in poverty, and stayers account for 31% to 73%. These boundaries do not permit us to assess whether the actual contribution from ‘movers’ was greater than or less than that of ‘stayers’. In Nepal and Peru the ‘movers’ probably contributed more than ‘stayers’ to poverty reduction, as in all cases their lowest effect is above 50%. Given these wide-ranging upper and lower bounds, empirically we are unable to answer questions such as whether the intensity of the ongoing poor decreased, or whether it was the barely poor or the deeply poor who moved out of poverty. While this can seem disappointing, for policy purposes, as Sen stresses, it may be better to be ‘vaguely right than precisely wrong’, and repeated cross-sectional data simply do not permit us, at this time, to bound ahead with precision.

9.3.3  Theoretical Incidence-Intensity Decompositions

Whereas monitoring and policy inputs must be based on empirical analyses, some research topics utilize theoretical analyses. This section introduces two theory-based approaches to decomposing changes in repeated cross-sectional data according to what we call ‘incidence’ and ‘intensity’. In each approach assumptions are made regarding the intensity of those who exit or remain poor. As we have already noted, the task implies some challenges because the empirical accuracy of the underlying assumptions is completely unknown, and as Table 9.6 showed, the actual range may be quite large. These techniques are thus offered in the spirit of academic inquiry.

For simplicity of notation, in this subsection, we denote the , , and  for period  by ,  and  and that for period  by ,  and . The first approach consists in the additive decomposition proposed by Apablaza and Yalonetzky (2013), which is illustrated in Panel A of Figure 9.2. Since , they propose to decompose the change in  by changes in its partial indices as follows

figure9_17Note that the illustration in Figure 9.2 assumes reductions in , , and  over time, but the graph can be adjusted to incorporate situations where they do not necessarily fall. This approach involves two assumptions. First, the intensity among those who left poverty is assumed to be the same as the average intensity in period . Second, the intensity change among the ongoing poor is assumed to equal the simple difference in intensities of the poor across the two periods. The decomposition is completed using an interaction term, as depicted in Panel A of Figure 9.2, below. This is indeed an additive decomposition of changes in the Adjusted Headcount Ratio  Apablaza and Yalonetzky interpret these changes as reflecting: (1) changes in the incidence of poverty , (2) changes in the intensity of poverty , and (3) a joint effect that is due to interaction between incidence and intensity (.

Figure 9.2 Theoretical Decompositions

figure9_2A second theoretical approach corresponds to a Shapley decomposition proposed by Roche (2013). This builds on Apablaza and Yalonetzky (2013) and performs a Shapley value decomposition following Shorrocks (1999).[19] It provides the marginal effect of changes in incidence and intensity as follows:




Panel B of Figure 9.2 illustrates Roche’s application of Shapley decompositions, which focuses on the marginal effect without the interaction effect. Roche’s proposal assumes that the intensity of those who exited poverty (our terms) is the average intensity of the two periods , and calls this the ‘incidence effect’. He takes the other group as comprising the average headcount ratio between the two periods, and their change in intensity as the simple difference in intensities across the periods   and describes this as the ‘intensity effect’.

Roche’s masterly presentation systematically applies Shapley decompositions to each step of dynamic analysis using the AF method. For example, if the underlying assumptions are transparently stated and accepted, the theoretically derived marginal contribution of changes in incidence and marginal contribution of changes in intensity can be expressed as a percentage of the overall change in  so they both add to 100% and can be written as follows





To address demographic shifts Roche follows a similar decomposition of change as that used in FGT unidimensional poverty measures (Ravallion and Huppi, 1991) and Shapely decompositions (Duclos and Araar 2006, Shorrocks 1999). This approach, presented in Equation (9.21), attribute demographic effects to the average population shares and subgroup ’s across time. Roche argues that, if the underlying assumptions are accepted, the overall change in poverty level can be broken down in two components: 1) changes due to intra-sectoral or within-group poverty effect, 2) changes due to demographic or inter-sectoral effect. So the overall change in the adjusted headcount between two periods respectively, could be expressed as follows 




It is common to express the contribution of each factor as a proportion of the overall change, in which case equation (9.21) is divided throughout by

The last columns of Table 9.6 Panel B provide Shapley decompositions for the same five countries. We see that in all cases the Shapley decompositions lie, as anticipated, between the upper and lower bounds. The Shapley decompositions have the broad appeal of presenting point estimates that pinpoint the exact contribution of each partial index to changes in poverty, according to their underlying assumptions, and thus may be used in analyses when empirical accuracy is not required or the assumptions are independently verified. A full illustration of the Shapley decomposition methods using data on multidimensional child poverty in Bangladesh is given in Roche (2013).[20]

9.4     This Complementary Chronic Poverty with Multiple Time Periods

Panel datasets provide information on precisely the same individual or household at different periods of time. Good quality panel datasets are particularly rich and useful for analysing multidimensional poverty because their analysis provides policy-relevant insights that extend what time series data provide. For example, using panel data we can distinguish the deprivations experienced by the chronically poor from those experienced by the transitory poor and thus identify the combination of deprivations that trap people in long-term multidimensional poverty. Also, we can analyse the duration over which a person was deprived in each indicator—and the sequences by which their deprivation profile evolved. As section 9.3.1 showed, we can identify precisely the contributions to poverty reduction that were generated by changes entries and exits from poverty and by the ongoing poor.

The following section very briefly presents a counting-based class of chronic multidimensional poverty measures that use a triple-cutoff method of identifying who is poor. We give prominence here to the measure that can be estimated using ordinal data. The partial indices associated with the chronic multidimensional poverty methodology include the headcount ratio and intensity, as well as new indices related to the duration of poverty and of dimensional deprivation. We also present a linked measure of transient poverty. As in other sections, we presume that interested readers will master standard empirical and statistical techniques that are appropriate for studies using panel data, and apply these in the analyses of poverty transitions and chronic poverty here described.

The closing section on poverty transitions informally sketches revealing analyses that can be undertaken without generating a chronic poverty measure. Rather, people are identified as multidimensionally poor or non-poor in each period, then population subgroups are identified that have differing sequences of multidimensional poverty. For example, one group might include non-poor people who ‘fell’ into multidimensional poverty, a second might include multidimensionally poor people who ‘rose’ out of poverty, a third might contain those who were poor in all periods, and a fourth might contain those who ‘churned’ in and out of poverty across periods. Naturally the number of ‘dynamic subgroups’ depends upon the sample design, the number of waves of data and the precise definition of each group.

9.4.1  Chronic Poverty Measurement Using Panel Data

Multiple approaches to measuring chronic poverty in one dimension exist, many of which have implications for the measurement of chronic multidimensional poverty.[21] Alongside important qualitative work, multiple methodologies for measuring chronic multidimensional poverty have also been proposed.[22] This section combines the AF methodology with the counting-based approach to chronic poverty measurement proposed in Foster (2009), which has a dual-cutoff identification structure and aggregation method that are very similar to the AF method. Foster (2009) provides a methodology for measuring unidimensional chronic poverty in which each time period is equally weighted for all a An  matrix is constructed in which each entry takes a value of one if person  is identified as poor in period and a value of 0 otherwise. A -dimensional ‘count’ vector is constructed in which each entry shows the number of periods in which person was poor. A second time cutoff is applied such that each person is identified as chronically poor if he or she has been poor in  or more periods. Associated FGT indices and partial indices are then constructed from the relevant censored matrices.  Order of Aggregation

This combined chronic multidimensional poverty measure applies three sets of cutoffs: deprivation cutoffs, a multidimensional poverty cutoff , and a duration cutoff . It is possible to analyse multidimensional poverty using panel data by combining the AF methodology and the Foster (2009) chronic poverty methods following two different orders of aggregation, which we call chronic deprivation or chronic poverty. These alternatives effectively interchange the order in which the poverty and duration cutoffs are applied.  In both cases, we first apply a fixed set of deprivation cutoffs to the achievement matrix in each period.

In the chronic deprivation option ( before ), we first consider the duration of deprivation in each indicator for each person and then compute a multidimensional poverty measure which summarizes only those deprivations that have been experienced by the same person across or more periods. This approach aggregates all ‘chronic’ deprivations into a multidimensional poverty index, regardless of the period in which those deprivations were experienced. This approach would provide complementary information that could enrich analyses of multidimensional poverty, but cannot be broken down by time period, nor does it show whether the deprivations were experienced simultaneously (Alkire, Apablaza, Chakravarty, and Yalonetzky 2014).

In the chronic multidimensional poverty option ( before ), we first identify each person as multidimensionally poor or non-poor in each period using the poverty cutoff. We then count the periods in which each person experienced multidimensional poverty. We identify as chronically multidimensionally poor those persons who have experienced multidimensional poverty in  or more periods.  Deprivation matrices

We observe achievements across  dimensions for a set of  individuals at  different time points. Let  stand for the achievement in attribute  of person  in period , where  Let  denote a  matrix whose elements reflect the dimensional achievements of the population in period . The deprivation cutoff vector  is fixed across periods. As before, a person is deprived with respect to deprivation  in periods if  and non-deprived otherwise. By applying the deprivation cutoffs to the achievement matrix for each period, we can construct the period-specific deprivation matrices . For simplicity, this section uses the non-normalized or numbered weights notation across dimensions, such that  Time periods are equally weighted. When achievement data are cardinal we can also construct normalized gap matrices  and squared gap matrices  or more generally, powered matrices of normalized deprivation gaps  where . In a similar manner as previously, we generate the -dimensional  column vector, which reflects the weighted sum of deprivations person  experiences in periods .  Identification

To identify who is chronically multidimensionally poor we first construct an identification matrix. The same matrix can be used to identify the transient poor in each period and to create subgroups of those who exhibit distinct patterns of multidimensional poverty (for analysis of poverty transitions).

Identification Matrix

Let be an  identification matrix whose typical element  is one if person is identified as multidimensionally poor in period  using the AF methodology, that is, using a poverty cutoff  which is fixed across periods, and 0 otherwise.

The typical column  reflects the identification status  for the th person in period , whereas the typical row displays the periods in which person  has been identified as multidimensionally poor (signified by an entry of 1) or non-poor (0). Thus we might equivalently consider each column of  to be an identification column vector for period such that  if and only if person is multidimensionally poor in period  according to the deprivation cutoffs , weights , and poverty cutoff ;  and  otherwise.

Episodes of Poverty Count Vector

From the  matrix we construct the -dimensional column vector whose th element = sums the elements of the corresponding row vector of  and provides the total number of periods in which person  is poor, or the total episodes of poverty, as identified by poverty cutoff . Naturally, , that is, each person may have from 0 episodes of poverty to  episodes, the latter indicating that a person was poor in each of the periods.

Chronic Multidimensional Poverty: Identification and Censoring

We apply the duration cutoff where  to the  vector in order identify the status of each person as chronically multidimensionally poor or not. We identify a person to be chronically multidimensionally poor if . That is, if they have experienced  or more periods in multidimensional poverty. A person is considered non-chronically poor if . We doubly censor the  vector such that it takes the value of 0 (non-chronically poor) if  and takes the value of  otherwise. The notation  indicates the censored vector of poverty episodes—just as the notation  indicated the censored deprivation count vector. Positive entries reflect the number of periods in which chronically poor people experienced poverty; entries of 0 mean that the person is not identified as chronically poor.

Among the non-chronically poor, we could (as we will elaborate) identify two subgroups: the non-poor and the transient poor. A person is considered transiently poor if . And naturally a person for whom , that is, who is non-poor in all periods, is considered non-poor.

An alternative but useful notation for the identification of chronic multidimensional poverty uses the identification function: we apply the identification function  to censor the  matrix and the vector. The doubly censored matrix reflects solely those periods in poverty that are experienced by the chronically poor (censoring all periods of transient poverty) and is denoted by . After censoring, the typical element  is defined by . The entry takes a value of  if person  is chronically multidimensionally poor and 0 otherwise.

Censored  Deprivation Matrices and Count Vector

To identify the censored headcounts, as well as the dimensional composition of poverty in each period, we censor the  sets of  deprivation matrices by applying the twin identification functions and . We denote the censored matrices by  and the censored deprivation count vectors for each period  .

Deprivation Duration Matrix

Finally, to summarize the overall dimensional deprivations of the poor, as well as the duration of these deprivations, it will be useful to create a duration matrix based on the censored deprivation matrices . Let  be an  matrix whose typical element  provides the number of periods in which is chronically poor and is deprived in dimension . Note that  We can use the duration matrix to obtain the deprivation-specific duration indices, which show the percentage of periods in which, on average, poor people were deprived in each indicator.  Aggregation

The measure of chronic multidimensional poverty when some data are ordinal may be written as follows:



Thus the Adjusted Headcount Ratio of chronic multidimensional povertyis the mean of the set of  deprivation matrices that have been censored of all deprivations of persons who are not chronically multidimensionally poor.  Alternative notation for this measure can be found in Alkire, Apablaza, Chakravarty, and Yalonetzky (2014).

When data are cardinal, the  class of measures are, like the AF class, the means of the respective powered matrices of normalized gaps.



9.4.2  Properties

For chronic multidimensional poverty, as for multidimensional poverty, the specification of axioms is, in some cases, a joint restriction on the (triple-cutoff) identification and aggregation strategies and, hence, on the overall poverty methodology. The properties are now defined with respect to the chronically multidimensionally poor population. The class of measures present respects the key properties that were highlighted as providing policy relevance and practicality to the AF measures, such as subgroup consistency and decomposability, dimensional monotonicity, dimensional breakdown, and ordinality. In addition, this class of measures satisfies a form of time monotonicity as highlighted in Foster (2009) in the unidimensional case. The intuition is that if a person who is chronically poor becomes poor in an additional period, poverty rises.

A full definition of the properties that this chronic multidimensional poverty measure fulfils is provided in Alkire, Apablaza, Chakravarty, and Yalonetzky (2014). The methodology of multidimensional chronic poverty measurement fulfils the appropriately stated properties of anonymity, time anonymity, population replication invariance, chronic poverty focus, time focus, chronic normalization, chronic dimensional monotonicity, chronic weak monotonicity, time monotonicity, chronic monotonicity in thresholds, monotonicity in multidimensional poverty identifier, chronic duration monotonicity, chronic weak transfer, non-increasing chronic poverty under association-decreasing switch, and additive subgroup decomposability for all . The class of measures also satisfies chronic strong monotonicity for  and chronic strong transfer when .

9.4.3  Consistent Partial Indices

Like , the chronic multidimensional poverty measure is the product of intuitive partial indices that convey meaningful information on different features of a society’s experience of chronic multidimensional poverty. In particular,  where:

·         is the headcount ratio of chronic multidimensional poverty—the percentage of the population who are chronically multidimensionally poor according to  and .

·         is the average intensity of poverty among the chronically multidimensionally poor—the average share of weighted deprivations that chronically poor people experience in those periods in which they are multidimensionally poor.

·         reflects the average duration of poverty among the chronically poor — the average share of  periods in which they experience multidimensional poverty.

These partial indices can also be calculated directly. In particular,



That is, the headcount ratio of chronic multidimensional poverty is the number of people who have been identified as chronically multidimensionally poor divided by the total population. We denote the number of chronically multidimensionally poor people by

The intensity of chronic multidimensional poverty is the sum of the weighted deprivation scores of all poor people over all time periods, divided by the number of dimensions times the total number of people who are poor in each period summed across periods. Note that



The average duration of chronic multidimensional poverty—the percentage of periods on average in which the chronically person was poor—can be easily assessed using the  vector.



The duration is the sum of the total number of periods in which the chronically poor experience multidimensional poverty, divided by the number of periods and the number of chronically poor. Note that  Box 9.2 illustrates these with a simple example.

Box 9.2 Computing Incidence and Duration of Chronic Poverty

Consider three people and four periods, with = 2.

Person 1 is multidimensionally poor in period 1

Person 2 is multidimensionally poor in periods 2, 3, and 4

Person 3 is multidimensionally poor in periods 1, 2, 3, and 4


Two people are chronically poor because they experience multidimensional poverty in = 2 or more periods. So the percentage of people identified as chronically poor  is 67%.

In this case, the vector =(0,3,4);  = 2 and our duration index is. That is, on average, chronically poor persons are poor during 87.5% of the time periods.  Dimensional Indices

For chronic multidimensional poverty, it is possible and useful to generate the standard dimensional indicators for each period: the censored headcount and percentage contribution. It is also possible and useful to generate the period-specific Adjusted Headcount Ratio (), headcount ratio (), and intensity () figures, which are different from, but can be consistently related to, the chronic poverty headcount and intensity values presented in 9.4.3. Finally, and of tremendous use, it is possible to present the average duration of deprivation in each dimension and to relate this directly to the overall duration of chronic poverty. Box 9.3 presents the intuition of this set of consistent indices; for their precise definition see Alkire, Apablaza, Chakravarty, and Yalonetzky (2014).

Box 9.3 Computing Incidence and Duration of Chronic Poverty

Cross-period indices reflecting chronic poverty:

: Adjusted Headcount Ratio of chronic multidimensional poverty

: Headcount ratio, showing the percentage of the population who are chronically poor

: Intensity, showing the average percentage of deprivations experienced by the chronically multidimensionally poor in those periods in which they are poor

: Average duration of chronic poverty, expressed as a percentage of time periods

: Average censored headcount of dimension  among the chronically poor in all periods in which they are poor and are deprived in dimension

: Average duration of deprivation in dimension  among the chronically poor, expressed as a percentage of time periods

: Percentage contribution of dimension  to the deprivations of the chronically poor.

Single-period indices reflecting the profiles of the chronic poor in that particular period of poverty:

: Headcount ratio, showing the percentage of the population who are chronically poor in period

: Intensity, showing the average percentage of deprivations experienced by the chronically multidimensionally poor in period

: Censored headcount of dimension  among the chronically poor in period

: Percentage contribution of dimension  to the deprivations of the chronically poor in period .

Cross-period averages of the unidimensional indices can also be constructed, such as  and  and analysed in conjunction with the relevant duration measure. Censored Headcount Ratios

The censored headcount ratios for each period  are constructed as the mean of the dimensional column vector for each period and represent the proportion of people who are chronically poor in time period  and are deprived in dimension :



We can also describe the average censored headcount ratios of chronic multidimensional poverty across  periods in each dimension as simply the mean of the censored headcounts in each period:



The Adjusted Headcount Ratio of chronic multidimensional poverty across all periods is simply the mean of the average weighted censored headcount ratios:


(9.29) Percentage Contributions of Dimension

The percentage contributions show the (weighted) composition of chronic multidimensional poverty in each period and across periods.

We may seek an overview of the dimensional composition of poverty across all periods. The total percentage contribution of each dimension to chronic poverty across all periods is given by



We may also be interested in analysing the percentage contributions of each dimension across various periods and thus in comparing the percentage contributions of dimensions across periods. The total percentage contribution in period  is


(9.31) Censored Dimensional Duration

We are also able to construct a new set of statistics that provide more detail regarding the duration of dimensional deprivations among the chronically poor. We use the  deprivation duration matrix , constructed in, in which each entry reflects the number of periods in which person  was chronically poor (by  and ) and was deprived in dimension . Recall that for the chronically poor,  in each dimension. The value of  is, naturally, 0 for non-poor persons in all dimensions. Thus the matrix will have a positive entry for  persons and an entry of 0 for all persons who were never chronically poor.

For each dimension we can then define a dimensional duration index for dimension  as follows



The value of  provides the percentage of periods in which the chronically poor were deprived in dimension on average.

The relationship between the mean across all  and the chronic multidimensional poverty figure provided earlier is also elementary





(9.34) Period-Specific Partial Indices

From the  censored identification matrix  we can also compute the period-specific headcounts of chronic multidimensional poverty. The headcount  for period  is the mean of the column vector of  for period . The average headcount across all periods is. The average headcount across all periods and the chronic multidimensional poverty headcount are related by the average duration of poverty thus:








(9.37) Illustration using Chilean CASEN

We present an example in Table 9.7 using three variables: schooling, overcrowding, and income in Chile’s CASEN (Encuesta de Caracterización Socioeconómica Nacional) dataset for three periods: 1996, 2001, and 2006. The table reports the Adjusted Headcount Ratio of chronic multidimensional poverty () and its three partial indices: the headcount ratio (), the average chronic intensity () and the average duration () for three poverty cutoffs  and three different duration cutoffs . All dimensions and periods are equally weighted for both identification and aggregation. When  and , the identification follows a double union approach. In this case, 49% of people are identified as chronically multidimensionally poor. However, a double-union approach does not appear to capture people who are either chronically or multidimensionally poor in any meaningful sense.

Table 9.7 Cardinal Illustration with Relevant Values of  and

table9_7We thus consider the cutoffs where . In this situation, 5% of people are chronically multidimensionally poor. The average chronic intensity is 72%, meaning that people experience deprivations in 72% of dimensions in the periods in which they are poor. The average chronic duration is 72% also, meaning that the average poor person is deprived in 72% of the three periods. The overall chronic Adjusted Headcount Ratio of 0.028 shows that Chile’s population experiences only 2.8% of the deprivations it could possibly experience. All possible deprivations occur if all people are multidimensionally poor in all dimensions, and in all periods.

9.4.4  Poverty Transitions Using Panel Data

Using the identification matrix and the associated doubly censored deprivation matrices that have been constructed above, it is also possible to analyse poverty transitions. Comparisons can be undertaken – for example, between subgroups experiencing different dynamic patterns of multidimensional poverty – to ascertain different policy sequences or entry points that might have greater efficacy in eradicating multidimensional poverty. This section very briefly describes the construction of dynamic subgroups and some of the descriptive analyses that can be undertaken.  Constructing Dynamic Subgroups

The chronic multidimensional poverty measures constructed previously respect the property of subgroup consistency and subgroup decomposability; thus, they can be decomposed by any population subgroup for which the data are arguably representative. In addition, it can be particularly useful to describe multidimensional poverty for what we earlier called ‘dynamic subgroups’ – the definition of which can be extended when panel data cover more than two periods.

By ‘dynamic subgroups’, we mean population subgroups that experience different patterns of multidimensional poverty over time. These include the groups mentioned in section 9.3.1 who exited poverty (1,0), entered poverty (0,1), or were in ongoing poverty (1,1). The possible patterns will vary according to the number of waves in the sample as well as the observed patterns in the dataset. With three waves, there are four basic groups: falling—people who were non-poor and became multidimensionally poor; rising—people who were multidimensionally poor and exited poverty; churning—people who both enter and exit multidimensional poverty in different periods, and long-term—people who remain multidimensionally poor continuously.[23]

The dynamic subgroups are formed by considering the  identification matrix . Note that we use the matrix that is censored by the poverty cutoff  but we do not, in this section on poverty transitions, apply the duration cutoff.  Consider a matrix of four persons and three periods in which each person experiences one of the four categories mentioned above. Recall that an entry of one indicates that person  is multidimensionally poor in period  and a 0 indicates they are non-poor.

            Falling:           0  0  1              (and     0  1  1)

            Rising:            1  0  0              (and      1  1  0)

            Churning:      1  0  1              (and     0  1  0)

            Long-term:     1  1  1

For more than three periods, additional categories can be formed. Note that the categories can and must be mutually exhaustive. Each person who is multidimensionally poor in any period (whether chronically or transiently poor) can be categorized into one of these four groups. Descriptive Analyses

Having decomposed the population into the non-poor and these (or additional) dynamic subgroups of the population, it can be useful to provide the standard partial indices for each subgroup, both per period and across all three periods:

·         , , and  (and standard errors);

·         Percentage composition of poverty by dimension (often revealing) and censored headcounts;

·         Intensity profiles across the poor (or inequality among the poor—see section 9.1).

It can also be useful to provide details regarding the sequences of evolution. For example, from the  matrix, isolate the subgroup of the poor who ‘fell into’ poverty between period 1 and period 2 (that is, whose entries are 0,1 for the respective periods  and .). Compare their evolution with those who stayed poor (1,1) and those who stayed non-poor (0,0), in the following ways:

·         At the individual level, compare the raw headcount in period 1 with the raw headcount in period 2.

·         Identify the dimensions in which deprivations were (a) experienced in both periods, (b) only experienced in period 1, and (c) only experienced in period 2.

·         Summarize the results, if relevant and legitimate, further decomposing the population into relevant subgroups whose compositional changes follow different patterns.

·         Repeat for each adjacent pair of periods. Analyse whether the patterns are stable or differ across different adjacent periods.


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[1] For inequality-adjusted poverty measures in the unidimensional context, see Thon (1979), Clark, Hemming, and Ulph (1981), Chakravarty (1983b), Foster, Greer, and Thorbecke (1984), and Shorrocks (1995). For inequality-adjusted multidimensional poverty measures, i.e., those that satisfy transfer and/or strict rearrangement properties, see section section 3.6.

[2] This section summarizes these two papers.

[3] In order to say that one multidimensional distribution is more equal than another, each must be smoothed using the same bistochastic matrix.

[4] See Sen (1976) and Foster and Sen (1997).

[5] Bourguignon and Chakravarty (2003) and Datt (2013) also propose a similar class of indices but using a union identification criterion.

[6] We have already shown that our multidimensional measure  satisfies weak transfer, the first type of transfer property, for  and the second type of transfer property, weak rearrangement, for .  Chakravarty, Mukherjee, and Ranade (1998), Tsui (2002), Bourguignon and Chakravarty (2003), Chakravarty and D’Ambrosio (2006), Maasoumi and Lugo (2008), Aaberge and Peluso (2012), Bossert, Chakravarty, and D’Ambrosio (2013), and Silber and Yalonetzky (2014).

[7] For empirical examples, see Alkire, Roche, and Seth (2013), who compare countries across four gradients of poverty.

[8] There are many variations. For example, if data are relatively accurate one might consider inequality using the uncensored deprivation score vector , and alternatively if one only wishes to capture inequality within some subgroup, then  and .

[9] If one is interested in decomposing (9.6) into within and between group components, then the total within-group inequality term can be computed as  and the total between-group inequality term can be computed as , where  the intensity of each subgroup  such that  and  is the share of all poor in subgroup .

[10] This section draws on Alkire, Roche, and Vaz (2014).

[11] Tables of the absolute levels and absolute rates of change make this feature visible; reporting the relative rate of change underscores this more precisely.

[12] For small samples, one needs to conduct hypothesis tests using the Student-t distribution which are very similar to the hypothesis tests described in Chapter 8 that use the Standard Normal distribution.

[13] Normative issues in assigning weights were discussed in details in Chapter 6.

[14] Comparisons of reductions in both raw and censored headcounts may be supplemented by information on migration, demographic shifts, or exogenous shocks, for example.

[15] Suitable adjustments can be made for demographic shifts when the population is not fixed across two periods.

[16]The corresponding considerations apply if poverty has increased and is expected to be small

[17] Naturally it is also possible to create estimates for where the upper bound was the overall change in intensity, and the lower bound was zero, and solve for . However this would not permit an increase in intensity (which would happen if the barely poor left poverty and the others stayed the same, for example), nor for an even stronger reduction in intensity. For example, in the example in Table 9.2, Nepal’s A reduced by five percentage points, whereas in our upper bound, intensity among the ‘stayers’ increased by 4% and in the lower bound it decreased by 13%.

[18]  These bounds are theoretically possible lower and upper bounds. Further research using panel datasets is required to investigate the likelihood of these bounds.

[19] The Shapley value decomposition was initially applied to decomposition of income inequality by Chantreuil and Trannoy (2011, 2013) and Morduch and Sinclair (1998). Shorrocks (1999) showed that it can be applied to any function under certain assumptions.

[20] The multidimensional poverty index implemented in Roche (2013) focuses on children under 5. The choice of dimensions and indicators are similar to Gordon et al. (2003).

[21] This section draws upon Alkire, Apablaza, Chakravarty, and Yalonetzky (2014). See also McKay and Lawson (2003), Dercon and Shapiro (2007), Foster (2009), Foster and Santos (2013), Jalan and Ravallion (1998), Calvo and Dercon (2013), Hoy and Zheng (2011), Gradín, Del Rio, and Canto (2011), and  Bossert, Chakravarty, and D’Ambrosio (2012).

 [22]Hulme and Shepherd (2003), Chakravarty and D’Ambrosio (2006), Calvo (2008), Addison, Hulme, and Kanbur (2009), Baluch and Masset (2003), Bossert, Ceriani, Chakravarty, and D’Ambrosio (2012), Nicholas and Ray (2011), Nicholas, Ray and Sinha (2013), and Porter and Quinn (2013).

[23] See Hulme, Moore and Shepherd (2001), Hulme and Shepherd (2003), and Narayan and Petesch (2007).

8 Robustness Analysis and Statistical Inference

Chapter 5 presented the methodology for the Adjusted Headcount Ratio poverty index  and its different partial indices; Chapter 6 discussed how to design multidimensional poverty measures using this methodology in order to advance poverty reduction; and Chapter 7 explained novel empirical techniques required during implementation. Throughout, we have discussed how the index and its partial indices may be used for policy analysis and decision-making. For example, a central government may want to allocate resources to reduce poverty across its subnational regions or may want to claim credit for strong improvement in the situation of poor people using an implementation of the Adjusted Headcount Ratio. One is, however, entitled to question how conclusive any particular poverty comparisons are for two different reasons.

One reason is that the design of a poverty measure involves the selection of a set of parameters, and one may ask how sensitive policy prescriptions are to these parameter choices. Any comparison or ranking based on a particular poverty measure may alter when a different set of parameters, such as the poverty cutoff, deprivation cutoffs or weights, is used. We define an ordering as robust with respect to a particular parameter when the order is maintained despite a change in that parameter.[1] The ordering can refer to the poverty ordering of two aggregate entities, say two countries or other geographical entities, which is a pairwise comparison, but it can also refer to the order of more than two entities, what we refer to as a ranking. Clearly, the robustness of a ranking (of several entities) depends on the robustness of all possible pairwise comparisons. Thus, the robustness of poverty comparisons should be assessed for different, but reasonable, specifications of parameters. In many circumstances, the policy-relevant comparisons should be robust to a range of plausible parameter specifications. This process is referred as robustness analysis. There are different ways in which the robustness of an ordering can be assessed. This chapter presents the most widely implemented analyses; new procedures and tests may be developed in the near future.

The second reason for questioning claimed poverty comparisons is that poverty figures in most cases are estimated from sample surveys for drawing inferences about a population. Thus, it is crucial that inferential errors are also estimated and reported. This process of drawing conclusions about the population from the data that are subject to random variation is referred as statistical inference. Inferential errors affect the degree of certainty with which two and more entities may be compared in terms of poverty for a particular set of parameters’ values. Essentially, the difference in poverty levels between two entities – states for example – may or may not be statistically significant. Statistical inference affects not only the poverty comparisons for a particular set of parameter values but also the robustness of such comparisons for a range of parameters’ values.

In general, assessments of robustness should cohere with a measure’s policy use. If the policy depends on levels of , then the robustness of the respective levels (or ranks) of poverty should be the subject of robustness tests presented here. If the policy uses information on the dimensional composition of poverty, robustness tests should assess these—which lie beyond the scope of this chapter, but see Ura et al. (2012).  Recall also from Chapter 6 people’s values may generate plausible ranges of parameters. Robustness tests clarify the extent to which the same policies would be supported across that relevant range of parameters. In this way, robustness tests can be used for building consensus or for clarifying which points of dissensus have important policy implications.

This chapter is divided into two sections. Section 8.1 presents a number of useful tools for conducting different types of robustness analysis; section 8.2 presents various techniques for drawing statistical inferences and section 8.3 presents some ways in which the two types of techniques can be brought together.

8.1     Robustness Analysis

In monetary poverty measures, the parameters include (a) the set of indicators (components of income or consumption); (b) the price vectors used to construct the aggregate as well as any adjustments such as for inflation or urban/rural price differentials; (c) the poverty line; and (d) equivalence scales (if applied). The parameters that influence the multidimensional poverty estimates and poverty comparisons based on the Adjusted Headcount Ratio are (i) the set of indicators (denoted by subscript ); (ii) the set of deprivation cutoffs (denoted by vector ); (iii) the set of weights or deprivation values (denoted by vector ); and (iv) the poverty cutoff (denoted by ). A change in these parameters may affect the overall poverty estimate or comparisons across regions or countries.

This section introduces tools that can be used to test the robustness of pairwise comparisons as well as the robustness of overall rankings with respect to the initial choice of the parameters. We first introduce a tool to test the robustness of pairwise comparisons with respect to the choice of the poverty cutoff. This tool tests an extreme form of robustness, borrowing from the concept of stochastic dominance in the single-dimensional context (section 3.3.1).[2] When dominance conditions are satisfied, the strongest possible results are obtained. However, as dominance conditions are highly stringent and dominance tests may not hold for a large number of the pairwise comparisons, we present additional tools for assessing the robustness of country rankings using the correlation between different rankings. This second set of tools can be used with changes in any of the other parameters too, namely, weights, indicators and deprivation cutoffs.

8.1.1     Dominance Analysis for Changes in the Poverty Cutoff

Although measurement design begins with the selection of indicators, weights, and deprivation cutoffs, we begin our robustness analysis by assessing dominance with respect to changes in the poverty cutoff, which is applied to the weighted deprivation scores constructed using other parameters. We do this because as in the unidimensional context, it is the poverty cutoff that finally identifies who is poor, thereby defining the ‘headcount ratio’ and effectively setting the level of poverty. It is arguably most visibly debated.[3] We have introduced the concept of stochastic dominance in the uni- and multidimensional context in section 3.3.1. This part of the chapter builds on that concept and technique, focusing primarily on the first-order stochastic dominance (FSD) and showing how it can be applied to identify any unambiguous comparisons with respect to the poverty cutoff for our two most widely used poverty measures—Adjusted Headcount Ratio () and Multidimensional Headcount Ratio (). Recall from section 3.3.1 the notation of two univariate distributions of achievements  and  with cumulative distribution functions (CDF)  and , where  and  are the shares of population in distributions  and  with achievement level less than . Distribution  first-order stochastically dominates distribution  (or  FSD  if and only if  for all  and  for some . Strict FSD requires that  for all .[4] Interestingly, if distribution  FSD , then  has no lower headcount ratio than  for all poverty lines.

Let us now explain how we can apply this concept for unanimous pairwise comparisons using  and  between any two distributions of deprivation scores across the population. For a given deprivation cutoff vector  and a given weighting vector , the FSD tool can be used to evaluate the sensitivity of any pairwise comparison to varying poverty cutoff . Following the notation introduced in Chapter 2, we denote the (uncensored) deprivation score vector by . Note that an element of  denotes the deprivation score and a larger deprivation score implies a lower level of well-being.

The FSD tool can be applied in two different ways: one is to convert deprivations into attainments by transforming the deprivation score vector  into an attainment score vector 1  and the other option is to use the tool directly on the deprivation score vector . The first approach has been pursued in Alkire and Foster (2011a) and Lasso de la Vega (2010). In this section, because it is more direct, we present the results using the deprivation score vector and thus avoid any transformation. A person is identified as poor if the deprivation score is larger than or equal to the poverty cutoff , unlike in the attainment space where a person is identified as poor if the person’s attainment falls below a certain poverty cutoff. To do that, however, we need to introduce the complementary cumulative distribution function (CCDF)—the complement of a CDF.[5] For any distribution  with CDF , the CCDF of the distribution is  1 , which means that for any value , the CCDF  is the proportion of the population that has values larger than or equal to . Naturally, CCDFs are downward sloping. The first-order stochastic dominance condition in terms of the CCDFs can be stated as follows. Any distribution  first order stochastically dominates distribution  if and only if  for all  and  for some . For strict FSD, the strict inequality must hold for all .

Now, suppose there are two distributions of deprivation scores,  and  with CCDFs  and . For poverty cutoff , if , then distribution  has no lower multidimensional headcount ratio  than distribution  at . When is it possible to say that distribution  has no lower  than distribution  for all poverty cutoffs? The answer is when distribution  first order stochastically dominates distribution . Let us provide an example in terms of two four-person vectors of deprivation scores:  and . The corresponding CCDFs  and  are denoted by a black dotted line and a solid grey line, respectively, in Figure 8.1. No part of  lies above that of  and so  first-order stochastically dominates  and we can conclude that  has unambiguously lower poverty than , in terms of the multidimensional headcount ratio.

Figure 8.1 Complementary CDFs and Poverty Dominance


Let us now try to understand dominance in terms of . In order to do so, first note that the area underneath a CCDF of a deprivation score vector is the average of its deprivation scores. Consider distribution  with CCDF  as in Figure 8.1. The area underneath  is the sum of areas I, II, III, and IV. Area IV is equal to , Area III is , and Areas I+II  is , so essentially each area is a score times its frequency in the population. The sum of the four areas , is simply the average of all elements in  and it coincides with the  measure for a union approach. When an intermediate or intersection approach to identification is used, then the  is the average of the censored deprivation score vector . In other words, the deprivation scores of those who are not identified as poor are set to 0. For example, for a poverty cutoff , the censored deprivation score vector corresponding to  is . Obtaining the average of censored deprivation scores is equivalent to ignoring areas III and IV in Figure 8.1. The  of  for  is the sum of the remaining area I, II which is .[6]

We now compute the area underneath the censored CCDF for every  and plot the area on the vertical axis for each  on the horizontal axis and refer to it as an  curve, depicted in Figure 8.2. We denote the  curves of distributions  and  by  and , respectively. Given that the  curves are obtained by computing the areas underneath the CCDFs, the dominance of  curves is referred as second-order stochastic dominance. Given that first-order stochastic dominance implies second-order dominance, if first-order dominance holds between two distributions, then  dominance will also hold between them. However, the converse is not necessarily true, that is, even when there is  dominance there may not be  dominance. Therefore, when the CCDFs of two distributions cross—i.e. there is not first-order () dominance—it is worth testing  dominance between pairs of distributions, to which we refer as pairwise comparisons from now on, using the  curves. Batana (2013) has used the  curves for the purpose of robustness analysis while comparing multidimensional poverty among women in fourteen African countries.

Figure 8.2 The Adjusted Headcount Ratio Dominance Curves


The dominance requirement for all possible poverty cutoffs may be an excessively stringent requirement. Practically, one may seek to verify the unambiguity of comparison with respect to a limited variation the poverty cutoff, which can be referred to as restricted dominance analysis. For example, when making international comparisons in terms of the MPI, Alkire and Santos (2010, 2014) tested the robustness of pairwise comparisons for all poverty cutoffs  [0.2, 0.4], in addition to the poverty cutoff of  1/3. In this case, if the restricted FSD holds between any two distributions, then dominance holds for the relevant particular range of poverty cutoffs for both  and .

8.1.2     Rank Robustness Analysis

In situations in which dominance tests are too stringent, we may explore a milder form of robustness, which assesses the extent to which a ranking, that is, an ordering of more than two entities, obtained under a specific set of parameters’ values, is preserved when the value of some parameter is modified. How should we assess the robustness of a ranking? One first intuitive measure is to compute the percentage of pairwise comparisons that are robust to changes in parameters – that is the proportion of pairwise comparisons that have the same ordering. As we shall see in section 8.3, whenever poverty computations are performed using a survey, the statistical inference tools need to be incorporated into the robustness analysis.

Another useful way to assess the robustness of a ranking is by computing a rank correlation coefficient between the original ranking of entities and the alternative rankings (i.e. those obtained with alternative parameters’ values). There are various choices for a rank correlation coefficient. The two most commonly used rank correlation coefficients are the Spearman rank correlation coefficient () and the Kendall rank correlation coefficient ().[7]

Suppose, for a particular parametric specification, the set of ranks across  population subgroups is denoted by , where  is the rank attributed to subgroup . The subgroups may be ranked by their level of multidimensional headcount ratio, the Adjusted Headcount Ratio, or any other partial indices. We present the rank correlation measures using population subgroups, but they apply to ranking across countries as well. We denote the set of ranks for an alternative specification of parameters by , where  is the rank attributed to subgroup . The alternative specification may be a different poverty cutoff, a different set of deprivation cutoffs, a different set of weights, or a combination of all three. If the initial and the alternative specification yield exactly the same set of rankings across subgroups, then  for all . In this case, we state that the two sets of rankings are perfectly positively associated and the association is highest across the two specifications. In terms of the previous approach, 100% of the pairwise comparisons are robust to changes in one or more parameters’ values. On the other hand, if the two specifications yield completely opposite sets of rankings, then  for all . In this case, we state that the two sets of rankings are perfectly negatively associated and the association is lowest across the two specifications. In terms of the previous approach, 0% of the pairwise comparisons are robust to changes in one or more parameters’ values.

The Spearman rank correlation coefficient can be expressed as



Intuitively, for the Spearman rank correlation coefficient, the square of the difference in the two ranks for each subgroup is computed and an average is taken across all subgroups. The  is bounded between  and . The lowest value of  is obtained when two rankings are perfectly negatively associated with each other whereas the largest value of  is obtained when two rankings are perfectly positively associated with each other.

The Kendall rank correlation coefficient is based on the number of concordant pairs and discordant pairs. A pair () is concordant if the comparisons between two objects are the same in both the initial and alternative specification, i.e.  and . In terms of the previously used terms, a concordant pair is equivalent to a robust pairwise comparison. A pair, on the other hand, is discordant if the comparisons between two objects are altered between the initial and the alternative specification such that  but . In terms of the previously used terms, a discordant pair is equivalent to a non-robust pairwise comparison. The  is the difference in the number of concordant and discordant pairs divided by the total number of pairwise comparisons. The Kendall rank correlation coefficient can be expressed as



Like ,  also lies between  and . The lowest value of  is obtained when two rankings are perfectly negatively associated with each other whereas the largest value of  is obtained when two rankings are perfectly positively associated with each other. Although both  and  are used to assess rank robustness the Kendall rank correlation coefficient has an intuitive interpretation. Suppose the Kendall Tau correlation coefficient is 0.90, from equation (8.2), it can be deduced that this means that 95% of the pairwise comparisons are concordant (i.e. robust) and only 5% are discordant. Equations (8.1) and (8.2) are based on the assumption that there are no ties in the rankings. In other words, both expressions are applicable when no two entities have equal values. When there are ties, Kendall (1970) offers two adjustments in the denominator of both rank correlation coefficients ( and ) to correct for tied ranks; these adjusted Kendall coefficients are commonly known as tau-b and tau-c.

Table 8.1 Correlation among Country Ranks for Different Weights[8]


Let us present one empirical illustration showing how rank robustness tools may be used in practice. The first illustration presents the correlation between 2011 MPI rankings across 109 countries and the rankings for three alternative weighting vectors (Alkire et al. 2011). The MPI attaches equal weights across three dimensions: health, education, and standard of living. However, it is hard to argue with perfect confidence that the initial weight is the correct choice. Therefore, three alternative weighting schemes were considered. The first alternative assigns a 50% weight to the health dimension and then a 25% weight to each of the other two dimensions. Similarly, the second alternative assigns a 50% weight to the education dimension and then distributes the rest of the weight equally across the other two dimensions. The third alternative specification attaches a 50% weight to the standard of living dimension and then 25% weights to each of the other two dimensions. Thus, we now have four different rankings of 109 countries, each involving 5,356 pairwise comparisons. Table 8.1 presents the rank correlation coefficient  and  between the initial ranking and the ranking for each alternative specification. It can be seen that the Spearman coefficient is around 0.98 for all three alternatives. The Kendall coefficient is around 0.9 for each of the three cases, implying that around 80% of the comparisons are concordant in each case.

The same type of analysis has been done to changes in other parameters’ values, such as the indicators used and deprivation cutoffs (Alkire and Santos 2014).

8.2     Statistical Inference

The last section showed how the robustness of claims made using the Adjusted Headcount Ratio and its partial indices may be assessed. Such assessments apply to changes in a country’s performance over time, comparisons between different countries, and comparisons of different population subgroups within a country. Most frequently, the indices are estimated from sample surveys with the objective of estimating the unknown population parameters as accurately as possible. A sample survey, unlike a census that covers the entire population, consists of a representative fraction of the population.[9] Different sample surveys, even when conducted at the same time and despite having the same design, would most likely provide a different set of estimates for the same population parameters. Thus, it is crucial to compute a measure of confidence or reliability for each estimate from a sample survey. This is done by computing the standard deviation of an estimate. The standard deviation of an estimate is referred to as its standard error. The lower the magnitude of a standard error, the larger the reliability of the corresponding estimate. Standard errors are key for hypothesis testing and for the construction of confidence intervals, both of which are very helpful for robustness analysis and more generally for drawing policy conclusions.  In what follows we briefly explain each of these statistical terms.

8.2.1     Standard Errors

There are different approaches to estimating standard errors. Two approaches are commonly followed:

  • Analytical Approach: Formulas that provide either the exact or the asymptotic approximation of the standard error and thus confidence intervals[10]
  • Resampling Approach: Standard errors and the confidence intervals may be computed through the bootstrap or similar techniques (as performed for the global MPI in Alkire and Santos 2014).

The Appendix to this chapter presents the formulas for computing standard errors with the analytical approach depending on the survey design.

The analytical approach is based on two assumptions. Such assumptions are based on the premise that the sample surveys used for estimating the population parameters are significantly smaller in size compared to the population size under consideration.[11] For example, the sample size of the Demographic and Health Survey of India in 2006 was only 0.04% of the Indian population. The first assumption is that the samples are drawn from a population that is infinitely large, so that even the finite population under study is a sample of an infinitely large superpopulation. This philosophical assumption is based on the superpopulation approach, which is different from the finite population approach (for further discussion see Deaton 1997). A finite population approach requires that a finite population correction factor should be used to deflate the standard error if the sample size is large relative to the population. However, if the sample size is significantly smaller than the finite population size, the finite population correction factor is approximately equal to one. In this case, the standard errors based on both approaches are almost the same.

The second assumption is that we treat each sample as drawn from the population with replacement. The practical motivation behind the assumption is the size of the sample survey compared to the population. The sample surveys are commonly conducted without replacement because, once a household is visited and interviewed, the same household is not visited again on purpose. When samples are drawn with replacement, the observations are independent of each other. However, if the samples are drawn without replacement, then the samples are not independent of each other. It can be shown that in the absence of multistage sampling, a sampling without replacements needs a Finite Population Correction (FPC) factor for computing the sampling variance. The FPC factor is of the order , where  is the sample size and  is the size of the population. The use of an FPC factor allows us to get a better estimate of the true population variance. However, when the sample size is small with respect to the population, i.e. , the use of an FPC factor will not make much difference to the estimation of the sampling variance as the FPC factor is closer to one (Duclos and Araar 2006: 276). These assumptions would be required in order to justify our assumption that each sample is independently and identically distributed.

We now illustrate relevant methods using the Adjusted Headcount Ratio () denoting its sample estimate by  and standard error of the estimate by . However, the methods are equally applicable to inferences for the multidimensional headcount ratio, the intensity, and the censored headcount ratios as long the standard errors are appropriately computed, as outlined in the Appendix of this chapter.

8.2.2    Confidence Intervals

A confidence interval of a point estimate is an interval that contains the true population parameter with some probability that is known as its confidence level. A significance level that is used is the complement of the confidence level. Let us denote the significance level[12] by , which by definition ranges between 0 and 100%. The level of confidence is  percent. Thus, for a given estimate, if one wants to be 95% confident about the range within which the true population parameter lies, then the significance is 5%. Similarly, if one wants to be 99% confident, then the significance level is 1%.

By the central limit theorem, we can say that the difference between the population parameter and the corresponding sample average divided by the standard error approximates the standard normal distribution (i.e. the normal distribution with a mean of 0 and a standard deviation of 1). Using the standard normal distribution one can determine the critical value associated with that significance level, which is given by the inverse of the standard normal distribution at . In other words, the critical value is the value at which the probability that the statistic is higher than that is precisely .[13] The critical values to be used when one is interested in computing a 95% confidence interval are: . If instead one is interested in computing a 99% or a 90% confidence interval, the corresponding critical values are  and , respectively.

For example, Table 8.2 presents the sample estimate of the Adjusted Headcount Ratio (), the multidimensional headcount ratio (), and the average deprivation share among the poor () from the Demographic and Health Survey of 2005–2006. India’s sample estimate of the population-Adjusted Headcount Ratio is , with a standard error . The % confidence interval is then . This means that with 95% confidence, the true population  lies between  and . Similarly, the 99% confidence interval of India’s  is (). The more one wants to be confident about the range within which the true population parameter lies, the larger the confidence interval will be.

Confidence Intervals for , , and


Similar to , the confidence interval for  is , for  is , and for  is  for all . It can be seen from the table that the standard error of  is 0.41%, whereas that of  is 0.20%.

8.2.3    Hypothesis Tests

Confidence intervals are useful for judging the statistical reliability of a point estimate when the population parameter is unknown. However, suppose that, somehow, we have a hypothesis about what the population parameter is. For example, suppose the government hypothesizes that the Adjusted Headcount Ratio in India is 0.26. Thus, the null hypothesis is . This has to be tested against any of the three alternatives  or  or .[14] This is a one-sample test. Note that the first alternative requires a so-called two-tailed test, and each of the other two alternatives requires a so-called one-tailed test. Now, suppose a sample (either simple random or multistage stratified)  of size  is collected. We denote the estimated Adjusted Headcount Ratio by . By the law of large numbers and by the central limit theorem, as , , where  is the population variance of . The standard error  of  can be estimated using either equation (8.11) or (8.30) in the Appendix, whichever is applicable.

In a two-tail test, the null hypothesis can be rejected against the alternative  with () percent confidence if .; in words, if the absolute value of the statistic is greater than the absolute value of the critical value. An equivalent procedure to reject or not the null hypothesis entails, rather than comparing the test statistic against the critical value, comparing the significance level against the so-called -value. The -value is defined as the actual probability that the test statistic assumes a value greater than the value observed, i.e. it is the probability of rejecting the null hypothesis when it is true.

Let us consider the example of India’s Adjusted Headcount Ratio, reported in Table 8.2, where  and . Now, . Thus, with % confidence, the null hypothesis can be rejected with respect to the alternative   and the corresponding -value is , where  stands for the cumulative standard normal distribution. Similarly, in a one-tail test to the right, the null hypothesis can be rejected against the alternative  with () percent confidence if . The corresponding -value is . Finally, in a one tail test to the left, the null hypothesis can be rejected against the alternative  with () percent confidence, if , where the relevant -value is .[15]

Note that the conclusions based on the confidence intervals and the one-sample tests are identical. If the value at the null hypothesis lies outside of the confidence interval, then the test will also show that null hypothesis is rejected. On the other hand, if the value at the null hypothesis lies inside the confidence interval, then the test cannot reject the null hypothesis.

Formal tests are also required in order to understand whether a change in the estimate over time—or a difference between the estimates of two countries—has been statistically significant. The difference is that this is a two-sample test. We assume that the two estimates whose difference is of interest are estimated from two independent samples.[16] For example, when we are interested in testing the difference in  across two countries, across rural and urban areas, or across population subgroups, it is safe to assume that the samples are drawn independently. A somewhat different situation may arise with a change over time. It is possible that the samples are drawn independently of each other or that the samples are drawn from the same population in order to track changes over time, as, for example, in panel datasets. This section restricts its attention to assessments in which we can assume independent samples.

Suppose there are two countries, Country I and Country II. The population achievement matrices are denoted by  and  respectively, and the population Adjusted Headcount Ratios are denoted by  and , respectively. We seek to test the null hypothesis , which implies that poverty in country I is not significantly different from poverty in country II in any of the three alternatives:  which means that one of the two countries is significantly poorer than the other; or , which means that country I is significantly poorer than country II; or , which means the opposite. For the first alternative, we need to conduct a two-tailed test, and for the other two alternatives, we need to conduct a one-tailed test.

Now, suppose a sample (either simple random or multistage stratified)  of size  is collected from  and a sample  of size  is collected from , where samples in  and  are assumed to have been drawn independently of each other. We denote the estimated Adjusted Headcount Ratios from the samples by  and , respectively. By the law of large numbers and the central limit theorem,  and . The difference of two normal distributions is a normal distribution as well. Thus,



where . Note that, as we have assumed independent samples, the covariance between the two Adjusted Headcount Ratios is zero. Hence, the standard error of , denoted by , may be estimated using equations (8.11) or (8.30) in the Appendix, whichever is applicable, as:



where  is the standard error of  and  is the standard error of . Like the one-sample test discussed above, in the two-tail test, the null hypothesis can be rejected against the alternative  with () percent confidence, if . Given that at the null hypothesis , this implies requiring . Similarly, in order to reject the null hypothesis against , we require  and against , we require  The corresponding -values can be computed as discussed in the one-sample test.

Table 8.3 presents an example of an estimation of MPI (an adaptation of ) in four Indian states: Goa, Punjab, Andhra Pradesh, and Tripura, with their corresponding standard errors, confidence intervals and hypothesis tests.[17] These results are computed from the Demographic and Health Survey of India for the years 2005–2006. In the table we can see that the MPI point estimate for Goa is 0.057, and with 95% confidence, we can say that the MPI estimate of Goa lies somewhere between 0.045 and 0.069. Similarly, we can say with 95% confidence that Punjab’s MPI is not larger than 0.103 and no less than 0.073, although the point estimate of MPI is 0.088. We can also state, after doing the corresponding hypothesis test, that Punjab is significantly poorer than Goa. However, we cannot draw the same kind of conclusion for the comparison between Andhra Pradesh and Tripura, although the difference between the MPI estimates of these two states (0.032) is similar to the difference between Goa and Punjab.

Table 8.3 Comparison of Indian States Using Standard Errors


It is vital to understand that in two sample tests, conclusions about the statistical significance obtained with confidence intervals do not necessarily coincide with conclusions obtained using hypothesis testing. Let us formally examine the situation. Suppose, . If the confidence intervals do not overlap, then the lower bound of  is larger than the upper bound of , i.e.  or . Given that for two independent samples, , if the confidence intervals do not cross, a statistically significant comparison can be made. However, if the confidence intervals overlap, it does not necessarily mean that the comparison is not statistically significant at the same level of significance. It is thus essential to conduct statistical tests on differences when the confidence intervals overlap.

8.3     Robustness Analysis with Statistical Inference

In practice, the robustness analyses discussed in section 8.1 are typically performed with estimates from sample surveys. In at least two cases, it is necessary to combine the robustness analyses with the statistical inference tools just described. This section describes how to do so in practice.

The dominance analysis presented in section 8.1.1 assesses dominance between two CCDFs or two  curves in order to conclude whether a pairwise ordering is robust to the choice of all poverty cutoffs. But it is also crucial to examine if the pairwise dominance of the CCDFs or  curves are statistically significant. For two entities in a pairwise ordering, one should perform one-tailed hypothesis tests of the difference in the two estimates for each possible  value, as described in section 8.2.3. This will determine whether the two countries’ poverty estimates are not significantly different or whether one is significantly poorer than the other regardless of the poverty cutoff.[18] One may also construct confidence interval curves around each CCDF curve (or  curve) and examine whether two corresponding confidence interval curves overlap or not, in order to conclude dominance. More specifically, if the lower confidence interval curve of a unit does not overlap with the upper confidence interval curve of another unit, then one may conclude that statistically significant dominance holds between two entities. However, as explained at the end of section 8.2.3, no conclusion on statistical significance can be made when the confidence intervals overlap. Thus a hypothesis test for dominance should be preferred.[19]

This need to combine methods also applies to the other type of robustness analysis presented in section 8.1.2, in the sense that one can implement this analysis to a ranking of entities and report the proportion of robust pairwise comparisons across the different  values. Moreover, the analysis described in section 8.2.3 (hypothesis testing or comparison of confidence intervals by pairs of entities) can be implemented not only with respect to the poverty cutoff but also with respect to changes in the other parameters, such as weights, deprivation cutoffs or alternative indicators.

As Alkire and Santos observe (2014: 260), the number of robust pairwise comparisons may be expressed in two ways. One may report the proportion of the total possible pairwise comparisons that are robust. A somewhat more precise option is to express it as a proportion of the number of significant pair-wise comparisons in the baseline measure, because a pairwise comparison that was not significant in the baseline cannot, by definition, be a robust pairwise comparison.

To interpret results meaningfully, it can be helpful to observe that the proportion of robust pairwise comparisons of alternative  specifications is influenced by: the number of possible pairwise comparisons, the number of significant pairwise comparisons in the baseline distribution, and the number of alternative parameter specifications. Interpretation of the percentage of robust pairwise comparisons in light of these three factors illuminates the degree to which the poverty estimates and the policy recommendations they generate are valid across alternative plausible design specifications.

Alkire and Santos (2014) perform both types of robustness analysis with the global MPI (2010 estimates) for every possible pair of countries with respect to: (a) a restricted range of  values, namely, 20% to 40%; (b) four alternative sets of plausible weights; and (c) to subgroup-level MPI values.[20] The country rankings seem highly robust to alternative parameters’ specifications.[21]

Chapter 9 further develops the techniques of multidimensional poverty measurement and analysis. Specifically, we present techniques for analysing poverty over time (with and without panel data) and for exploring distributional issues such as inequality among the poor.

Appendix: Methods for Computing Standard Errors

This appendix provides a technical outline of how standard errors may be computed. We first present the analytical approach and then the bootstrap method using the notation in Method I presented in Box 5.7. For the multidimensional and censored headcount ratios, we use the notation in Box 5.4. The  and its partial indices are written as













Note that  is the logical ‘and’ operator. The standard errors of the subgroups’ s and partial indices may be computed in the same way and so we only outline the standard errors of equations (8.5)–(8.8).

Simple Random Sampling with Analytical Approach

Suppose  samples have been collected through simple random sampling from the population. We denote the dataset by  and its th element by  for all  and . We denote the deprivation status score for  by . For statistical inferences, our analysis focuses on the censored deprivation scores. The score, defined at the population level, becomes a random variable while performing statistical inference. We assume that a random sample (of size ) of censored deprivation scores { is a sequence of independently and identically distributed random variables with an expected value  and Var. Then as  approaches infinity, the random variable  converges in distribution to , where . That is



The unbiased sample estimate of  is



and the standard error of the Adjusted Headcount Ratio is



The analytical approach based on the central limit theorem (CLT) also applies to the calculation of the standard errors of , which leads to



where  and . Note that unlike ,  is an average across 0s and 1s, i.e. the mean is a proportion and  is estimated as



and so the unbiased standard error is



With the same logic, the standard error for , can be estimated as



where, .

The formulation of  is analogous to the formulation of , and so the standard error of ) is computed as



where  and  is the number multidimensionally poor in the sample.

Note that if the number of multidimensionally poor is extremely low, the sample size for estimating  may not be large enough. This may affect the precision of  using (8.16). It may then be accurate to treat  as a ratio of  and  for computing . By the Taylor series expansion (see the discussion in Casella and Berger 1990: 240–245),  can be approximated as  and  can be estimated as



where  and  are based on (8.10) and (8.13), respectively, and  can be estimated as



By combining (8.17) and (8.18), the alternative formulation becomes



Stratified Sampling with an Analytical Approach

We next discuss the estimation of standard errors when samples are collected through two-stage stratification.[22] Using information on the population characteristics, the population is partitioned into several strata. The first stage, from each stratum, draws a sample of PSUs with or without replacement. The second stage draws samples either with or without replacement, from each PSU.

We suppose that the population is partitioned into  strata and there are  PSUs in the th strata for all . The population size of the th PSU in the th stratum is  so that . We denote the total number of poor by  and the number of poor in the th PSU in the th strata by . The population  measure and its partial indices are presented in (8.20)–(8.23) with the same notation for the identity function as in (8.5)–(8.8).









Note that  if the th person from the th PSU in the th stratum is deprived in the th dimension and  otherwise; and  and  are the deprivation score and the censored deprivation score of the th person from the th PSU in the th stratum, respectively. Thus, ; and  if  and  otherwise.

Now, suppose a sample of size  is collected through a two-stage stratified sampling. The first stage selects  PSUs from the th stratum for all . The second stage selects  samples from the th PSU in thstratum . So, . Each sample  in the th PSU in the th stratum is assigned a sampling weight , which are summarized by an -dimensional vector . The achievements are summarized by matrix , which is a typical sample dataset.

In order to estimate the measure from the sample, first, the total population and the total number of poor should be estimated from the sample. We denote the estimates of the population  by and the estimate of the poor population  by . Then,





The sample estimates of the population averages in (8.20)–(8.23) are presented in (8.26)–(8.29).









As each sample estimate is a ratio of two estimators, their standard errors are approximated using (8.17) and using equations (1.31) and (1.63) in Deaton (1997). The standard error for  in (8.26) is



where ,  and .

The standard errors of  and  are






where and . Terms  and  are the same as in (8.30).

Finally, we present the standard error for  in (8.27), where the denominator is  instead of  as



where ,   and . Intuitively,  is the estimated average intensity for stratum ,  is the average of sampling weights in stratum  across the poor, and  is the sum of all sampling weights in PSU  of stratum  also across the poor.

As a reasonably smaller sample size may affect the precision of the standard error of  the variance  can be approximated as in (8.17), but using (8.30) and (8.31) as






Hence, combining (8.34) and (8.35), we have



Note that the analytical standard errors and confidence intervals may not serve too well when the sample sizes are small or when the estimates are too close to the natural upper or lower bounds.[23] In these cases, resampling non-parametric methods, such as bootstrap, may be more suitable for computing standard errors and confidence intervals.

The Bootstrap Method

An alternative approach for statistical inference is the ‘bootstrap’, which is a data-based-simulation method for assessing statistical accuracy. Introduced in 1979, it provides an estimate of the sampling distribution of a given statistic , such as the standard error, by resampling from the original sample (cf. Efron 1979; Efron and Tibshirani 1993). It has certain advantages over the analytical approach. First, the inference on summary statistics does not rely on CLT as the analytical approach. In particular, for reasonably small sample size, standard errors/confidence intervals computed through the CLT-based asymptotic approximation may be inaccurate. Second, the bootstrap can automatically take into account the natural bounds of the measure. Confidence intervals using the analytical approach can lie outside natural bounds, which can be prevented when the bootstrap re-sampling distribution of the statistic is directly used.

Third, the computation of standard errors may become complex when the estimator and its standard error have a complicated form or have a no-closed expression. These types of complexities are common both in the context of statistical inference of inequality or poverty measurement and in tests where comparisons of group inequality or poverty (across gender or region) are of particular interest (Biewen 2002). Although, the delta-method can handle these analytical standard errors from stochastic dependencies, but when the number of time periods or groups increases, computing the standard errors analytically can easily become cumbersome (cf. Cowell 1989, Nygard and Sandström 1989). In practice, Monte Carlo evidence suggests that bootstrap methods are preferred for these analyses and shows that the simplest bootstrap procedure achieves the same accuracy as the delta-method (Biewen 2002; Davidson and Flachaire 2007). In developing economics,  bootstrap has been used to draw statistical inferences for poverty and inequality measurement (Mills and Zandvakili 1997; Biewen 2002).

Here we briefly illustrate the use of the bootstrap for computing standard errors. Readers interested in using the bootstrap for confidence interval estimation and hypothesis testing can refer to Efron and Tibshirani (1993), chapters 12 and 16, respectively.

The bootstrap algorithm can be described as a resampling technique, which is conducted  number of times by generating a random artificial sample each time, with replacement from the original sample, which is our dataset . The th resample produces an estimate  for all . Thus, we have a set of  resample estimates of : . If the artificial samples are independent and identically distributed (i.i.d.), the bootstrap standard error estimator of , denoted , is defined as



where stands for the arithmetic mean over the artificial samples. Even if the artificial sample is drawn from a more complex but known sampling framework, the bootstrap standard error can be easily estimated from standard formulas (cf. Efron 1979; Efron and Tibshirani 1993). If the resampling is conducted on an empirical distribution of a given dataset , then it is referred to as a non-parametric bootstrap. In this case, each observation is sampled (with replacement) from the empirical distribution, with probability inversely proportional to the original sample size. However, the resampling can also be selected from a known distribution chosen on an empirical or theoretical basis.  In this case, it is referred to as a parametric bootstrap.


Box 8.1 illustrates the use of the bootstrap for computing standard errors of the  and its partial indices. In this case, the statistic  comprises , , , and . Thus, the estimate  includes , ,  or . To obtain the bootstrap standard errors, we need to pursue the following steps.

  1. Draw bootstrap resamples from the empirical distribution function.
  2. Compute the set of  relevant bootstrap estimates of , ,  or  from each bootstrap sample.
  3.  Estimate the standard errors by the sampling standard deviation of the  replications: , , , or  (cf. Efron and Tibshirani 1993: 47).

We have already discussed that the bootstrap approach has certain advantages—especially that it does not rely on the central limit theorem. Although the non-parametric bootstrap approach does not depend on any parametric assumptions, it does involve certain choices. The first is the number of replications. Indeed a larger number of replications increas the precision of the estimates, but is costly in terms of time. There are different approaches for selecting the appropriate number of replications (see Poi 2004). The second involves the choice of the bootstrap sample size being selected from the original sample. The third involves the choice of the resampling method. The bootstrap sample size in Efron’s traditional bootstrap is equal to the number of observations in the original sample, but the use of smaller sample sizes has also been studied (for further theoretical discussions; see Swanepoel (1986) and Chung and Lee (2001).


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Alkire, S. and Foster, J. (2011a). ‘Counting and Multidimensional Poverty Measurement’. Journal of Public Economics, 95(7–8): 476–487.

Alkire, S. and Santos, M. E. (2010). ‘Acute Multidimensional Poverty: A New Index for Developing Countries’. OPHI Working Paper 38, Oxford University; also published as Human Development Research Paper 2010/11.

Alkire, S. and Santos, M. E. (2014). ‘Measuring Acute Poverty in the Developing World: Robustness and Scope of the Multidimensional Poverty Index’. World Development, 59: 251–274.

Alkire, S. and Seth, S. (2013b). ‘Multidimensional Poverty Reduction in India between 1999 and 2006: Where and How?’ OPHI Working Paper 60, Oxford University.

Anderson, G. (1996). ‘Nonparametric Tests of Stochastic Dominance in Income Distributions’. Econometrica, 64(5): 1183–1193.

Barrett, G. and Donald, S. G. (2003). ‘Consistent Tests for Stochastic Dominance’. Econometrica, 71(1): 71–103.

Batana, Y. M. (2013). ‘Multidimensional Measurement of Poverty Among Women in Sub-Saharan Africa’. Social Indicators Research, 112(2): 337–362.

Biewen, M. (2002). ‘Bootstrap Inference for Inequality, Mobility and Poverty Measurement’. Journal of Econometrics, 108(2): 317–342.

Boland, P. J. and Proschan, F. (1988). ‘Multivariate Arrangement Increasing Functions with Applications in Probability and Statistics’. Journal of Multivariate Analysis, 25(2): 286–298.

Cherchye  et al. (2007): Cherchye L., Moesen, W., Rogge, N., Puyenbroeck, T.V., Saisana, M., Saltelli, A., Liska, R., and Tarantola, S. (2007). ‘Creating Composite Indicators with DEA and Robustness Analysis: The Case of the Technology Achievement Index’. Journal of the Operational Research Society, 59(2): 239–251.

Cherchye et al. (2008): Cherchye L., Ooghe, E., and Puyenbroeck, T. V. (2008). ‘Robust Human Development Rankings’. Journal of Economic Inequality, 6(4): 287–321.

Chung, K. H. and Lee, S. (2001). ‘Optimal Bootstrap Sample Size in Construction of Percentile Confidence Bounds’. Scandinavian Journal of Statistics, 28(1): 225–239.

Cowell, F. (1989). ‘Sampling Variance and Decomposable Inequality Measures’. Journal of Econometrics 42(1): 27–41.

Davidson, R. and Duclos, J.-Y. (2000). ‘Statistical Inference for Stochastic Dominance and for the Measurement of Poverty and Inequality’. Econometrica, 68: 1435–1464.

Davidson, R. and Duclos, J.-Y. (2012). ‘Testing for Restricted Stochastic Dominance’. Econometric Reviews, 32(1): 84–125.

Davidson, R. and Flachaire, E. (2007). ‘Asymptotic and Bootstrap Inference for Inequality and Poverty Measures’. Journal of Econometrics, 141(1): 141–166.

Deaton, A. (1997). The Analysis of Household Surveys. A Microeconometric Approach to Development Policy. John Hopkins University Press.

Duclos, J.-Y. and Araar, A. (2006). Poverty and Equity: Measurement, Policy and Estimation with DAD. Springer.

Efron, B. (1979). ‘Bootstrap Methods: Another Look at the Jackknife’. The Annals of Statistics, 7(1): 1–26.

Efron, B. and Tibshirani, R. (1993). An Introduction to the Bootstrap. Chapman & Hall/CRC.

Foster J., McGillivray, M., and Seth, S. (2009). ‘Rank Robustness of Composite Indices’. OPHI Working Paper 26, Oxford University.

Foster, J. E., McGillivray, M., and Seth, S. (2013). ‘Composite Indices: Rank Robustness, Statistical Association and Redundancy’. Econometric Reviews, 32(1): 35–56.

Høyland et al. (2012): Høyland, B., Moene, K., and Willumsen, F. (2012). ‘The Tyranny of International Index Rankings’. Journal of Development Economics, 97(1): 1–14.

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Kendall, M. G. and Gibbons, J. D. (1990). Rank Correlation Method. E. Arnold.

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Mills, A.M. and Zandvakili, S. (1997). ‘Statistical Inference via Bootstrapping for Measures of Inequality’. Journal of Applied Econometrics, 12(2): 133–150.

Nardo et al. (2005): M., Saisana, M., Saltelli, A., Tarantola, S., Hoffman, A., and Giovannini, E. (2005). ‘Handbook on Constructing Composite Indicators: Methodology and User’s Guide’. OECD Statistics Working Papers 2005/3. OECD Publishing.

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[1] This chapter is confined to assessing the robustness of rank ordering across groups. Naturally it is essential also to assess the sensitivity of key values (such as the values of  and dimensional contributions) to parameter changes, in situations in which policies use these cardinal values.

[2] There is a well-developed literature on robustness and sensitivity analyses for composite indices rankings with respect to relative weights, normalization methods, aggregation methods, and measurement errors. See Nardo et al. (2005), Saisana et al. (2005), Cherchye et al. (2007), Cherchye et al. (2008), Foster, McGillivray and Seth (2009, 2013), Permanyer (2011, 2012), Wolff et al. (2011), and Høyland et al. (2012). These techniques may require adaptation to apply to normative, counting-based measures using ordinal data.

[3] More elaborative dominance analysis can be conducted with respect to the deprivation cutoffs and weights. For multivariate stochastic dominance analysis using ordinal variables, see Yalonetzky (2014).

[4] In empirical applications, some statistical tests cannot discern between weak and strong dominance and thus assume  first order stochastically dominates distribution , if  for all . See, for example, Davidson and Duclos (2012: 88–89).

[5] This is also variously known as survival function or reliability function in other branches of studies.

[6] Technically,  for poverty cutoff  can be expressed as . In our example, Area I is computed as  and Area II as .

[7] In this book, we only focus on bivariate rank correlation coefficients, but there are various methods to measure multivariate rank concordance that we do not cover. For such examples, see Boland and Proschan (1988), Joe (1990), and Kendall and Gibbons (1990). For an application of some of the multivariate concordance methods to examine multivariate concordance of MPI rankings, see Alkire et al. (2010).

[8] The computations of the Spearman and Kendall coefficients in the table have been adjusted for ties. For the exact formulation of tie-adjusted coefficients, see Kendall and Gibbons (1990).

[9] Various sampling methods, such as simple random sampling, systematic sampling, stratified sampling, and proportional sampling, are used to conduct a sampling survey.

[10] Yalonetzky (2010).

[11] If the particular condition under study does not justify the assumptions made here, then these assumptions need to be relaxed and the standard error formulations are adjusted accordingly.

[12] The significance level is also referred to as the Type I error, which is the probability of rejecting the null hypothesis when it is true. See section 8.2.3 for the notion of null hypothesis. By statistical convention, the significance level is denoted with . However, to avoid confusion with the use of this symbol for other purposes in this book, we denote it .

[13] Whenever the population standard deviation is unknown or when the sample size is small, one needs to use the Student-t distribution to compute the critical values rather than the standard normal distribution.

[14] We present the tests for country-level estimates but they are equally applicable to other population subgroups. Also, we only present the tests in terms of the  measure, but again they are also applicable to ,  and  for all  and so we have chosen not to repeat the results.

[15] See Bennett and Mitra (2013) for an exposition of hypothesis testing of  and other AF partial sub-indices using a minimum p-value approach.

[16] See chapters 14 and 16 of Duclos and Araar (2006) for further discussion on non-independent samples for panel data analysis.

[17] Alkire and Seth (2013b) use an MPI harmonized for strict comparability of indicator definitions across time.

[18] For formal tests on stochastic dominance in unidimensional poverty and welfare analysis, see Anderson (1996), Davidson and Duclos (2000), and Barrett and Donald (2003).

[19] Other new ways of testing robustness may be developed in the near future.

[20] They compute the MPI for four population subgroups: children 0–14 years of age, women 15–49 years of age, women aged 50 years and older, and men 15 years and older, and test the rankings of subgroup MPIs across countries.

[21] Further methodological work is needed to propose overall robustness standards for measures that will be used for policy.

[22] Appendix D of Seth (2013) gives an example of standard error estimation for one-stage sample stratification in the multidimensional welfare framework; for consumption/expenditure see Deaton (1997).

[23] When the estimate is too close to the natural upper and lower bounds (0 and 1), the confidence intervals using analytical standard error may fall outside these bounds. Different methods for adjustments are available. For a discussion of such methods, see Newcombe (1998).

7 Data and Analysis

Chapter 6 transitioned from considerations about selecting a measurement methodology (Chapters 3–5), to issues met in implementing real-world measures that undergird and reinforce policies to fight poverty. It mentioned in passing the desiderata criteria that indicators be ‘technically solid’—a criterion that entails rigorous consideration of properties and also of empirical techniques. To take this forward, we now switch focus to emphasize the practice of empirical poverty measurement. In particular, this chapter introduces empirical issues that are distinctive to counting-based multidimensional poverty methodologies. Novel issues include the requirement that indicators accurately reflect deprivations at the individual level—not just on average—and that all indicators be transformed to reflect deprivations in the chosen unit of identification (person, household).

This chapter is not exhaustive. It presumes readers have a sound understanding of household survey data and micro-economic analysis, and also of various assessments of indicator validity. It supplements a presumed solid foundation with new elements that pertain to multidimensional poverty measurement design and analysis in particular. [1] A more extensive and detailed version is available at

Section 7.1 introduces very briefly the different types of data sources available, namely, censuses, administrative records, and household surveys. Section 7.2 discusses issues to be considered when selecting the indicators to include in a multidimensional poverty measure. Finally, section 7.3 presents some basic descriptive analytical tools that can prove helpful in exploring the relationships between different indicators and informing the process of the measure design. Box 7.1 discusses the different fronts on which data collection should be improved in the near future in order to permit the design of better poverty measures.

7.1 Data for Multidimensional Poverty Measurement

As stated in Chapter 1, the initial step in poverty measurement, even prior to identification, is to select the space of analysis (resources, capabilities, utility) and the purpose of the measure to be constructed. The choice of space, as well as the feasible options for measurement design, will necessarily be shaped by data availability. We briefly review the main types of data used for multidimensional measurement and considerations of when to use each.

Multidimensional poverty is measured using micro data. By micro data we mean the unit-level data containing responses that each unit of analysis (such as person or household) provided. This contrasts with macro data, or aggregate indicators or marginal measures such as the mortality rate, literacy rate, mean household income, or the enrolment ratio, which summarize the achievements of a society. The three most common micro data sources are censuses, household surveys, and administrative records—also called register data. New relevant data sources such as mobile telephony and satellite imagery are rising sharply, and will deeply enrich future multidimensional poverty analyses.

A census is an enumeration of all households in a well-defined territory at a given point in time (Mather 2007). National censuses are typically conducted every five to ten years, and contain information on a strictly limited number of variables: demographic variables, such as nationality, age, gender, marital status, place of birth, location, ethnicity or religion, and language; social variables, such as literacy, educational attainment, and housing conditions; and economic variables, such as activity condition and employment (UNSD 2008; cf. United Nations 2008: 112–113, Table 1). Special censuses may be implemented for targeting and monitoring certain programmes, again using a few simple variables.

Censuses provide information with negligible sampling error (the whole population is considered) at highly disaggregated levels (municipalities–neighbourhoods). Census variables are used in the construction of multidimensional poverty maps using (and were previously used in the UBN tradition). And census data is essential for multidimensional measures that target individuals or households. The disadvantages of censuses are that (a) they have low frequency, (b) they offer information on a small set of indicators, and (c) micro data may not be available to researchers.

Administrative data refers to information typically collected by a government department or agency primarily for administrative purposes (birth registration, customs, administration of a social benefit, etc.). One prominent example is population registers constructed through a civil registration system (United Nations 2001). Population registers consist of an inventory of each member of the resident population of a country augmented continuously by information on vital events (births, deaths, adoptions, marriages, divorces, among others) (UNSD 2001). Other examples are tax, education, police, and health records.

Some advantages of using administrative datasets are that (a) they typically cover virtually 100% of the population of interest in a continuous form, (b) there are no added data collection costs, (c) there is data for individuals who might not normally respond to surveys, (d) when linked to other unit-level data sources, administrative data can produce a powerful resource for multidimensional poverty. However, there are also some disadvantages: (a) the information collected in administrative records is limited and may not match the research purpose, (b) any changes in data collection procedures or definitions may prevent comparability over time, (c) serious data quality issues may compromise accuracy, (d) metadata is usually not available,[2] (e) access to administrative (micro) data varies by country, and (f) linking data sources is rarely straightforward.

Household surveys are the most commonly used data source to study poverty. These collect on a sample or subset of the population. The sample may be representative of the population of interest, which can be the total population of a country or of a particular region or children under 15, or some other group.

The respondents for the survey are selected from what is called a ‘frame’ or list, which is usually obtained from the most recent census and is typically a list of households. Different sampling methods such as simple random sampling and complex multi-stage sampling are used in order for the sample to be representative of the population. Deaton (1997) offers a valuable introduction to each method.[3]

Household surveys were collected as early as the eigthteenth century in England (Stigler 1954). After World War II household surveys expanded internationally with India being a pioneer. Since the 1970s, several international programmes have promoted and supported the collection of household survey data in developing countries. These include the World Fertility Surveys, which were introduced in the 1970s and became the Demographic and Health Surveys (DHS) in 1984, and the World Bank Living Standards Measurement (LSMS) survey programme, ongoing since 1980 (Grosh and Glewwe 2000: 6). In 1995, the United Nations Children’s Fund (UNICEF) began its Multiple Indicators Cluster Surveys (MICS), and in 2000 the World Bank launched the Core Welfare Indicator Questionnaire (CWIQ). The World Bank also intensively supported the development and widespread fielding of household budget surveys and income and expenditure surveys that are used for income and consumption poverty measures and may contain other topics. Multidimensional poverty measures typically rely on multi-topic household surveys, which collect information with one survey instrument using a sample frame that has been defined to capture a diverse set of topics.[4]

Box 7.1 The Mile Ahead in Data Collection

As Chapter 1 mentioned, enormous progress has been made in data collection worldwide since the 1940s. International institutions, universities, national institutes of statistics, and census bureaus have played a crucial role in this progress. Now virtually every country in the world has a periodic census, administrative data, and at least one multi-topic household survey being conducted periodically, usually more. However, there is still a long way to go. Data remain limited in terms of frequency, population coverage, dimensional coverage, representation of vulnerable subgroups, international comparability, interconnectedness, and the unit of analysis.

In terms of frequency, poverty data continues to lag behind most other economic information. The lack of frequent data makes it impossible to inform policies responding to the impact of certain events such as financial crises and natural disasters on the poor.

In terms of population coverage, household surveys typically exclude certain groups such as nomadic people, recent or illegal migrants or refugees, and the homeless—as well as institutionalized groups such as prisoners, those hospitalized or in nursing homes, the military, and members of religious orders. They may also overlook the elderly within households. Some excluded groups may be particularly marginalized thus should be considered in poverty measures.

A different though related problem consists of the falling response rates in household surveys over subsequent rounds, even when they are not panel surveys. Such a problem is being observed in some – usually developed – countries, such as the UK, particularly with respect to indicators such as wealth or assets. While such a problem can be partially overcome by reweighting the sample, this is not ideal, and creative ways to deal with falling response rates will need to be devised.

Dimensional coverage is limited, often in ways that would be relatively easy to address. Common missing dimensions that may be relevant for poverty studies include health functionings, safety from violence, the quality of work, empowerment, social connectivity, and potentially time use. Limited dimensional coverage hampers studies of interconnectedness, in the sense that it does not allow researchers to analyse the joint distribution of violence and other dimensions of poverty, and identify high-impact policy sequences and causal links across these.

Many surveys seek to define household-level achievements, but do not elucidate intra-household inequalities, gender inequalities, and age inequalities, nor do they cover overlooked topics such as the care economy and household duties.

The nature of the indicators that are collected is another area of potential improvement. Paraphrasing the Report by the Commission on the Measurement of Economic Performance and Social Progress (Stiglitz, Sen, and Fitoussi 2009), the time is ripe to move from the space of resources to the space of functionings. Functionings, as argued in Sen’s capability approach, seem to be central to poverty reduction and are of intrinsic importance. Yet even such a central dimension as health functioning is absent from most good surveys that collect income or consumption data.

Last, but not least, given that the aim remains to reduce poverty not to measure it, improved channels of complementarity are required between censuses, administrative records, household surveys, and other information such as from satellites and cell phones, in order to advance towards an integrated programme of data collection and compilation. Merging GIS data on environmental conditions with household surveys, for example, greatly strengthens poverty measurement and the monitoring and impact evaluation of sustainable poverty reduction programmes. In sum, despite incredible progress in data collection, there is still a ways to go so that poverty reduction can be informed by a sufficient depth and frequency of data as are available for societies’ other priorities.

7.2 Issues in Indicator Design

All data sources, however rich, impose constraints on a poverty measurement and analysis exercise. These are navigated via a number of important decisions that are made when designing a multidimensional poverty measure and choosing its component indicators. The following subsections address the ‘new’ empirical considerations that are relevant both in designing new surveys for multidimensional poverty and in implementing measures using existing data.

7.2.1 Unit-level Indicator Accuracy

A first essential and distinctive criterion for counting-based multidimensional poverty measures is that each indicator must be a relatively accurate reflection of the achievements enjoyed by each unit of identification across the relevant period and not simply the achievements enjoyed ‘on average’ by some group, as we will clarify in this section.

The unit of identification refers to the entity who is identified as poor or non-poor, (determination of the poverty status)—usually the individual or the household. In the literature of poverty measurement, this is typically referred to as the unit of analysis. However, the term unit of identification is more accurate because the identification may take place at a more aggregated level, but the analysis can still be performed at a more disaggregated level. For example, if the unit of identification is the household, analyses can still generate the percentage of the population who are poor.

Household surveys are usually designed to create indicators that are representative of the achievements and/or distributions of some population subgroups, such as states or ethnic groups, at the time of the survey. For example, there are indicators typically collected which have short reference periods in order to increase indicator precision, and are judged to be accurate ‘on average’ (Deaton and Kozel 2005). Prominent examples include consumption in the last seven days, illness in the last two weeks, and time use in the past twenty-four hours. The assumption is that people with unusually high consumption/health problems in the reference period will balance others with unusually low values, with the average (and in some cases, the distribution) being assumed to be accurate across the representative unit at the level of the single indicator. However achievements may not be accurate at the individual level. For example, if the last seven days’ consumption included a family wedding, if the respondent had a rare and brief bout of the flu in the past fortnight, or if the last twenty-four hours was a major public holiday, then a person’s response will not provide a good indication of his or her average consumption, morbidity, or time use over the past year. In contrast, indicators used for targeting people or households are always required to be accurate at the individual level.

Multidimensional measures require the joint distribution of deprivations to be accurate on average. This is straightforward to achieve if, as in the case of targeting, each indicator accurately identifies each person’s deprivations, and so the deprivation scores, which reflect joint deprivations and will determine whether or not the person is identified as poor, are similarly accurate at the individual level. Selected indicators ideally balance indicator precision and unit-level accuracy in order to justify the assumption that responses reflect individual or household achievement or deprivation status during the relevant period. A related requirement, which occurs when multidimensional poverty measures (and, incidentally, other measures) are used to track trends in poverty over time, is that the indicators should ideally reflect individual achievement levels across the relevant period so that the comparisons are not unduly distorted by seasonal effects or short-term shocks.

Many indicators are relatively unproblematic in this regard. Indicators such as child vaccination, completed years of schooling, child mortality, housing materials, chronic disability, or long-term unemployment, for example, are likely to reflect individual or household achievements accurately. Moreover, a non-union identification strategy in the poverty cutoff can partially, as we discussed in Chapter 6, ‘clean’ data of errors.

That being said, in the case of other indicators, particularly those with short reference periods, the ideal may be more difficult to obtain. In practice, when collecting primary data, the unit-level accuracy issue should be considered when creating the questionnaire. When using secondary data, unit-level accuracy may guide the choice between different indicators, when such an option exists.

7.2.2 Indicator Transformation to Match Unit of Identification

Alkire-Foster multidimensional poverty measures reflect the joint distribution of deprivations for a given unit of identification. While the advantages of this and related approaches have been much discussed in Chapter 3, its empirical implementation actually requires novel techniques. In particular, relevant data may be available for individuals, for the household, and, perhaps, also at the community level. But it is necessary to transform all indicators such that they reflect deprivations of just one (the chosen) unit of identification. Consideration of the unit-level profile of joint deprivations leads to its identification as poor or non-poor (based on the poverty cutoff and deprivation score) and hence to the inclusion or censoring of its deprivations in the resultant multidimensional poverty measure.

To begin with an elementary case, consider a child poverty measure covering children aged six to twelve. Suppose that there are data on children’s education, health, and nutrition; household-level data on income and housing; and village-level data on the quality of primary school facilities. In the achievement matrix, each child will naturally have their own achievement levels in health, education, and nutrition. All children from the same household may take the household’s achievements in income and housing. And all school-going children in the same village may have the same achievements in quality of school facilities. The proposed transformations of household- and community-level data imply the assumption that the household- and community-level variables affect each child in the same way—which may require justification. The deprivation cutoffs for each indicator will be applied to this matrix, and each child will be identified as deprived or non-deprived in each indicator and, subsequently, as poor or non-poor, based on their weighted deprivation score. Thus each child’s deprivation profile will draw on data from individual-, household-, and community-level sources. In this way, household- and community-level indicators can be appropriately ‘transformed’ so they inform the deprivation profile of each child. But many cases are not so simple. The next section will set out how to proceed in those cases.

7.2.3 The Unit of Identification and Applicable Population

We begin with an important definition. The applicable population of a certain achievement refers to the group of people for which such an achievement is relevant; namely, it can be measured and has been effectively measured, in this case, to inform poverty measurement. Note that both conditions need to hold for the population group to be applicable.[5] In some cases, the achievement is conceptually applicable to the whole population (with appropriate adjustments in the levels considered adequate by age group or gender), but, despite this, data is typically not collected at the individual level. An example of this is the case of anthropometric indicators (nutritional indicators). While there are anthropometric indicators for all age groups and genders, these are often collected only for children under 5 years of age and for women of reproductive age. Thus for those anthropometric indicators, the remaining population groups are non-applicable because of data constraints. In other cases, the achievement is conceptually inapplicable to certain groups of the population, as in the case of income earned by infants. In either of the two cases, the existence of non-applicable populations poses a problem to be resolved when constructing a poverty measure, if that measure is to reflect their poverty also.

The choice of the unit of identification and the treatment of non-applicable populations are often constrained by data availability. In many cases, the ideal unit of identification would be the individual; another commonly used unit is the household. In this case, the household is judged to be poor or not, and all its members are then identified as poor. ‘The standard apparatus of welfare economics and welfare measurement concerns the wellbeing of individuals. Nevertheless, a good deal of data have to be gathered from households…’ (Deaton 1997: 23). Both units have advantages, as discussed in Chapter 6.

If the unit of identification of the poor is the individual, then all the considered achievements need to be available at the individual level, and thus all the achievements need to be applicable to the whole population for which the poverty measure is defined. If the unit of identification is the household, then one data point on common achievements like housing and heating may be applied to all household members. But not all the considered achievements are equally applicable to the whole population—income being a case in point (see below, this section). In these cases, certain (explicit) assumptions need to be made regarding the sharing and impact of the achievements of certain household members with respect to the others. Other achievements are individual—like health status or educational level—although they may affect household members also.

Strictly speaking, few household-level indicators are applicable to each household member. Most vary by age and some by gender. Housing conditions are an example of indicators that satisfy the ‘universality’ of applicable population. Any person, regardless of age or gender needs a clean source of drinking water; adequate flooring, walls, and roof; clean cooking fuel; and adequate sanitation, for example. Such dwelling conditions are jointly consumed by the household. Household surveys collect information on the dwelling conditions, and for poverty analysis it is then assumed that all members equally share them—although this may or may not be accurate.

Another universally relevant and applicable indicator for all people, now at the individual level, is nutritional status. In this case, one will need to combine at least two indicators: one for the nutritional status of children under 5 years of age (which can be weight-for-age, weight-for-height, or height-for-weight) and another for the nutritional status of adults, typically the Body Mass Index (BMI).[6]

Food consumption is another universal indicator. Although nutritional requirements vary significantly by age and gender, it may be possible to define the relevant consumption range for each type of individual.[7] However, data availability usually poses a limitation: ‘Household surveys nearly always collect data on household consumption (or purchases), not on individual consumption, and so cannot give us direct information about who gets what’ (Deaton 1997: 205). Thus, even though conceptually food is a private consumption good, applicable to each member of the population, in practice, it is often assumed that total consumption is distributed within the household in proportion to the nutritional requirements of each member. This assumption may or may not be accurate.

Income is not a universal indicator, as it is inapplicable to people who lack income from non-labour sources like financial assets and do not earn an income (for example, babies and children, housewives, non-working students, and some elderly members) or are unemployed. Standard income poverty measures aggregate all forms of household members’ income (labour or capital) and divide by the total household size (including the members who do not earn an income) to obtain the household per capita income. Equivalence scales may be applied to obtain the household adult equivalent income. The per capita or adult equivalent income is compared to the poverty line, and the household is identified as poor or not and thus all its members. Here, as in the case of consumption, assumptions regarding equal or proportional sharing have been made.

Some other relevant indicators for poverty measurement which are inapplicable for certain groups include child vaccination (adults and children older than the relevant age group do not qualify), child mortality (people who have not had children do not qualify), and employment status. Even education as measured, for example, by years of schooling is not universally applicable, as children below the official mandatory age for starting school do not qualify.

Given that some achievements relevant for poverty measurement are either conceptually or empirically applicable only for certain population groups, the selection and definitions of indicators may take one of the following three options, depending on the purpose of the measure. They may also work as complement measures. The options are :[8]

(a) To restrict consideration to universally applicable achievements

(b) To construct group-specific poverty measures

(c) To combine achievements that are not universally applicable and test assumptions regarding intra-household distribution and/or impact.

(a) Universal measures

One option is to include ‘universal’ achievements, that is, achievements which are applicable to the whole population. This approach narrows the set of possible indicators, although it will include housing, consumption, nutrition, and access to services, if data permit.

(b) Group-specific measures

A second option is to construct different poverty measures for each relevant group, such as child poverty measures, or measures of female or elder poverty. This is an attractive option and suited to group-based policies such as for children or women or elders.

However, three issues need to be considered. First, discriminating by groups may lessen but not eliminate applicability issues. For example, the relevant nutritional indicators change at the age of five, whereas schooling becomes mandatory typically at the age of six or seven, and vaccination ages vary. Thus no group is perfectly homogeneous and indicators may need to be adjusted within population subgroups. Second, it may be critically important to have detailed poverty analyses by population subgroups and these may inform group-based policies profoundly. Yet ‘general’ measures may also be required to track national poverty or to target households. Finally, using unlinked poverty measures for different population subgroups may miss the overlaps of disadvantaged groups and fail to fully exploit possible synergies in policy design.

(c) Combined measures

The third option is to use achievements drawn from a subset of household members (or other unit of identification), and make explicit assumptions about the distribution of such achievements and potential positive or negative intra-household externalities.

This has been the route taken when constructing the global Multidimensional Poverty Index (MPI) (Alkire and Santos 2010, 2014). Assume for example that one wants to include nutrition but information was only collected for children under five and women of reproductive age. Given these data constraints, a household nutrition deprivation indicator might be defined as ‘having at least one child or woman undernourished’. Hence, any household member is considered deprived in nutrition if—despite being well nourished herself—any child or woman is undernourished in her household. Similarly, if one wants to include an indicator of child mortality in the poverty measure, one way to proceed is to consider all household members as deprived if even one child in the household has died. Analogously, one can consider all household members as deprived if there is at least a child of school age who is not attending school. In all of the three examples above there is an obvious assumption of a negative intra-household externality produced by the presence of an undernourished person, the experience of a child’s death, or a child being out of school. Indicators based on the assumption of a positive intra-household externality are also possible. For example, in the global MPI all household members are considered non-deprived if at least one person has five years of schooling, assuming that there is interaction and mutual sharing of cognitive skills within the household as suggested by Basu and Foster (1998).

Two practical situations must be addressed in order to compute combined measures. First, there are households where not even one person qualifies for the achievement under consideration. For example, a household may not have any children in the relevant age bracket. What indicator is used to define deprivation in these households? There is no perfect procedure. Dropping these households from the sample would bias the estimates, as the inapplicability of a certain achievement to a particular population subgroup is not a random issue and households without that population subgroup would be systematically excluded.[9] Dropping any indicator for which no applicable person is present in a household and reweighting the remaining indicators would violate dimensional breakdown and compromise comparability across people. A viable option is to consider all households lacking the relevant data as non-deprived (or deprived) in that particular indicator, then scrutinize this assumption. For example, a household with no children of school age cannot be deprived in child school attendance, so they could be considered non-deprived in this indicator.

A second situation occurs when the survey has not collected information from all applicable members. The example of nutrition above illustrated this case. Nutrition is applicable to all household members, yet some surveys only collect information on children and women in reproductive age, for example. So households that do not have children or women in reproductive age lack data altogether As above, all such households may be considered as non-deprived (or deprived) in that indicator. Clearly, considering them as non-deprived is a heroic assumption, as we simply do not know because information is not available. But it could be seen as a ‘conservative’ approach (in the same spirit of the ‘presumption of innocence’ principle in legal arenas), and will lead to a ‘lower bound’ poverty estimate. That is, it will offer the minimum possible estimate of the proportion of people in households with an undernourished member, which could be improved later.

Naturally the assumptions used in measurement design should be examined empirically as well as debated normatively. Special studies of omitted populations (such as the elderly), including qualitative studies, should be considered to enrich measurement design and analysis, and results should be compared with group-based results to cross-check conclusions.

7.2.4 Assessing Combined Measures

Most multidimensional poverty measures are ‘combined measures’, hence some assumptions are made to permit analysis. This section describes how to assess key assumptions. Two particular kinds of analysis may be especially relevant. The first is related to the household composition effect; the second is related to the prevalence of the indicators.

First, by including indicators referring to achievements that are specific to certain groups (such as school attendance), the composition of the household will affect the probability that the household will be identified as poor. Taken to the extreme, if all the indicators in the poverty measure refer to a deprivation that can only occur among children, then clearly households without children would never be identified as poor. Obviously, a balance regarding the relevant populations for the considered achievements is required. The potential effect of household composition need not prevent the inclusion of an indicator if its importance is normatively clear. For example, it may be the case that there is a national concern about child nutrition. Inclusion of such indicators can be made provided (a) not all of the indicators in the poverty measure refer only to a particular specific group (if so, then a group-specific poverty measure is a better alternative); (b) the specific group for which the achievement is relevant is big enough so that a known and significant proportion of households have at least one member for whom the achievement is relevant; and (c) an empirical assessment of the impact of household composition is performed to test the sensitivity of the measure to its specification.[10]

For example, Alkire and Santos (2014) present two assessments of the influence of household composition on the probability of being poor. The first are hypothesis tests of differences in means. In each country they test whether MPI-poor households have a significantly different average size, average number of children under 5, number of females, number of members 50 years and older, prevalence of female-headed households, and proportion of school-aged children, compared to non-poor households. The second analysis decomposes each country’s MPI by age and gender and compares the rankings, correlations, and the proportion of robust pairwise country comparisons across subgroup MPIs.

Regarding requirement (b), some deprivations are intrinsically important yet are either rare events or pertain to population subgroups that are very small, and thus the presence of a household member for whom the deprivation is relevant is quite unlikely. Under such circumstances, it is best to keep this indicator separate from the multidimensional poverty measure. Examples of such indicators vary by context but might include prevalence of a rare condition that affects pregnant women.

7.2.5 Handling Missing Values

The construction of household variables (for example), from data pertaining to a subset of household members described above is completely separate from the need to address missing values, to which we now turn.

Missing values are particular cases for which a variable that is collected by the survey is not available. For example, if there is a woman of reproductive age for whom the information on BMI should have been collected, given the survey design, but for whom this information is incidentally not available, that is a case of a missing value. Missing values require attention in all poverty measures; multidimensional poverty measures have the advantage of using fewer variables than many monetary measures, but the treatment of missing values also differs slightly.

There are essentially two ways of dealing with missing values. One is to drop that observation from the entire sample. That is, if the unit of identification is the household, households with a missing value in any indicator for a multidimensional measure are dropped from the sample.

The other option is to create a rule that may assign a value for the missing data, particularly in a combined measure in which, for example, data are missing only for some individuals for whom the indicator is applicable. For example, in the case of the global MPI, if at least one member has five or more years of education (although other members have missing values), the household was classified as non-deprived. If there was information on at least two-thirds of household members, each having less than five years of education, the household was classified as deprived; otherwise it was dropped from the sample (Alkire and Santos 2014).

If the observations with missing values systematically differ from those with observed values, the reduction in the sample leads to biased estimates. To assess whether the sample reduction creates biased estimates, the group with missing values is compared to the rest, using the indicators for which values are present for both groups. If the two groups are not (statistically) significantly different, then one may proceed with an estimation using the reduced sample. If they are (statistically) significantly different, then one may still use the reduced sample but should explicitly signal whether the poverty estimate is likely to be a ‘lower’ or an ‘upper’ bound, based on the results of the bias analysis (Alkire and Santos 2014).[11]

One might consider whether to use imputation to assign the observation with a missing value an estimated value of the indicator under consideration. This is commonly done in income poverty measurement, but further research is required before it is applied multidimensionally. Imputation techniques entail using the observations for which there is information to estimate a model with the achievement under consideration as the dependent variable against a set of explanatory variables. The estimated parameters are then used to predict the achievement for the cases with missing values, given their values in the explanatory variables. However, imputation techniques are not problem-free. First, the estimated model needs to be accurate. Second, in the case of multidimensional poverty measurement, the issue of missing values is multiplied because what is of interest is each person’s joint distribution of deprivations. This poses a significant challenge for imputation. One could specify a different model for each indicator. However, when that option is taken, it is likely to incur endogeneity problems. Moreover, this option is blind to varying profiles of joint deprivations. The more accurate route to take would be to specify a model that could predict a vector of deprivations. This imputation would also have to be performed such that it was accurate for each unit of identification, not merely on average. But this has not been done yet. Third, and connected to the previous section, it is also worth noting that imputation techniques cannot solve the problem of non-applicable populations as there are no observations that can be used to estimate a model. For example, one cannot impute the BMI of elderly people using a survey that collects BMI information only for women of reproductive age. As the field of multidimensional poverty measurement progresses, appropriate imputation techniques may be developed for this context.

7.3 Relationships among Indicators

Before implementing any measure empirically it is helpful to understand the variables that may be entered into the measure: by looking at univariate and bivariate statistics such as measures of central tendency, dispersion and association. In the presence of multiple dimensions it is helpful to view their joint distribution, to scrutinize the associations across dimensions and explore similarities or redundancies that may exist.[12] Such analysis may lead one to drop or re-weight an indicator, to combine some set of indicators into a subindex, or to adjust the categorization of indicators into dimensions. It can also inform the selection of indicators and their robustness checks, the setting of deprivation values, and the interpretation of results.

Statistical approaches are relevant for multidimensional poverty measures, but Chapter 6 argued, value judgements also constitute a fundamental prior element. Thus, information on relationships between indicators is used to improve rather than determine measurement design. For example, if indicators are very highly associated in a particular dataset, that is not sufficient grounds to mechanically drop either indicator; both may be retained for other reasons—for example if the sequence of their reduction over time differs, or if both are important in policy terms. So the normative decision may be to retain both indicators, with or without adjustments to their weights, but the analysis of redundancy will have clarified their justification and treatment.

The techniques commonly used to assess relationships between indicators include many of those already presented in section 3.4—that is, principle component analysis, multiple correspondence analysis, factor analysis, cluster analysis, and confirmatory structural equation models, as well as cross-tabulations and correlations. This section is limited to explaining the limitations of correlation analysis between deprivations and introducing a distinctive indicator of redundancy. Both of these draw on contingency tables presented in section 2.2.3. It is further limited in that we restrict information to the dichotomized deprivation matrix, using only uncensored or censored headcount ratios for each indicator.

7.3.1 Cross-Tabulations

As was mentioned earlier, cross-tabulations or contingency tables are a basic way to view the joint distribution between two dichotomous variables—which could be the uncensored or censored headcounts. We return to these to consider matters of correlation and redundancy. A two-way contingency table (Table 7.1) provides information on two kinds of matches:

: The percentage of people simultaneously not deprived in any two indicators and

: The percentage of people simumtaneously deprived in any two indicators and )

It also shows two kinds of mismatches:

: The percentage of people deprived in indicator but not in indicator

: The percentage of people deprived in indicator but not in indicator .

Finally, it shows the ‘marginal’ distributions as and so on. Note that two of these marginals will correspond to the uncensored or censored headcount ratios of the two indicators.


We show this familiar building block to remind readers that correlations between dichotomous variables (which generate the same coefficient as the Cramer’s V), draw on all elements of the cross-tab: the matches, the mismatches, and the marginal entries. In words, the correlation is the product of the matches minus the product of the mismatches, divided by the square root of the product of the marginals




What is important to notice is that while the correlation is affected by the extent to which deprivations between variables match (which is key for redundancy), it is also affected by values of the headcount ratios and their difference. This, as we will show, somewhat dilutes the insights that correlations offer for redundancy, so that the correlation coefficients are best interpreted alongside the contingency table for each indicator pair. Similarly, PCA, MCA, and FA also use all elements of the cross-tab.

Instead of using the correlation (Cramer’s V) alone, we propose another measure of association, which has some attractive characteristics for a direct assessment of redundancy.[13] This measure shows the matches between deprivations as a proportion of the minimum of the marginal deprivation rates. If two deprviation measures are not independent, and if at least one of the headcount ratios is different from zero, then the measure of Redundancy or Overlap is defined as



That is, the measure of Redundancy displays the number of observations which have the same deprivation status in both variables, which reflects the joint distribution, as a proportion of the minimum of the two uncensored or censored headcount ratios. By using the ‘minimum’ of the uncensored or censored headcounts in the denominator we ensure that the maximum value of is 100%.

If takes the value of 80%, this shows that 80% of the people who are deprived in the indicator having the lower marginal headcount ratio are also deprived in the other indicator. Thus a high level of is a more direct signal that a further assessment of redundancy is required than a correlation measure might be.

An example will clarify and close this section.

1. Contingency Tables for Mozambique and Bangladesh


Consider the Contingency Tables in Panel I and II of Table 7.2, which draw on 2011 DHS surveys for each country. In Mozambique, 38% of the population are deprived in years of schooling and 31% in school attendance. Only 16% are deprived in both indicators. For Bangladesh, 20% and 15% are deprived in years of schooling and school attendance respectively, and 6% are deprived in both. How do we assess the association between these indicators? Consider first the correlation or Cramer’s V coefficients, computed using equation (7.1). Using the values in Table 7.2 it can be easily verified that the Cramer’s V between attendance and schooling is 0.199 for Mozambique and 0.196 for Bangladesh. They are quite similar. But when we compute the measure using equation (3.30), we find that 52.8% of possible matched deprivations overlap for Mozambique, but only 37.9% match for Bangladesh. focuses on the precise relationship of interest.


Table 7.3 gives the Cramer’s Vs (correlation coefficients) and the measures of overlap/redundancy for three pairs of indicators for Mozambique. The highest redundancy values correspond to those between cooking fuel and other indicators. These are exceedingly high and might suggest that cooking fuel is redundant in these datasets, unless it is retained for other normative reasons (sequencing, policy). Yet the Cramer’s Vs between for cooking fuel and other dimensions are not particularly high and would not show this—indeed the correlation between water and schooling is much higher. As explained above, the divergence between these two values reflects the different components of the cross-tab that they draw upon. Although correlations are often used, we consider the measure of overlap to provide clear and precise information that should be considered alongside other kinds of information in evaluating indicator redundancy.

Chapter 8 addresses robustness analysis and statistical inference, which are required to draw conclusions or guide policies based on estimated poverty measures.


Alkire, S. and Ballon, P. (2012). Understanding Association across Deprivation Indicators in Multidimensional Poverty. Paper presented at the Research Workshop on ‘Dynamic Comparison between Multidimensional Poverty and Monetary Poverty’. OPHI, University of Oxford.

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Alkire, S. and Santos, M. E. (2014). ‘Measuring Acute Poverty in the Developing World: Robustness and Scope of the Multidimensional Poverty Index’. World Development, 59: 251–274.

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Deaton, A. and Kozel, V. (2005). ‘Data and Dogma: The Great Indian Poverty Debate’. World Bank Research Observer, 20(2): 177–199.

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[1] Deaton (1997) remains in our view an unsurpassed and essential guide for all analysts.

[2] Metadata refers to comprehensive information about the dataset (population on which the data was collected, definition of the variables, etc.).

[3] Whenever the sampling procedure departs from simple random sampling, survey weights must be used for estimations to be representative of the population under analysis. Metadata should be consulted in order to thoroughly understand the survey structure and the weights to be used.

[4] The substantial growth in the collection of household surveys towards the end of the twentieth century has been part of what Ravallion (2011b) called the ‘Second Poverty Enlightenment’ (the ‘First Poverty Enlightenment’ occurred near the end of the eighteenth century).

[5] Note that ‘applicable population’ differs from ‘eligible population’, a term frequently used in the metadata of household surveys to refer to the population that has been defined as eligible to collect information on a specific indicator (say, nutrition) in a particular survey instrument.

[6] The nutritional status of older children and adolescents (children 5–19 years old) can be measured with height-for-age or BMI-for-age.

[7] The differences in nutritional requirements are the basis for the construction of adult equivalence scales on which there is a broad literature including Deaton and Muellbauer (1980), OECD (1982), Morales (1988), Browning (1992), Nelson (1993), Hagenaars et al. (1994), Lanjouv and Ravallion (1995), Ravallion (1996), and Deaton (1997).

[8] As multidimensional poverty measurement is a field that is rapidly expanding, researchers may devise other innovative options in the near future.

[9] Precisely because a group of households with the same characteristics (for example, absence of children) is excluded, reweighting the sample would not compensate for the problem. The measure becomes a form of group-specific poverty measure.

[10] Alkire and Santos (2014) present two types of tests of the influence of household composition on the probability of being poor. One consists of hypothesis tests of differences in means. In each country they test whether MPI-poor households have a significantly higher average size, a higher average number of children under five, a higher number of females, a lower average number of members fifty years and older, a higher prevalence of female-headed households, and higher proportion of school-aged children, compared to non-poor households. A second set of analysis decomposes each country’s MPI by respondents’ age and gender and compares the rankings, correlations, and the proportion of robust pairwise country comparisons across the subgroup MPIs.

[11] Naturally, retained sample sizes should be reported and issues of representativeness and sampling weights reassessed.

[12] See Alkire and Ballon (2012) for a fuller discussion, on which this section draws.

[13] For a constructive review of measures of both association and similarity see Alkire and Ballon (2012). This particular measure was first proposed by Simpson (1943).

6 Normative Choices in Measurement Design

The modern field of inequality measurement grew out of the intelligent application of quantitative methods to imperfect data in the hope of illuminating important social issues. The important social issues remain, and it is interesting to see the ways in which modern analytical techniques can throw some light on what it is possible to say about them (Cowell 2000: 133).

Human beings are diverse in many and important ways: they vary in age, gender, ethnicity, nationality, location, religion, relationships, abilities, personalities, occupations, leisure activities, interests, and values. Poverty measures seek to identify legitimate, accurate, and policy-relevant comparisons across people, whilst fully respecting their basal diversity. Further, they seek to do so using data that are affected by several kinds of errors and limitations. This is no straightforward task.

After a measurement methodology has been chosen, the design of poverty measures—whether unidimensional or multidimensional—also require a series of choices. We turn to these now. The choices relate to the space of the measure, its purpose, unit of identification and analysis, dimensions, indicators, deprivation cutoffs, weights, and poverty line. Of these, ‘purpose’ is particularly influential in shaping the measure. As Ravallion states succinctly, ‘One wants the method of measurement to be consistent with the purpose of measurement’ (1998: 1). This chapter describes each of these normative choices in the context of multidimensional poverty measurement design and outlines alternative ways that these choices might be understood, made, and justified. Many normative theories or approaches might be used to inform measurement design, including human needs, objective lists, subjective wellbeing, human rights, and preference-based approaches, as well as many other less formally defined approaches.[1] Whichever are used, the normative contribution is not simply philosophical; it has a practical aim: to motivate poverty reduction.

Taken together, normative choices link the data and measurement design back to poor people’s lives and values and forward to the policies that, informed by poverty analysis, will seek to improve these. For example, dimensions which contribute disproportionately to poverty might become policy ‘priorities’. Do these reflect poor people’s values? Regions showing high poverty levels may be targeted geographically: does these accord with poor communities', taxpayers' and experts' understandings of who is poor? Programmes such as cash transfers may target households: do the poorest households benefit? The headlines (and political leaders) celebrate when multidimensional poverty falls—is this situation also applauded by those they seem to have assisted? It goes without saying that if the measure of poverty is unhinged from people’s voices and values, poverty policies are unlikely to hit the mark.

The normative choices inherent in monetary and multidimensional poverty design appear to cause consternation, particularly if measurement conventions have not yet been established.[2] In a section of The Idea of Justice named ‘The fear of non-commensurability’, Sen describes ‘non-commensurability’ as ‘a much-used philosophical concept that seems to arouse anxiety and panic’ (2009: 240). Yet setting priorities is no weakness. As Sen points out, ‘the need for selection and discrimination is neither an embarrassment, nor a unique difficulty, for the conceptualization of functionings and capabilities’ (1992: 46 and 44).

Building on Sen in their extremely relevant work Disadvantage, Wolff and De-Shalit elaborate additional conceptual insights that are relevant to address the ‘indexing problem’ in measurement. Defining poverty as clustered disadvantage, their policy goal is: ‘a society in which disadvantages do not cluster, a society where there is no clear answer to the question of who is the worst off. To achieve this, governments need to give special attention to the way patterns of disadvantage form and persist, and to take steps to break up such clusters’ (2007: 10). They argue that because disadvantages are interconnected and must be solved by policies that break up such clusters, and also because key policy decisions such as budget allocation require “some sort of overall assessment of disadvantage,” then “an overall index of disadvantage seems inescapable” (95, 89). They then proceed to address how such an index could be legitimately constructed, and we will return to their work in following sections.

Given that multidimensional poverty measurement remains a relatively new field of endeavour, a clear overview of the judgments and comparisons that normative choices draw upon, using the capability approach as a springboard, may prove useful.[3] To motivate the discussion we begin by sharing a birds-eye view of how the Adjusted Headcount Ratio measure can—if a set of assumptions about the normative choices are fulfilled—reflect capability poverty.[4], [5]

6.1 The Adjusted Headcount Ratio: A Measure of Capability Poverty?

Suppose that there are a set of dimensions, each of which represent functionings or capabilities that a person might or might not have—things like being well-nourished, being able to read and write, being able to drink clean water, and not being the victim of violence. The deprivation profile for each person shows which functionings they have attained and in which they are deprived, and weights are applied to these dimensions that reflect the relative value of each among the set of dimensions. Suppose that there is considerable agreement regarding the value of achieving the deprivation cutoff level of these functionings, such that most people would achieve at least that level if they could. Furthermore, suppose that we can anticipate what percentage of people would refrain from such achievements in certain functionings—those who might be fasting to the point of malnutrition at any given time, for example. It is convenient but not necessary to assume that these indicators are equally weighted.[6] And let us assume that in identifying who is poor, the calibration of poverty cutoff reflects these predictions of voluntary abstinence, as well as anticipated data inaccuracies, while recognizing that a sufficient battery of deprivations probably signifies poverty. Setting the cutoff in this way permits a degree of freedom for people to opt out if they so choose while seeking, insofar as is possible, to identify as poor people for whom the deprivations are unchosen. Applying such a poverty cutoff reduces errors in identification—for example by permitting people who would voluntarily abstain not to be identified as poor and avoiding identifying people as poor because of data inaccuracies. Among the poor, the more deprivations they experience, the poorer they are. Having identified who is poor, we construct the Adjusted Headcount Ratio ().

How might such an reflect capability poverty? The key insight is this: in such a measure, a higher value of represents more unfreedom, and a lower value, less. Given that the set of indicators will be unlikely to represent everything that constitutes poverty, if each element is widely valued, and if people who are poor and are deprived in a dimension would value being non-deprived in it, then we anticipate that deprivations among the poor could be interpreted as showing that poor people do not have the capability to achieve the associated functionings. Thus would be a (partial) measure of unfreedom, or capability poverty.

As noted above, such an interpretation of relies on assumptions regarding the parameters:

a)      Indicators measure or proxy functionings or capabilities

b)      People generally value attaining the deprivation cutoff level of each indicator

c)      The weights reflect a defensible set or range of relative values on the deprivations

d)      The cross-dimensional poverty cutoff reflects ‘who is capability poor’.

Such an interpretation implicitly also relies upon assumptions about data quality and accuracy, and empirical techniques (that measures are implemented accurately). It has quite a restricted and uniform approach to values: for example, using a non-union poverty cutoff to permit ‘some’ abstinence from functionings.[7] But it might at least signal an avenue worth pondering.

In fact, as Box 6.1 elaborates more formally, under these conditions, our identification strategy and Adjusted Headcount Ratio can be related to Pattanaik and Xu’s signature work (1990), except that we now focus on unfreedoms rather than on freedoms. In their lucid and illuminating paper, Pattanaik and Xu elaborate on Sen’s claim that freedom has intrinsic value, thus that the extent of freedom in an opportunity set matters—independently of its relationship to preferences and utility. In developing this claim axiomatically they propose that the ranking of two opportunity sets in terms of freedom should depend only on the number of options present in each set.[8]

Sen, responding to Pattanaik and Xu (1990), observed that not every additional option (singleton) would contribute to an expansion of freedom–only those options that a person values and has reason to value. ‘The evaluation of the freedom I enjoy from a certain menu must depend to a crucial extent on how I value the elements included in that menu’. For example, ‘if a set is enlarged by including an alternative which no one would choose in relevant circumstances (e.g., being beheaded at dawn), the addition of that alternative may not necessarily be seen as a strict enhancement of freedom…’ (Sen 1991: 21 and 25). Nor would a deprivation in that negatively valued alternative be seen as impoverishing.

Our assumptions regarding the choice of parameters avoid Sen’s critique if each dimension of poverty reflects something that people value and have reason to value. Further, we follow Anand and Sen (1997), who argued that it may be easier to obtain agreement on the value of a small set of unfreedoms than an ample set of freedoms.[9] As Sen points out, ‘in the context of some types of welfare analysis, e.g. in dealing with extreme poverty in developing economies, we may be able to go a fairly long distance in terms of a relatively small number of centrally important functionings (and corresponding basic capabilities, e.g. the ability to be well nourished and well sheltered, the capability of escaping preventable morbidity and premature mortality, and so forth)’ (1992: 44–45; cf. 1985b).

Note that this capability interpretation of does not directly represent ‘unchosen’ sets of capabilities in a counterfactual sense (Foster 2010). Nor does it necessarily incorporate agency (Alkire 2007). Rather, in a manner parallel to Pattanaik and Xu, it interprets the deprivations in at least a minimum set of widely valued achieved functionings as unfreedom, or capability poverty (Box 6.1).

Naturally capability poverty measures that have different specifications and reflect different purposes could be constructed for the same society. There might be a child poverty measure or a capability poverty measure reflecting the values of a specific cultural group such as nomadic populations, or there might be a national capability poverty measure that reflects important deprivations about which there is widespread agreement across social groups. Thus the decision to measure capability poverty does not generate one unique measure; decisions as to the scope and purpose of the measure and the data sources guide measurement design even if the choice of space has been settled.

We also hasten to point out that many legitimate and tremendously useful measures could be constructed using but located in a different space or in a mixture of spaces. These would not be measures of capability poverty but could be powerful tools for reducing capability poverty. For example, the dimensions might be resources such as service delivery (hopefully identifying whether marginalized groups have real access and clarifying the quality of the services). The point is that our measurement framework can be used for different purposes including those unrelated to capabilities. So it is vitally important (and not terribly difficult) to articulate and explain the purpose of each application and to justify the choices and calibration of parameters.

This section set out the circumstances in which may measure capability poverty. The assumptions regarding values and data that must be fulfilled to do so were transparently stated. Under this interpretation, embodies a rather rudimentary kind of freedom; there could be many interesting extensions — for example, incorporating agency and process freedoms. Also, measures that do not reflect capability poverty will fulfil some purposes splendidly. Still, if the assumptions articulated here are fulfilled, we can indeed offer as a measure of capability poverty. For what is needed in this context is not a quixotic search for the perfect measure but rather methodologies that may be sufficient to advance critical ethical objectives. Most empirical outworkings of the capability approach have used drastic simplifications, and these can often be cheered as true advances, even while their limitations are borne in mind. ‘In all these exercises, clarity of theory has to be combined with the practical need to make do with whatever information we can feasibly obtain for our actual empirical analyses. The Scylla of empirical overambitiousness threatens us as much as the Charybdis of misdirected theory’ (Sen 1985: 49). In this sense, our methodology may be a step forward in operationalizing the measurement of capabilities.

Box 6.1 Unfreedoms and

Let be a poverty methodology satisfying decomposability, weak monotonicity, non-triviality, and ordinality.

The first three properties are satisfied by all members of methodology ; however, is the only adjusted Foster–Greer–Thorbecke (FGT) measure that satisfies ordinality, and it is this property that ensures that its poverty levels and comparisons are meaningful when the dimensional variables are ordinal.

By decomposability, the structure of depends entirely on the way that measures poverty over singleton subgroups; and by dichotomization, this individual poverty measure can be expressed as a function of the individual’s deprivation vector (which is any row of deprivation matrix ). In the case of , we have , where is the censored distribution defined as if and if . We will now explore the possible forms that can take for dichotomized measures. Note that while the definition of is based in part on the dimensional cutoff , we have not specified the identification method employed by the general index . Hence a second question of interest is what forms of identification might be consistent with various properties satisfied by .

The individual poverty function for has two additional properties of interest. First, it satisfies anonymity or the requirement that , where is any permutation matrix. This property implies that all dimensions are treated symmetrically by the poverty measure. Second, it satisfies semi-independence, which states that if , and , then .[10] Under this assumption, removing the same dimensional deprivation from two deprivation vectors should preserve the (weak) ordering of the two. We have the following result:

Let be the individual poverty function associated with a dichotomized poverty measure. satisfies anonymity and semi-independence if and only if there exists some such that for any deprivation vectors and we have: if and only if .

In other words, ranks individual deprivation vectors in precisely the same way that does for some . This result is especially powerful since it simultaneously determines both the individual poverty index ( associated with the Adjusted Headcount Ratio and the identification method (based on a dimensional cutoff ) consistent with the assumed properties. To establish the result we extend the generalization of Pattanaik and Xu given in Foster 2010. In particular, if full independence were required, so that the conditional in semi-independence were converted to full equivalence, then a direct analogue of the Pattanaik and Xu result would obtain, namely, if and only if . In this specification, would make comparisons of individual poverty the same way that the union-identified does—by counting all deprivations.

While our result uniquely identifies the individual poverty ranking, it leaves open a multitude of possibilities for the overall index —one for each specific functional form taken by . For example, the individual poverty function , when averaged across the entire population to obtain , would place greater emphasis on persons with many deprivations. It would be interesting to explore alternative forms for and the properties of the associated index . Note that because of dichotomization, each of these measures would provide a way of evaluating multidimensional poverty when the underlying variables are ordinal.

Given the arguments in Foster 2010, it is straightforward to establish the above result. In particular, let or for all be the set of all individual deprivation vectors, and let be an individual poverty function associated with a standard dichotomized poverty measure such that satisfies anonymity and semi-independence. By anonymity, all vectors with must satisfy . In other words, the value of depends entirely on the number of deprivations in . Weak monotonicity implies that for , and so the value of is weakly increasing in the number of deprivations in . By non-triviality and decomposability, it follows that for .[11] Let be the lowest deprivation count for which is strictly above ; in other words, for , and for . Semi-independence ensures that must be increasing in the deprivation count above . For suppose that for with . Then by repeated application of anonymity and semi-independence, we would have for some with , a contradiction. It follows, then, that is constant in for and increasing in for . Clearly, this is precisely the pattern exhibited by the function .

6.2 Kinds of Normative Choices

It may be asked whether choices underlying measurement design are normative and, if so, in what sense? If data are constrained and exactly one educational variable exists, in what sense is its selection normative? Similarly, if an indicator is redundant or invalid according to statistical assessments, how is its deselection normative? And if nutritional experts judge that an indicator of stunting is more accurate than wasting, in what way is a choice in its favour normative?

Normative considerations operate at different levels. Releasing a measure rather than not doing so may reflect a high-level normative judgement that releasing the measure is more likely to improve welfare than not releasing it.[12] This assessment may be made after consideration of what Sen (2009) terms a ‘comprehensive’ description of the situation. At a lower level, in each part of measurement design, value judgements are used to justify particular choices—like dimensions, weights, and poverty cutoffs. The value judgements may pertain to the content directly, or they may address the methodologies or processes by which to justify design choices, as later sections will illustrate.

At this higher meta-level, the comprehensive description and its normative assessment will draw upon different kinds of analyses—statistical, axiomatic, deliberative, practical, and policy-oriented, for example—to authorize the use of measures that fulfil a set of plural purposes reasonably well.

These higher level reasoned judgements that draw on a comprehensive description of the options often include the following types of assessments:

Expert (including qualitative) assessments of indicator accuracy—for example, in showing the level and changes of a key functioning like nutrition (Svedberg 2000) or the quality and legitimacy of a participatory process;

Empirical assessments, which could include analyses of measurement error, data quality, redundancy, robustness, statistical validity and reliability, or triangulation with other analyses and data sources;

Deliberative insights on people’s values from participatory discussions, social movements, consultations, and from documentation of similar recent processes;

Theoretical assessments, which could consider properties and principles, sets of dimensions, standards or conventions on indicators, or legal and policy frameworks;

Practicalities such as constraints of data, time, human resources, authority, political will, and political feasibility given the processes and authorities involved; and

Policy relevance—for example, how the timing and content of the measure could dovetail with resource allocation decisions or how a measure might support and monitor a set of planned interventions as set out in a national plan or a current campaign.

This section introduces this meta-coordination role of normative reasoning; later sections describe how particular kinds of assessment mentioned already may inform particular design choices.

The higher normative function is inextricably linked to the purpose(s) of the measure, which are often multiple and normally motivate policy and public action. As Foster and Sen put it, ‘The general conclusion that seems irresistible is that the choice of a poverty measure must, to a great extent, depend on the nature of the problem at hand’ (1997: 187).

A relevant example is Mexico’s move towards a multidimensional poverty measure. In his book Numbers that Move the World, Miguel Székely points out that:

Just like there are ideas that move the world, so too there are numbers and statistics that move the world. A number can awaken consciences; it can mobilize the reluctant, it can ignite action, it can generate debate; it can even, in the best of circumstances, lay to rest a pressing problem (2005: 13).[13]

In describing Mexico’s steps towards new options of poverty measurement, Székely describes how a committee was formed whose mandate was ‘to propose to the Secretary of Social Development a methodology that could be officially adopted as an instrument of the Mexican government to measure the magnitude of poverty, its intensity, and its characteristics’ (Székely 2005: 17). In 2001, the committee invited three international experts, including James Foster; and Foster, together with John Iceland and Robert Michael, identified the following desiderata that the proposed measurement methodology should fulfil — criteria that the committee adopted in its subsequent work:

  • It must be understandable and easy to describe
  • It must reflect ‘common-sense’ notions of poverty
  • It must fit the purpose for which it is being developed
  • It must be technically solid
  • It must be operationally viable—e.g. in terms of data requirements
  • It must be easily replicable (Székely 2005: 10 and 19).[14]

As these criteria suggest, there are usually plural desiderata for a measure, and these must be taken into account within a coordinating normative framework.[15] Consider the first purpose: a measure should be simple and easy to communicate. Earlier we observed that the widely used headcount ratio of income poverty lacks some very desirable properties. Indeed, because the headcount ratio ‘ignores the “depth” as well as the “distribution” of poverty’, Foster and Sen found it ‘remarkable that most empirical studies of poverty tend, still, to stop at the head-count ratio’ (1997: 168 and 169). On the other hand, when formulated as a criterion, it becomes evident that this characteristic — that a measure not only be axiomatically sound and empirically solid, but also easy to understand — is actually essential if the measure is to inform and engage public debate and policy.

Returning to the income poverty headcount ratio, it seems that the desirability of certain properties is balanced against ease of communication. For measures whose purpose is to incite public action, the choice to favour communication is comprehensible. Indeed the development of the Human Development Index, as Sen describes it, was largely driven by this need of communicability. Sen recounts how Mahbub ul Haq, the director of the then newly created Human Development Report Office of the United Nations Development Programme called for an index ‘of the same level of vulgarity as the GNP—just one number—but a measure that is not as blind to social aspects of human lives as the GNP is’ (Sen 1998). Properties vs. communicability is not the only trade-off: at times statistical accuracy and non-sampling measurement error may need to be balanced with ‘common sense’, or an ideal measure tempered by the need to use existing data.

The ‘higher’ or coordinating normative reasoning creates a ‘comprehensive’ description of possible measures according to the criteria, rules out options that are strictly worse than others, and identifies their relative strengths and challenges.[16] Even if, as is likely, the final choice of a multidimensional poverty measure is but one of a set of candidate measures, each of which is defensible and cannot be further ranked, the criteria will still have worked to eliminate measures that may have been less comprehensible and violated more key properties—or had higher measurement error, lower robustness and less policy salience than the ones that remain. They will also have identified the strengths and challenges of each candidate, and so the selection among them is essentially also a selection of which criteria to prioritize—a choice that will have been simplified by a clear analysis. For example, a society may wish to prioritize a measure that has legitimacy because it transparently draws on public consultations, which are important because recent history had discredited poverty statistics (so prioritizing criteria 2), or a measure that will incentivize policies because it is closely tied to a popular national plan (criteria 3) and so on.

In sum, multidimensional poverty measurement can seem rather bewildering at first because its justification may draw on axiomatic, statistical, ethical, data-related, deliberative/participatory, policy-oriented, political, and historical features. But in practice poverty measurement is considerably more concrete (Anand and Sen 1997, Alkire 2002b). The available resources and actual constraints – from timing to data to funding to political demand – for a given exercise often provide considerable structure and guidance. Thus, although normative engagement is required to coordinate various considerations, ‘there is no general impossibility here of making reasoned choices over combinations of diverse objects’ (Sen 2009: 241).

6.3 Elements of Measurement Design

The Alkire-Foster methodology is a general framework for measuring multidimensional poverty – an open-source technology that can be freely altered by the user to best match the measure’s context and evaluative purpose. As with most measurement exercises, it will be the designers who will have to make and defend the specific decisions underlying the implementation, limited and guided by the purpose of the exercise and other concrete constraints.

Traditional unidimensional measures require a set of parallel decisions with normative content.[17] For example, should the variable be expenditure or income? What indicators should comprise the consumption aggregate? How should ‘missing’ prices be set? What should the poverty line(s) be? If it reflect a food basket, how many calories should it total, and should it exclude cheap unhealthy foods? Choices to create comparability can likewise be important for final results, such as the construction of Purchasing Power Parity values or urban–rural adjustments, or adjustments for inflation. Robustness standards are crucial for all poverty measures, as they ensure that the results obtained are not unduly dependent upon the calibration choices (whether these are normatively based or not).[18]

The flexibility in AF measurement design means that measures at the country or subnational level can be designed to embody prevailing priorities or norms of what it means to be poor. For example, if dimensions, weights, and cutoffs are specified in a legal document such as the Constitution, the identification function might be developed using an axiomatic approach, as was done in Mexico.[19] Qualitative and participatory work can (and in our view should) triangulate other analyses.[20] The weights can also be developed by a range of processes: expert opinion, or coherence with a consensus document such as a national plan, focus groups, survey data, or human rights. And the poverty cutoff, which is analogous to poverty lines in unidimensional space, could be chosen so as to reflect poor people’s assessments of who is poor, as well as wider social assessments.

This section introduces the purpose of a poverty measure and the normative choices that inhere in measurement design.[21] We cover eight design elements. The first five serve to structure a poverty measure; the last three calibrate key parameters (cutoffs and weights).

  1. Purpose(s) of the measure: The purpose(s) of a measure may include its policy applications, the reference population, dimensions, and time horizon.
  2. The choice of space: The choice of space determines whether poverty is measured in the space of resources, inputs and access to services, outputs, or functionings and capabilities.
  3. The unit(s) of identification and analysis: These are unit(s) for which the AF method reflects the joint distribution of disadvantages, identifies who is poor, and analyses poverty.
  4. Indicators: Indicators are the building blocks of a measure; they bring into view relevant facets of poverty and constitute the columns of the achievement and deprivation matrices.
  5. Dimensions: Dimensions are conceptual categories into which indicators may be arranged (and possibly weighted) for intuition and ease of communication.
  6. Deprivation cutoffs: The deprivation cutoff for an indicator shows the minimum achievement level or category required to be considered non-deprived in that indicator.
  7. Weights: The weight or deprivation value affixed to each indicator reflects the value that a deprivation in that indicator has for poverty, relative to deprivations in the other indicators.
  8. Poverty cutoff: The poverty cutoff shows what combined share of weighted deprivations is sufficient to identify a person as poor.

In practice, these design choices are not made in a linear fashion but rather iteratively, and in combination with consultations and empirical work. Thus discussing them sequentially may seem rather tedious. Just like it is far more pleasant to hear a horse whinny than to transcribe its whinny painstakingly onto a musical staff to learn how it is done, so too, considering these choices one by one makes the task seem rather dull. One can only hope the transcription is a one-time task, whereas the skill of whinnying lasts awhile.

6.3.1 Purpose(s) of a Measure

The purpose(s) of a measure clarify the way(s) in which the measure will be used to describe and understand situations, to make comparisons across groups or across time, and to guide policy or monitor progress. The purpose shapes the choice of space and many of the calibration decisions that will follow and so should be explicitly formulated and stated. The Stiglitz–Sen–Fitoussi Commission drew attention, in the case of quality of life measures, to the fundamental importance of the purpose of the measure to the identification of dimensions and indicators. ‘The range of objective features to be considered in any assessment … will depend on the purpose of the exercise … the question of which elements should belong to a list of objective features inevitably depends on value judgements …’ (2009).

The purpose may also identify constraints and shape processes. For example, if the purpose includes legitimacy to the wider public, then public consultations may be essential; if it is performance monitoring, involvement with the concerned agencies and institutions may be useful. While a measure may have a single purpose, it is more common for measures to seek to fulfil multiple purposes.

For example, a national poverty measure might be used to assess the population-wide levels and trends in capability poverty across regions and population groups in ways that are regarded as legitimate and accurate by the citizenry. Note that this statement of purpose has scope (population-wide), space (capability), relevant comparisons (across population groups and time trends), and popular legitimacy (which affects procedures). A study may design a youth poverty measure in order to understand, profile, and draw attention to youth capabilities at a given point in time. A targeting measure may use census data to identify and target the poorest of the poor in terms of social rights for certain services. A performance monitoring measure may track changes over time across a set of indicators reflecting the goals of a programmatic intervention, such as improvement in the quality of education, or women’s empowerment across various domains. A local community development measure may monitor a village development plan in ways that community members have proposed and understand. Measures might be designed to inform the private sector and civil society about the state of poverty in their country and so encourage public debate. They might also clarify what value-added their analyses have in comparison with monetary poverty measures.

The purpose of the measure will often also include political economy and institutional issues and constraints that are pertinent to the measure fulfilling its purpose, such as timescale, data, budgetary resources, political and legal procedures, updating procedures and so on.[22] For example, will a given dataset be used or will a new survey be designed and implemented and if so what is its budget and frequency? Are particular committees, commissions, or institutional processes to be involved in measurement design and what is their authority? If a measure will be updated over time, what is its legal or institutional basis, which institution(s) or person(s) have the authority to update the measure, and when and how is occasional methodological updating to take place? Clarity on such issues can greatly streamline design procedures.

6.3.2 Choice of Space

As mentioned in section 6.1 another preliminary choice is the space in which that measurement is to proceed. Will it be in the space of income, of resources and access to resources, of functionings and capabilities, or of subjective utility? There are well-known arguments in favour of each space, and purposes for which each space might be appropriate. Conceptually it is vital to be clear about the choice of space prior to the selection of indicators. This is because the same indicators – such as years of schooling – may be used in empirical measures of both types, but the interpretation and, at times, the treatment of the data may vary.

Following Sen, we may take the space that is of central interest to be the space of functionings and capabilities (they are the same space). Functionings are the beings and doings that people value and have reason to value, and capabilities are the freedoms to achieve valuable functionings. This implies that measurement should focus on valuable activities and states of being that people actually achieve, given their values and their varying abilities to convert resources into functionings. The choice of space may have implications for the interpretation of variables’ scales of measurement. In some cases, measures use indicators that reflect achievements in other spaces (or subjective and self-reported states), if these can be justified empirically as proxies of functionings or capabilities.

An essential step at this stage is to revisit the scales of measurement introduced in section 2.3. To summarize, any achievement matrix may contain data having categorical, ordinal, or cardinal scales (which may be binary, interval, or ratio scale). In measures requiring cardinal data the indicator’s scale of measurement has to be reassessed after the space of measurement has been chosen. For example, years of schooling may seem to be a ratio-scale variable. But in terms of human functionings, is it? Or do the earlier years of education confer marginally more capabilities than later years, or the completion of the twelfth year (with a diploma) than the eleventh year (without a diploma)? In using , we dichotomize variables at a deprivation cutoff. This obviates the need to rescale indicators to construct an appropriate normalized gap in different spaces, but still requires that the deprivation cutoffs (discussed in section 6.3.6) reflect deprivations in the chosen space.

Not all measures focus on the capability space—or need to. They might reflect social rights, social exclusion, access to services, social protection, or the quality of services. And most poverty reduction requires, as intermediary steps, institutions that effectively deliver resources and services to people and communities. Thus even if the goal is capability expansion, this might be stimulated or monitored in part by a multidimensional poverty measure that is framed in an intermediary space of inputs or outputs. The choice of space specifies how a given measure will advance the purpose.

6.3.3 Units of Identification and Analysis

The unit(s) of identification or analysis[23] may be a person, a household, a geographic area, or an institution (such as a school or firm or clinic). A common unit of identification for a poverty measure is a person (any adult, a child, a worker, a woman, an elderly person). This permits a poverty measure to be decomposed by variables like gender, age, ethnicity, occupation, and other relevant individual characteristics. It may also permit analysis of intra-household patterns of poverty or of group-specific poverty (indigenous groups, youth unemployed, urban slums). Alternatively, household members’ information may considered together, which has advantages in terms of supporting intra-household sharing, and also at getting an overview of households. In this case, household members’ combined achievements are used as a unit of identification for a population-wide measure, and all household members receive the same deprivation score.

The unit of analysis – meaning how the results are reported and analysed – may still reflect each person. That is, even if the unit of identification is the household, one can report the percentage of people who are identified as poor (by using individual sampling weights), rather than the percentage of households that are identified as poor (which is used if the household is the unit of analysis).

Where data are not available at the household level or where the measure focuses on topics such as infrastructure, poverty can be computed for data zones or geographic areas, so long as there is a justification for overlooking within-region inequality. Other measures may use a particular institution such as a school or clinic or firm as a unit of identification and/or analysis.

What is essential is that data for all variables must be available for (or transformed to represent) each unit of identification (see also Chapter 7), that the definition of applicable populations be transparent and complete, and that the unit of analysis be explicitly and clearly stated and justified.

There are ethical considerations in choosing and justifying a unit of analysis. For example, using the person as the unit of identification coheres with human rights policies, can show gendered or age disparities, and may permit intra-household analysis (Alkire, Meinzen-Dick, et al. 2013). Yet using the household as a unit of identification acknowledges intra-household caring and sharing—for example, educated household members reading for each other, and multiple household members being affected by someone’s severe health conditions. Policies targeting or addressing the household may also strengthen or at least not weaken the household unit. Normative and policy assessments may be supplemented by participatory insights regarding the appropriate unit of identification and the ensuing focus of policy interventions.

The justification of a unit of identification may include empirical assessments of bias and comparability. For example, if the unit of identification is the household, then indicators that draw on individual-level achievements may be checked for biases according to household size and composition (Alkire and Santos 2014). If the unit of identification is the person, then the comparability of indicators across diverse groups requires analysis—as in the case of education and health indicators for people of different ages (infants and toddlers, school-aged children, adults, and the elderly). The scale of errors that could be introduced if household-level variables are presumed to be equally shared by all household members is a further fruitful topic of empirical scrutiny.

For some policy purposes it could be useful to construct a set of measures, in which each measure takes a nested unit of identification that includes or is included by related measures: for example, the person, the household, the village or neighbourhood, and the district. The ‘nested’ measures permit further analysis of the interactions between deprivations at different levels. For example, the individual-level data may have health and educational functionings, household data may have living standard and housing-related functionings, and village-level data may have environmental, infrastructure, and service delivery information. Analyses may explore the extent to which health- and education-deprived people live in living-standard-deprived households, for example, and whether these live in services-deprived villages. Alternatively one may study which poor children live in poor households. Analyses using nested measures can be compared to analyses of a multidimensional poverty index at the individual level that applies relevant household- and village-level deprivations to each individual.

In sum, although often data constraints will require that the household be the unit of identification, when other options are feasible, then this choice can be considered, made, and justified using different kinds of reasoning to assess the ‘fit’ between the proposed measure and its purpose.

6.3.4 Dimensions

When these structural choices have been established, poverty measures require the selection and valuation of deprivations. Sen introduces the task as follows: ‘In an evaluative exercise, two distinct questions have to be clearly distinguished: (1) What are the objects of value? (2) How valuable are the respective objects’ (Sen 1992: 42). These two tasks of selecting focal deprivations (using dimensions, indicators, and cutoffs) and setting relative values for them, recur in poverty measurement.

The term ‘dimensions’ in this chapter refers to conceptual categorizations of indicators for ease of communication and interpretation of results. By ‘indicators’ we mean the d variables that appear in columns of the achievement and deprivation matrices and are used to construct the deprivation scores and to measure poverty.

A multidimensional poverty measure is constructed using indicators. In some cases, these indicators may each represent distinct facets of poverty. In other cases, it may be useful to talk about several indicators as forming a ‘dimension’ of poverty. Why use dimensions? Dimensions may reflect the categories defined by some deliberative or synthetic processes. For example, a dimension might be children’s education; indicators might include a child’s years of completed schooling and their achievement scores last period. In this case the indicators may be the best possible approximation of those dimensions from an existing dataset. It may also arise from a theory or policy source. Noll (2002) develops a systematic conceptual framework for social indicators in Europe by reviewing concepts of welfare and common policy goals, then identifying fourteen dimensions that fit the measurement’s purposes. Grouping indicators into dimensions may facilitate the communication of results because there are likely to be fewer dimensions than indicators and they are likely to be intuitive and accessible to non-experts.

Grusky and Kanbur argue that the selection of dimensions merits active attention because ‘economists have not reached consensus on the dimensions that matter, nor even on how they might decide what matters’ (2006: 12). Yet the extensive and historic discussions about the post-2015 development agenda have been tremendously useful in illuminating areas of agreement across different interest groups with respect to widely varying national and international considerations.

Unusually, the selection of dimensions does not necessarily rely on empirical or technical analysis. Naturally sometimes analysts explore or confirm the extent to which dimensional grouping of indicators is corroborated statistically. Such statistical explorations should not determine the selection of dimensions or grouping of indicators; they may, however contribute to their justification and expose interesting relationships that should be considered.

In addition to the inevitable consideration of data constraints, there are at least three overlapping kinds of information that may inform the selection of dimensions: deliberation and public reasoning, legitimate consensus, and theoretical arguments (Alkire 2008a).

The first approach is a repeated deliberative or participatory exercise, which engages a representative group of participants as reflective agents in making the value judgements to select focal capabilities. Deliberation may involve online assessments as well as face-to-face focus groups; it may also consolidate the body of recent and similar participatory work that has been undertaken for other purposes. In a supportive, well-informed, and equitable environment, participatory processes seem to be ideal for choosing dimensions; not, however, if deliberation is dominated by conflict or inequality or misinformation, or coloured by the absence or dominance of certain groups. Furthermore, the process of aggregating the values of a diverse assembly of groups and people, whose deliberative processes may vary in quality, is neither elementary nor void of controversy (indeed it is an appropriate topic for further research). Even if a new set of deliberative exercises are not possible, it may be possible to consider documentation of previous such processes, be it from a previous measurement consultation, participatory exercises, a widely debated national plan, high-profile legal documents, the media, or a respectable set of life histories of disadvantaged people and communities (Leavy et al. 2013, Narayan et al. 2000). So it may be rare for a set of dimensions to be justified without any reference to participatory studies and public debates.

In writing on the selection of capabilities – which relate to dimensions and indicators[24]—Sen calls for deliberative engagement rather than using a pre-ordained list. ‘I have nothing against the listing of capabilities,’ Sen writes, ‘but must stand up against a grand mausoleum to one fixed and final list of capabilities’ (2004: 80). A central reason against promulgating a fixed list is that to be relevant, the dimensions (and indicators) should reflect the purpose of the measure. ‘What we focus on cannot be independent of what we are doing and why’ (Sen 2004: 79). The deprivations in international measures will rightly differ from a national measure or a measure of child poverty or of an indigenous community, for example. A further motivation for not fixing a list of capabilities even for a given purpose—including poverty measurement—is that a fixed list would crowd out debate and public reasoning, which can play an educational and motivational role. It also would not catalyze constructive debate that may influence people’s values. ‘To insist on a fixed forever list of capabilities would … go against the productive role of public discussion, social agitation, and open debates’ (Sen 2004: 80). Also, as technology advances and social values change, the list might become outdated (Sen 2004: 78). For example recent approaches to poverty often incorporate environmental and energy considerations that were lacking previously.

A second approach to the selection of dimensions is the use of an authoritative document or list that has attracted a kind of enduring consensus and associated legitimacy. Examples include a constitution, a national development plan, a declaration of human rights, or some time-bound international agreements such as the MDGs. Most official multidimensional poverty measures have some transparent link to such a policy process or document. The use of a set of dimensions that already have a kind of visibility and legitimacy is useful for international or global measures (where public deliberation is difficult), as well as for those that are clearly designed to monitor policy processes. It also naturally connects measurement to policy management.

A third potential source of dimensions is a conceptual framework or particular theory—which may range from Maslow’s hierarchy of human needs to a religious framework such as the Maqasid A-Sharia, to lists like Martha Nussbaum’s set of central human capabilities. These approaches are particularly relevant for communities where the theory enjoys widespread approval and/or is consistent with lists generated by alternative theories or processes (Alkire 2008a).

Comparing the lists that groups generate by these and additional processes, one finds a striking degree of commonality between them. Table 6.1 lists dimensions generated by these processes that pertain to multidimensional poverty measurement. Indeed there are a plethora of similar resources for the selection of deprivations, which may contribute towards standards supporting multidimensional poverty measurement design. And whilst the particular names and grouping of indicators differ, the universe of options is not too great, and this fact itself may be of no little comfort to those designing multidimensional poverty measures.

It should be borne in mind that the selection of dimensions and indicators affects the selection of weights. In a book supporting the development of national poverty plans in Europe, Tony Atkinson and colleagues point out the convenience of keeping weights in mind when selecting dimensions and indicators. In particular, they commend choosing indicators (or, possibly, dimensions) such that their weights are roughly equal to facilitate policy interpretations: ‘the interpretation of the set of indicators is greatly eased where the individual components have degrees of importance that, while not necessarily exactly equal, are not grossly different’ (Atkinson et al. 2002). Sen also emphasizes the interconnection between these choices: ‘There is no escape from the problem of evaluation in selecting a class of functionings in the description and appraisal of capabilities, and this selection problem is, in fact, one part of the general task of the choice of weights in making normative evaluation’ (Sen 2008). Elsewhere we observed that this linkage holds not only for dimensions that are selected but also for those that are omitted: ‘choosing one out of several possible variables is tantamount to assigning that dimension full weight and the remaining dimensions zero weight’ (Alkire and Foster 2011b). And it is to the selection of indicators that we now turn.


6.3.5 Indicators

Indicators are the backbone of measurement. Their quality, accuracy, and reach determine the informational content of a poverty measure. Given data constraints the process through which these are selected may include participatory and deliberative exercises, legal or political documents, statistical explorations, robustness tests, or theoretical guidelines.

While a considerable amount of attention, discussion, and practice has focused on the normative selection of dimensions of poverty and wellbeing, there is a paucity of comparable normative literature on the selection of indicators. The literature on indicator selection is, however, richly arrayed with a plethora of empirical considerations, which must be considered alongside normative and policy issues. Some of these will be raised in Chapter 7. These include:

  • statistical techniques to assess aspects such as the reliability, validity, robustness, and standard errors of economic and social indicators, [27]
  • indicators’ comparability across time and for different population subgroups,
  • dataset-specific issues such as data quality, sample design, seasonality, and missing values,
  • the justification of indicators as proxies for a hard-to-measure variable of interest.

Such analysis of each component indicator is fundamentally important for building rigorous measures, and, while these are not covered here, we presume readers will learn relevant techniques.

Alongside these, numerous guidelines seek to match indicator selection with policy purposes (IISD 2009, Maggino and Zumbo 2012). For example Atkinson and Marlier (2010: 8–14) provide an insightful overview of the purposes for which appropriate indicators should be stock or flow, subjective or objective, relative or absolute, static or dynamic, input or output or outcome, and so on. When statistics are used by the public, issues such as ease of interpretation also affect the choice. Still, as we saw in Chapter 4, the literatures on existing practices of addressing these technical, policy, and practical concerns are dispersed.

Naturally, the cost of data collection, cleaning, and preparing an indicator are also likely to influence indicator selection, especially when new surveys are fielded or regular updates are anticipated. This is a very important and underdocumented consideration, given the need both for better and more frequent data, and for timely, thorough analysis of new data (Alkire 2014).

The selection of indicators should be transparently justified—as many counting measures are. The criteria for selection will vary, however. But to give a useful example, Atkinson and Marlier (2010: 45) outline five criteria for internationally comparable indicators of deprivation in social inclusion:

  1. An indicator should identify the essence of the problem and have an agreed normative interpretation.
  2. An indicator should be robust and statistically validated.
  3. An indicator should be interpretable in an international context.
  4. An indicator should reflect the direction of change and be susceptible to revision as improved methods become available.
  5. The measurement of an indicator should not impose too large a burden on countries, on enterprises, or on citizens.

As the field of multidimensional poverty advances, we anticipate that conventions and standards will be developed to facilitate the selection of indicators and the calibration of parameters described in the following sections, much as has been done for monetary poverty.[28] These will reduce although not eliminate the value judgements in measurement design. In the case of monetary poverty, conventions did not make the creation of a consumption aggregate mechanical, imputation of housing costs easy, or the comparison of rural and urban monetary poverty lines uncontroversial. There remain animated debates, such as whether to include popular sugary drinks in the food poverty basket or elite goods in the consumer price index. Yet conventions still serve to streamline and legitimize key choices during the design process and reflect an ongoing and evolving technical consensus (or partial consensus) regarding sound measurement principles.

6.3.6 Deprivation Cutoffs

Deprivation cutoffs are fundamentally normative standards. They define a minimum level of achievement, below which a person is deprived in each indicator or subindex.[29] As we saw in Chapter 2, the deprivation cutoffs, together with the deprivation values, create cardinal comparability across indicators for measures and may be interpreted as creating a ‘natural zero’. Deprivation cutoffs for each indicator are a distinguishing feature of multidimensional poverty measures that reflect the joint distribution of deprivations (Bourguignon and Chakravarty 2003). This is because, by the property of deprivation focus, having more than the deprivation cutoff achievement level in one dimension – for example, clean water – does not erase’ the deprivation in another dimension (like malnutrition). This coheres with a human rights approach, among others.

Deprivation cutoffs may be justified with reference to international or national standards.[30] They may be set to reflect ‘basic minima’ or ‘aspirations’ that have arisen in participatory, consultative, or deliberative exercises. They might reflect the ‘targets’ of national development plans or of some international agreement or legal guidelines–for example, on compulsory schooling and social protection – or a social contract or, in some cases, medical standards (e.g. for anaemia, micro-nutrients, stunting, wasting, and so on).

Note that in indicators that use the household as the unit of identification, deprivation cutoffs must be defined such that they combine individual-level data when it is available for multiple household members. For example, if the household is the unit of identification, a deprivation cutoff for an educational variable may consider data for some or all household members. This can be done in many ways. Alternative deprivation cutoffs for the variable ‘years of schooling among adults aged 15 and above’ could be: if any household member has achieved a certain level, if any adult lacks a given level, if the women of the household reach a certain level, if a certain proportion of adults achieve that level, if (all or some) household members have levels that were appropriate when they were of school-going age, or if the educational achievements for at least one male and at least one female (or some other combination) each meet a certain standard. Empirical implementation and analysis of several definitions can be useful to understand the patterns of educational deprivations within households – and their accuracy, for example given the gender composition of households.

In other cases, deprivation cutoffs are set across subindices, such as defining housing deprivations if a person has substandard housing construction in terms of any two of: roof, flooring, walls. Again each subindice design requires independent and careful validation, which we do not cover.

Having fixed one set of deprivation cutoffs, a second vector of cutoffs may be constructed in which at least one indicator reflects more (less) extreme deprivation. This second vector can be implemented across the same indicators, weights, and poverty cutoff as previously to identify a subset of the poor who are in more (less) extreme poverty according to these more (less) exacting standards.[31]

In practice, it is common in multidimensional poverty design to construct indicators and candidate multidimensional poverty measures using various cutoff vectors, in order to assess the sensitivity of measures to a change in deprivation cutoffs, and also, in the case of uncertainty about which cutoff to choose, to clarify the implications of a choice to policy users. For example, Alkire and Santos (2010, 2014) implemented cutoffs such as ‘stunting’ ‘piped water into the dwelling’ or ‘flush toilet’ to understand whether country rankings changed dramatically if these standards were used instead of the chosen MDG cutoffs.

The selection of deprivation cutoffs enables the computation of uncensored headcount ratios for all indicators. Reasoned consideration of these ratios is quite important, to cross-check indicator selection and in weighting. For example, if the uncensored headcount ratio for an indicator is much lower than other indicators, that indicator will be unlikely to influence the measure; however, if changes in this indicator would be quite precise and if its normative importance is high, a large weight can be attributed to it, returning it to prominence. Also, suppose the indicators have been selected, following Atkinson et al. (2002), such that their importance and hence weights are ‘roughly equal’. If deprivation rates across indicators are exceedingly variable, then equal weights across indicators will produce a measure that is dominated by the indicators having the highest censored headcounts. We might do well to remind readers of the need to consider design issues iteratively rather than sequentially in practice.

6.3.7 Weights

Another key component of normative choices is the relative weight placed on dimensional deprivations. In multidimensional poverty measures weights could be applied (i) to each indicator (thus determining the relative importance of each indicator to the other as interpreted from the ratio of the weights); (ii) within an indicator (if a sub-index such as an asset index or housing index is constructed); and (iii) among people in the distribution, for example to give greater priority to the most disadvantaged. This section focuses solely on the first of these.

As people are diverse and our values differ both from each other and from ourselves at different points in time, the relative values that people place on different indicators of disadvantage vary.[32] This is no catastrophe. Sen observes, ‘It can, of course, be the case that the agreement that emerges on the weights to be used may be far from total’, but continues, ‘we shall then have some good reason to use ranges of weights on which we may find some agreement. This need not fatally disrupt evaluation of injustice or the making of public policy ... A broad range of not fully congruent weights could yield rather similar principal guidelines’ (2009: 243). Thus, as Chapter 8 suggests, robustness tests should be undertaken to assess whether the main policy prescriptions are robust to a range of weights or to show their sensitivity to alternative weighting structures.

The weights applied in the measure differ radically from weights in ‘composite’ indicators and are, for that reason, easier to set and to assess normatively. Critics of at times overlook the dramatic simplicity of weights in comparison with composite measures or multidimensional poverty measures that require cardinal data, so we begin by clarifying this important distinction.

Weights in composite measures are applied to quantities (achievement levels), and the marginal rates of substitution across indicators are usually assumed to be meaningful at all achievement levels.[33] We elsewhere clarified that, unlike , composite indices, including the Human Development Index (HDI), require ‘strong implicit assumptions on the cardinality and commensurability of the three dimensions of human development. The key implication is that after appropriate transformations, all variables are measured using a ratio scale in such a way that levels are comparable across dimensions’ (Alkire and Foster 2010a). This is rather stringent. To take a very straightforward case, in the original arithmetic HDI the weights govern the effect that an improvement in one dimension has on the overall HDI. The weights must accurately reflect the value of such a change, whether the increment occurs at the highest or the lowest level of achievement in that dimension. That is, changes from each starting level must be able to be justified separately and independently. Weights also govern trade-offs across all variables, for every increment of each variable. That is, the trade-off between an increment in variable A from any starting achievement level and corresponding increments in variables B and C would need to be justified—whatever the starting level those variables take (Ravallion 2012). Weights are thus used to compare changes in the same indicator at any level of achievement and trade-offs across variables. We might refer to them as ‘precision weights’.

Precision weights are also used in multidimensional poverty methodologies that require cardinal (normally ratio-scale) data, such as those proposed by Chakravarty, Mukherjee, and Ranade (1998), Tsui (2002), Bourguignon and Chakravarty (2003), Maasoumi and Lugo (2008), and Chakravarty and D’Ambrosio (2013), among others. Ratio-scale data are also required for measures when . In measures when , the deprivation cutoff creates a ‘natural zero’ and the normalized gaps for each indicator are understood to be cardinally meaningful. In this situation, in a manner similar to composite indices, the weights govern the impact that each increment or decrement in a deprived indicator has on poverty. Also similar to composite indices, weights govern trade-offs across all variables at all deprived levels of every variable.

In and other dichotomous counting approaches, weights are almost completely different. We may refer to them as deprivation values to mark this difference verbally. They are applied to the 0–1 deprivation status entry. Their function is to reflect the relative impact that the presence or absence of a deprivation has on the person’s deprivation score and thus on identification and, for poor people, on poverty. Correspondingly, the weights affect how much impact the removal of a particular deprivation has on . Thus they create comparability across dichotomized indicators (see section 2.4). But because deprivation values are applied to dichotomous 0 – 1 variables, they need not calibrate different levels of deprivations in a single variable. Further, because all indicators are dichotomous, the only possible trade-offs across deprivations (presence or absence) take the value of the relative weights. Put differently, because uses dichotomized deprivations rather than normalized gaps, deprivation values are not required to govern trade-offs across different levels of achievement in different variables as they are in the measures requiring precision weights. They only reflect the presence or absence of a deprivation. This greatly simplifies their selection and justification, and is worth noting clearly as the distinction between precision weights and deprivation values is often overlooked.

Due to an appreciation of democratic debate, and to permit values to evolve, as in the selection of capabilities, Sen does not commend any fixed-and-forever vector of weights: ‘The connection between public reasoning and the choice and weighting of capabilities in social assessment…also points to the absurdity of the argument that is sometimes presented, which claims that the capability approach would be usable—and ‘operational’—only if it comes with a set of ‘given’ weights on the distinct functionings in some fixed list of relevant capabilities’. In contrast, Sen advocates occasional public discussion for similar reasons to those given in the selection of dimensions: ‘The search for given, pre-determined weights is not only conceptually ungrounded, but it also overlooks the fact that the valuations and weights to be used may reasonably be influenced by our own continued scrutiny and by the reach of public discussion. It would be hard to accommodate this understanding with inflexible use of some pre-determined weights in a non-contingent form’ (2009: 242–3). In practice, for measures used to compute changes over time, it can be useful to fix the weights and other parameters for a given time period, such as a decade, and update them thereafter.

The selection of deprivation values also reflects the purpose of the exercise. For example, if the purpose is to evaluate changes in poverty, weights might reflect the fundamental importance people place on each indicator, whereas if the purpose is to monitor progress in the short or medium term, the relative weights might partly depict the relative priority of reductions in indicator deprivations. For example, if a region has very high levels of educational achievements but deeply rutted roads, then a long-term poverty measure may give a higher weight to education because of its importance and value. But a measure used for participatory planning may give higher priority weights to roads because of the pressing need for progress in this area.

The potential value of public discussion does not mean that weights must be created by participatory processes—although Sen would suggest that they be made transparent in order to catalyze such discussion. Thus Weights may be also be corroborated or justified using expert opinion; analyses of survey data, such as perceived necessities (see Chapter 4); subjective evaluations; or the input of policymakers and relevant authorities. They may reflect values implicit in a legal document or national plan, or use some socially accepted value structure that has been applied in poverty measurement or similar exercises previously.

The justification of weights is explored extensively both theoretically and practically by Wolff and De-Shalit, who reach the conclusion that, ‘… even though disadvantage is plural, indexing disadvantages is possible, despite various theoretical and practical problems …’ (2007: 181).

Their proposal is to use multiple methods and create measures whose key policy proposals are robust to them. Let us unpack this. In terms of setting weights, for example, they (like Sen) point out the need for a democratic procedure, but also recognize ‘that individual valuations might be liable to distortion, false consciousness, or the result of limited experience and thus ignorance of the real nature of various alternatives.’ Hence they justify including additional inputs. ‘Keeping both sides in play is sensitive to the fact that legitimacy in a democracy builds out of people’s voices’ while at the same time recognizing potential weaknesses of participatory processes (2007: 99).

In the end, Wollf and De-Shalit commend the creation of orderings that are robust to a range of plausible weights: ‘A social ordering is weighting sensitive to the degree that it changes with different weighting assignments to different categories. A social ordering, therefore, is weighting insensitive – robust – to the degree it does not change with different weighting assignments to the different categories’ (2007: 101–2).

So, first of all, the deprivation values that are used to create ‘relative weights’ across 0–1 deprivations are fundamentally straightforward, which simplifies matters. But there are plural ways to make and justify weights, which seems to reconstitute complexity. Happily, in fact, the plurality of potentially justifiable weights means that weights can be justified and cross-checked against different sources. Technically, given the legitimate pluralism in values, it would be desirable to implement a poverty measure with a range of weighting vectors and to release measures whose relevant policy implications were robust (Chapter 8 and Alkire and Santos 2014).

6.3.8 Poverty Cutoff k

The cross-dimensional poverty cutoff identifies each person as poor or non-poor according to the extent of deprivations they experience, which are summarized in their deprivation score. It establishes the minimum eligibility criteria for poverty in terms of breadth of deprivation. Normatively it reflects a judgement regarding the maximally acceptable set of deprivations a person may experience and not be considered poor. Thus the value of can only be justified after fully articulating the parameters described previously.

Like the income poverty line, the final choice of in most cases should be a normative one, with describing the minimum deprivation score associated with people who are considered poor and consider themselves to be poor (Sen 1979). For multidimensional measures the normative content could come from participatory processes in which poor people articulate the conditions and combinations of deprivations that constitute poverty. They may be informed by subjective poverty assessments and qualitative studies.[34] As noted by Tsui, ‘In the final analysis, how reasonable the identification rule is depends, inter alia, on the attributes included and how imperative these attributes are to leading a meaningful life’ (2002: 74). If, for example, deprivation in each dimension meant a terrible human rights abuse and data were highly reliable, then could be set at the minimal union level to reflect the fact that human rights are each essential, have equal status, and cannot be positioned in a hierarchical order (Alkire and Foster 2010b).

In some circumstances the value of could be chosen to reflect priorities and policy goals.[35] For example, if a subset of dimensions were essential while the rest may be replaced with one another, the weights and could be set accordingly. For example, suppose , and , and . A cutoff of would then identify as poor anyone who is either deprived in dimension 1 or in all the remaining dimensions while a slightly higher value of would require deprivation in the first dimension and in one other.[36] Alternatively one could select a cutoff whose resulting headcount identified the poorest segment of the population that the budgetary resources could address. Thus the weights and poverty cutoff allow for a range of identification constellations.

To justify and communicate the poverty cutoff, the relative values of deprivations (or possibly dimensions) should be explicitly considered. While technically the poverty cutoff can be set at any level, in practice a range of poverty cutoff values may identify the same group as poor. For example, if there are five equally weighted indicators, then a poverty cutoff of 21% will identify the same set of persons to be poor as a cutoff of 25%, 33%, or 40%. Given these weights, any person who has at least two deprivations will be identified as poor by any poverty cutoff taking the value 20<k<40%. Yet if communication is a priority, then a poverty cutoff value of 40% might be chosen, as it most intuitively conveys the fact that poor people are deprived in at least two out of the five (2/5) deprivations. When deprivations take different weights, of course, the distribution may be smoother, but communicating the poverty cutoff in terms of indicators or dimensions may remain relevant.

No matter which technique is finally employed in selecting the parameter , as in the case of income poverty lines, it should be routine to construct the poverty measure for a range of poverty cutoffs, to publish robustness results for alternative poverty cutoffs and/or to explore dominance tests across relevant values of . Techniques for doing so are set out in Chapter 8.

6.4 Concluding Reflections

Previous chapters drew readers’ attention to normative issues in poverty measurement by explaining and applying various axioms and properties. This chapter moved on to clarify the wider normative choices inherent in measurement design—after a methodology has been selected. These choices affect every step of measurement design. At a higher level, normative choices assess measures according to plural desiderata, and draw on empirical, political, and procedural considerations among others. More specific considerations can be drawn on, alongside empirical insights, to justify in each of the design choices.

Value judgements are not a one-shot game. In the interest of facilitating repeated and on-going self-critical consideration, we have argued that the design choices of multidimensional poverty measures should consider using deliberative processes, and that the normative issues and processes should be explicitly articulated in the public domain, in order that the public might both understand the existing parameters and be able to debate or improve upon them. To counterbalance and inform this flexibility, we suggest the use of empirical and statistical techniques, for example, to explore redundancy and to show whether key points of comparison are robust to a range of plausible parameter choices.[37] Clearly, the initial choice of parameters would be more difficult if important comparisons were sensitive to small adjustments in them. By applying robustness tests this sensitivity can be explored transparently. Before getting to robustness techniques in Chapter 8, the Chapter 7 addresses practical data considerations and relationships among indicators that are pertinent to multidimensional poverty measurement design and implementation, and suggests some specific methods to address these, even if imperfectly.


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[1] Chang (1997), Finnis (1997), Griffin (1986, 1996), Raz (1986), Brighouse and Robeyns (2010), Rawls (1971, 1993, 1999), Roemer (1996), Dworkin (1986, 2000), Galtung (1994), Nussbaum (2001), Reader (2005), Stewart (1985), Streeten et al. (1981), Wiggins (1998), Cohen (1989), Adler (2011), Fleurbaey and Maniquet (2011), Fleurbaey and Blanchet (2013), Parfit (1984), Wolff and De-Shalit (2007) and Deneulin (2014).

[2] Rawls (1971: 93) rather unfortunately referred to ‘weighting primary goods for the least advantaged, for those with the least authority and least income’ as the ‘index problem’ — a problem which critics allege that Rawlsians failed to ‘solve’. The title ‘index problem’ wrongly implied that value judgments are a ‘problem’ rather than an inherent component of measurement design.

[3] A clear and substantive part of Sen’s writing, which is not here reflected, is his criticism of prices as an alternative. For example, he writes, ‘While market prices certainly do exist for commodities, and do not for human functionings, the evaluative significance of the market prices has to be established. It is not obvious how, in making an evaluative judgement about progress, valuational decisions are to be avoided by simply “reading off” the market prices and the metric of exchange value. For one thing, the problems of externalities, missing markets, and other concerns will suggest that market prices be “adjusted”, and we have to decide whether such adjustments should be made, and if so, how this should be done’(1997: 30).

[4] Section 6.1 and Box 6.1 draw upon Alkire and Foster (2007).

[5] We are especially conscious in this chapter of being unable to cite or engage all the many scholars who have creative insights on measures of wellbeing and poverty that draw on the capability approach. Their work deserves its own in-depth constructive survey, building on other such surveys that already exist, including Chiappero-Martinetti and Roche (2009), Clark (2008), and Robeyns (2006), as well as references in recent applied work such as Arndt and Volkert (2011) and Van Ootegem and Verhofstadt (2012). We would refer interested readers to the Human Development and Capability Association (HDCA) and the bibliographies on capabilities published annually in the Journal of Human Development and Capabilities.

[6] This assumption can be relaxed to obtain more general results if required.

[7] Of course if direct information were available on coercion, this might also be either combined with functionings or as a separate indicator of agency deprivation. Sen writes, ‘Offering the opportunity to all to lead a minimally decent life need not be combined with an insistence that everyone makes use of all the opportunities that the state offers; for instance, making everyone entitled to an adequate amount of food does not have to be combined with a state ban on fasting’ (2009: 288). In the event that one had information on a ban on fasting that affected certain locations (or similar procedural unfreedoms), this might be used as an indicator alongside nutritional outcomes — with weights on each appropriately considered.

[8] Pattanaik and Xu subsequently extended their approach to reflect the quality of the option. A significant literature has further explored this theme, including Pattanaik and Xu (1990, 1998, 2000), Bavetta and Seta (2001), Fleurbaey (2002) Gekker (2001) Gravel (1994, 1998), and Klemisch-Ahlert (1993).

[9] Sen writes, ‘it can be argued that if we concentrate on certain basic general functionings and the corresponding capabilities, there may be much more agreement on their importance than there would be if we concentrated on particular commodity bundles and particular ways of achieving these functionings. For example, there is likely to be more intercultural—and also interpersonal—agreement on the importance of having the capability to avoid acute hunger or severe undernourishment than on the significance of having an adequate supply of particular food items (e.g. some specific types of meat or fish or grains or pulses) to serve those functionings’ (1992: 108–9). Such agreements may be incompletely theorized (Alkire 2002b, Ruger 2007, Sunstein 1996).

[10] The symbol refers to the th usual -dimensional basis vector whose th entry is equal to 1 and the rest of the elements are equal to 0. Note that semi-independence is a weakening of the property of ‘independence’ found in Pattanaik and Xu (1990).

[11] Symbol refers to a -dimensional vector of zeros.

[12] It must be acknowledged that the possibility (and desirability) that a poverty measure would reflect reasoned normative assessments across plural criteria does not mean that it will do so. Both income and multidimensional poverty measures can be manipulated, or carelesslyimplemented carelessly, and as a result be inaccurate. No measure is strategy proof. An advantage of the direct measure is that its construction is easy to communicate, and uses explicit weights and cutoffs, so an informed public and technical advisors can more readily identify and criticize serious shortcomings.

[13] Author’s translation.

[14] Many related sets of principles have been proposed, some of which include voice or processes. For example the Australian National Development Index (ANDI), which is described as: ‘Informed by experts but defined by Australians.’ It has the following principles: inclusive (everybody has a right to a say), independent (an apolitical index), accessible (easily understood information about well-being), trustworthy (best scientific practices in measurement and analysis), and balance (promote broad and integrated definitions of well-being).

[15] Naturally each of these may have subcomponents. For example, the need to be ‘technically solid may entail combining several component principles. ‘The possibility of combining is particularly important when some of the principles have a very narrow domain but are very persuasive over these domains. They may rank rather low in the context of ordering a much wider domain included in X, because of being silent on a great many comparisons. But since they are likely to be compatible with other principles with a narrow scope dealing with other types of comparisons, combining them may be both feasible as well as effective in generating highly valued rankings of X’ (Sen 1979: 126–7).

[16] This use of plural principles to identify that set of options ‘than which nothing is better’ is described in Sen (1985b, 1997b, 2009) and Alkire (2002b: ch. 3. 4.)

[17] The literature is vast. For perceptive overviews of different aspects, see Foster, Seth, et al. (2013), Haughton and Khandker (2009), Chen and Ravallion (2010), Jenkins and Micklewright (2007), Grusky and Kanbur (2006), Hentschel, and Lanjouw (2000), Klugman (2002), Banerjee and Duflo (2011), UNDESA (2010), World Bank (2000), Deaton (1992, 1997), Deaton and Grosh (2000), Ravallion (1992, 1996, 1998), and Anand (1983).

[18] See Chapter 8. For example, variable poverty line robustness methods such as those found in Foster and Shorrocks (1988b) helped make the fundamentally arbitrary $1.25-a-day poverty line more palatable.

[19] See Alkire and Foster (2009).

[20] Shaffer (2013) provides an excellent review of issues pertinent to multidimensional poverty measurement design in this extensive literature; cf. Addison Hulme and Kanbur (2009). Leavy et al (2013) is an examplary synthesis of participatory studies; cf. Narayan et al. (2000); Narayan and Petesch (2007).

[21] Many examples were already covered in Chapter 3 and Chapter 4. Additional relevant assessments discuss these issues for other measurement approaches, like Michalos et al. (2011), Japan (2011), Gunewardena (2004), and Wagle (2008: ch. 3).

[22] Sen’s argument that it could be useful to describe who is poor according to widespread judgements, even if resources do not suffice to address it (1979, 1997), supplements but does not rule out this option.

[23] Section 7.2.1 defines this as the entity that is identified as poor or non-poor—usually the individual or the household.

[24] The selection dimensions and/or indicators could reflect capabilities and functionings, with dimensions being broader (being educated) and indicators being more specific (finishing primary school). So the discussion on capabilities that follows should be considered as applying to both the present and the following sections. See Alkire (2002a, 2002b), Nussbaum (2003), Robeyns (2003, 2005), Sen (2004), Burchardt (2013), and Burchardt and Vizard (2007, 2011).

[25] Narayan et al. (2000).

[26] UNDG (2013).

[27] The field is large and thriving. For a recent overview see Land et al. (2012a,b); see also Chapters 3, 4 and 7 of this book and journals such as Social Indicators Research. Tests are applied to single and composite indicators. Hagerty (2001) was a pioneering assessment using fourteen criteria; cf. Sirgy et al. (2006), Nardo et al. (2008), Zumbo (1998) and Andrews (1986). For recent applications, see Guio et al. (2009, 2012), Whelan and Maître (2010), and Fucso et al. (2011).

[28] Foster, Seth, et al. (2013), Haughton and Khandker (2009), and Ravallion (2012.) Naturally standards too provoke debates. For a thoughtful exchange on the global poverty figures, see Anand, Segal, and Stiglitz (2010).

[29] Applications of the AF methodology use the same deprivation cutoffs for all units in a poverty measure. It could be an interesting research exercise to vary the cutoffs according to locally defined standards, although doing so would change certain properties of the measure.

[30] See Alkire and Santos (2010, 2014) for a description of how the global MPI cutoffs reflect international standards. For examples of cutoffs based on legal standards, see CONEVAL (2009); on cutoffs reflecting a national development plan, see Angulo et al. (2013).

[31] For example, Alkire, Conconi, and Seth (2014b) implement a measure of destitution for many countries in which eight of the global MPI cutoffs are lowered and the same weights and poverty cutoff of 33% is used. CONEVAL’s definition of extreme poverty changes the cutoff of income poverty from the basic needs poverty line to the food basket poverty line and simultaneously raises the cutoff from strictly greater than 50% to 75% (using a higher poverty cutoff still creates a strict subset of the extreme poor).

[32] It is empirically possible with AF measures, and would be of research interest, to apply individual or group-based weighting vectors to each deprivation vector or set of vectors, rather than applying the same weighting vector to all. Doing so will change the properties of the measure, as well as its interpretation and that of consistent indices. It would also imply that each individual had fixed consistent preferences regarding that indicator, which is a strong assumption. Yet such research could be of interest to those probing individual and group preferences (Adler 2011; Fleurbaey and Maniquet 2012; Decancq 2012).

[33] Decancq and Lugo (2012) clearly and thoroughly explain the role of what we call ‘precision’ weights and discuss eight approaches to setting these, which are grouped into data-driven, normative, or hybrid weight categories; see Decancq et al. (2011, 2013) and Ravallion (2011b).

[34] Ravallion presents the rationale for income poverty lines in terms of ‘subjective’ poverty lines which, it seems, reflect people’s values and views on poverty: ‘Arguably then, what one is doing in setting an objective poverty line in a given country is attempting to estimate the country’s underlying “subjective poverty line.” A close correspondence between subjective and objective poverty lines can then be expected, though arguably it is the subjective poverty line which can then claim to be the more fundamental concept for poverty analysis’ (1998: 30). Given the potential for manipulation and adaptation (Clark 2012), as well as issues of comparability due to different aspirations (Narayan et al. 2007), purely subjective assessments of poverty should be combined with qualitative and participatory insights.

[35] Care must be taken in this case: Sen argued early on that ‘the non-availability of public resources to help eliminate severe deprivations should not make us redefine poverty itself’ (1992: 108; see also 1979).

[36] Such a weighting structure is used in Mexico (CONEVAL 2010); cf. Nolan and Whelan (1996).

[37] For example, see Alkire and Foster (2011a) and Lasso de la Vega (2010); see also Alkire and Santos (2010, 2014) and Alkire, Santos, Seth, and Yalonetzky (2010). A stringent and full set of dominance conditions that ensure the robustness of comparisons to widely varying weights, deprivation cutoffs, and poverty cutoffs, has been derived in Yalonetzky (2012). Statistical tests for these conditions are available in Yalonetzky (2014) for discrete variables and in Anderson (2008) for continuous variables. Bennett and Mitra (2013) propose a test for multiple hypotheses that allows the researcher to check the robustness of the cutoffs and compare two groups.

5 The Alkire-Foster Counting Methodology

This chapter provides a systematic overview of the multidimensional measurement methodology of Alkire and Foster (2007, 2011a), with an emphasis on the first measure of that class: the Adjusted Headcount Ratio or . It builds on previous chapters, which demonstrated the importance of adopting a multidimensional approach (Chapter 1), introduced the general framework (Chapter 2), and reviewed the different alternative methods for multidimensional measurement and analysis (Chapter 3). Chapter 3 also highlighted the advantages of certain axiomatic measures that consider the joint distribution of deprivations and exhibit a transparent and predictable behaviour with respect to different types of transformations. The fourth chapter reviewed counting methods to identify the poor (Chapter 4), which are frequently used in axiomatic measures.

Why focus on the AF methodology and on  in particular? As argued in 1.3, we focus on the AF methodology for a number of technical and practical reasons. From a technical perspective, being an axiomatic family of measures, the AF measures satisfy a number of desirable properties introduced in section 2.5, detailed in this chapter. From a practical perspective, the AF family of measures uses the intuitive counting approach to identify the poor, and explicitly considers the joint distribution of deprivations. Among the AF measures, the  measure is particularly applicable due to its ability to use ordinal or binary data rigorously and because the measure and its consistent partial indices are intuitive. The technical and practical advantages of  make it a particularly attractive option to inform policy.

It is worth noting from the beginning that the AF methodology is a general framework for measuring multidimensional poverty, although it is also suitable for measuring other phenomena (Alkire and Santos 2013). With the AF method, many key decisions are left to the user. These include the selection of the measure’s purpose, space, unit of analysis, dimensions, deprivation cutoffs (to determine when a person is deprived in a dimension), weights or values (to indicate the relative importance of the different deprivations), and poverty cutoff (to determine when a person has enough deprivations to be considered poor). This flexibility enables the methodology to have many diverse applications. The design of particular measures—which entail value judgements—is the subject of Chapter 6.

As described in section 2.2.2, the methodology for measuring multidimensional poverty consists of an identification and an aggregation method (Sen 1976). This chapter first describes how the AF methodology identifies people as poor using a ‘dual-cutoff’ counting method, standing on the shoulders of a long tradition of counting approaches that have been used in policy making (Chapter 4). The aggregation method builds on the unidimensional axiomatic poverty measures and directly extends the Foster–Greer–Thorbecke (1984) class of poverty measures introduced in section 2.1. The main focus of this chapter is the Adjusted Headcount Ratio , which reflects the incidence of poverty and the intensity of poverty, capturing the joint distribution of deprivations. The chapter shows how to ‘drill down’ into  in order to unfold the distinctive partial indices that reveal the intuition and layers of information embedded in the summary measure, such as poverty at subgroup levels and its composition by dimension.  Examples illustrate the methodology and also present standard tables and graphics that are used to convey results.

This chapter proceeds as follows. Section 5.1 presents the overview and practicality of the AF class of poverty measures, focusing especially on the Adjusted Headcount Ratio. Section 5.2 sets out the identification of who is poor using the dual-cutoff approach. Section 5.3 outlines the aggregation method used to construct the Adjusted Headcount Ratio. Section 5.4 presents the main distinctive characteristics of the Adjusted Headcount Ratio and section 5.5 presents its useful, consistent partial indices or components. We present a case study of the Adjusted Headcount Ratio using the global Multidimensional Poverty Index in section 5.6. Section 5.7 presents the members of the AF class of measures that can be constructed in the less common situations where data are cardinal, along with their properties and partial indices. Finally, section 5.8 reviews some empirical applications of the AF methology.

5.1     The AF Class of Poverty Measures: Overview and Practicality

The AF methodology of multidimensional poverty measurement creates a class of measures that both draws on the counting approach and extends the FGT class of measures in natural ways. Before proceeding with a more formal description of the AF methodology, we first provide a stepwise synthetic and intuitive presentation of how to obtain the Adjusted Headcount Ratio (), which is our focal measure. We also introduce the Adjusted Poverty Gap () and the Adjusted Squared Poverty Gap (or FGT) Measure (). For clarity, we distinguish the steps that belong to the identification step and those that belong to the aggregation step.

One constructs these  measures as follows:


  1. Defining the set of indicators which will be considered in the multidimensional measure. Data for all indicators need to be available for the same person.
  2. Setting the deprivation cutoffs for each indicator, namely the level of achievement considered sufficient (normatively) in order to be non-deprived in each indicator.
  3. Applying the cutoffs to ascertain whether each person is deprived or not in each indicator.
  4. Selecting the relative weight or value that each indicator has, such that these sum to one.[1]
  5. Creating the weighted sum of deprivations for each person, which can be called his or her ‘deprivation score’.
  6. Determining (normatively) the poverty cutoff, namely, the proportion of weighted deprivations a person needs to experience in order to be considered multidimensionally poor, and identifying each person as multidimensionally poor or not according to the selected poverty cutoff.


  1. Computing the proportion of people who have been identified as multidimensionally poor in the population. This is the headcount ratio of multidimensional poverty , also called the incidence of multidimensional poverty.
  2. Computing the average share of weighted indicators in which poor people are deprived. This entails adding up the deprivation scores of the poor and dividing them by the total number of poor people. This is the intensity of multidimensional poverty , also sometimes called the breadth of poverty.
  3. Computing the  measure as the product of the two previous partial indices: . Analogously,  can be obtained as the mean of the vector of deprivation scores, which is also the sum of the weighted deprivations that poor people experience, divided by the total population.

When all indicators are ratio scale, one may also compute  and  as follows:

  1. Computing the average poverty gap across all instances in which poor persons are deprived, . This entails computing the normalized deprivation gap as defined in equation (2.2): () for each person and indicator. The normalized gap is the difference between the deprivation cutoff and the poor person’s achievement for each indicator, divided by its deprivation cutoff. If a person’s achievement does not fall short of the deprivation cutoff, the normalized gap is zero. The average poverty gap is the mean of poor people’s weighted normalized deprivation gaps in those dimensions in which poor people are deprived and is one of the partial indices. This depth of multidimensional poverty is denoted by .
  2. Computing the  measure as the product of three partial indices: . Analogously,  can be obtained as the sum of the weighted deprivation gaps that poor people experience, divided by the total population.
  3. Computing the average severity of deprivation across all instances in which poor persons are deprived, . This entails computing the squared deprivation gap, that is, squaring each normalized gap computed in step 10. The average severity of deprivation is the mean of poor people’s weighted squared deprivation gaps in those dimensions in which they are deprived. This is the severity of multidimensional poverty, .
  4. Computing the  measure as the product of the following partial indices: . Analogously,  can be obtained as the sum of the weighted squared deprivation gaps that poor people experience, divided by the total population.

Note that in all three cases (,  and the deprivations experienced by people who have not been identified as poor (i.e. those whose deprivation score is below the poverty cutoff) are censored, hence not included; this censoring of the deprivations of the non-poor is consistent with the property of ‘poverty focus’ which—analogous to the unidimensional case—requires a poverty measure to be independent of the achievements of the non-poor. For further discussion see Alkire and Foster (2011a).

These three measures of the AF family, as well as any other member, satisfy many of the desirable properties introduced in section 2.5. Several properties are key for policy. The first is decomposability, which allows the index to be broken down by population subgroup (such as region or ethnicity) to show the characteristics of multidimensional poverty for each group. All AF measures satisfy population subgroup decomposability. So the poverty level of a society—as measured by any —is equivalent to the population-weighted sum of subgroup poverty levels, where subgroups are mutually exclusive and collectively exhaustive of the population.

All AF measures can also be unpacked to reveal the dimensional deprivations contributing the most to poverty for any given group. This second key property—post-identification dimensional breakdown (section 2.2.4) — is not available with the standard headcount ratio and is particularly useful for policy.

The AF measures also satisfy dimensional monotonicity, meaning that whenever a poor person ceases to be deprived in a dimension, poverty decreases. The headcount ratio does not satisfy this.  Dimensional monotonicity and breakdown both use the partial index of intensity.

A few comments on the AF class before we turn to the final key property for policy. All AF measures also have intuitive interpretations. The Adjusted Headcount Ratio reflects the proportion of weighted deprivations the poor experience in a society out of the total number of deprivations this society could experience if all people were poor and were deprived in all dimensions. The Adjusted Poverty Gap reflects the average weighted deprivation gap experienced by the poor out of the total number of deprivations this society could experience. The Adjusted Squared Poverty Gap Measure  reflects the average weighted squared gap or poverty severity experienced by the poor out of the total number of deprivations this society could experience. In all cases, the term ‘adjusted’ refers to the fact that all measures incorporate the intensity of multidimensional poverty—which is key to their properties.

Additionally, while each AF measure offers a summary statistic of multidimensional poverty, they are related to a set of consistent and intuitive partial indices, namely, poverty incidence (), intensity (), and a set of subgroup poverty estimates and dimensional deprivation indices (which in the case of the  measure are called censored headcount ratios) and their corresponding percent contributions. Each  measure can be unfolded into an array of informative indices.

Among the AF class of measures, the  measure is particularly important because it can be implemented with ordinal data. This is critical for real-world applications. It is relevant when poverty is viewed from the capability perspective, for example, since many key functionings are commonly measured using ordinal variables. The  measure satisfies the ordinality property introduced in section 2.5.1. This means that for any monotonic transformation of the ordinal variable and associated cutoff, overall poverty as estimated by  will not change. Moreover,  has a natural interpretation as a measure of ‘unfreedom’ and generates a partial ordering that lies between first- and second-order dominance (Chapter 6). Because of its intuitiveness and practicality, this book mainly focuses on .

The remaining sections present the AF method more precisely yet, we hope, intuitively.

5.2 Identification of the Poor: The Dual-Cutoff Approach

Poverty measurement requires some identification function, which determines whether each person is to be considered poor. The unidimensional form of identification, discussed in section 2.2.1,  entails a host of assumptions that restrict its applicability in practice and its desirability in principle.[2] From the perspective of the capability approach, a key conceptual drawback of viewing multidimensional poverty through a unidimensional lens is the loss of information on dimension-specific shortfalls; indeed, aggregation before identification converts dimensional achievements into one another without regard to dimension-specific cutoffs. In situations where dimensions are intrinsically valued and dimensional deprivations are inherently undesirable, there are good reasons to look beyond a unidimensional approach to identification methods that focus on dimensional shortfalls.

In the multidimensional measurement setting, where there are multiple variables, identification is a substantially more challenging exercise. As explained in section 2.2.2, a variety of methods can be used for identification in multidimensional poverty measurement. Here we follow a censored achievement approach. This approach first requires determining who is deprived in each dimension by comparing the person’s achievement against the corresponding deprivation cutoff and thus considering only deprived achievements (and ignoring—or censoring—achievements above the deprivation cutoff) for the identification of the poor. One prominent method used within the censored achievement approach is the counting approach, which is precisely the identification approach followed in the AF methodology, among others (Chapter 4).

As we have seen, a counting approach first identifies whether a person is deprived or not in each dimension and then identifies a person as poor according to the number (count) of deprivations she experiences. Note that ‘number’ here has a broad meaning as dimensions may be weighted differently. As reviewed in Chapter 4, the use of a counting approach to identification in multidimensional poverty measurement is not new. However, the value added of the AF methodology is threefold. In the first place, the AF methodology has formalized the counting approach to identification into a dual-cutoff approach, clarifying the requirement of two distinct sets of thresholds to define poverty in the multidimensional context. One is the set of deprivation cutoffs, which identify whether a person is deprived with respect to each dimension. Then, a (single) poverty cutoff delineates how widely deprived a person must be in order to be considered poor.

Second, as a consequence of using a dual-cutoff approach, the AF methodology considers the joint distribution of deprivations at the identification step and not just at the aggregation step, as previous methodologies did (almost all non-counting methodologies used the union criterion). Third, the AF methodology has integrated the counting approach to identification with an aggregation methodology that extends the unidimensional FGT measures, overcoming the limitations of the headcount ratio (which most counting methods used) yet allowing intuitive interpretations.[3]

Thus the AF methodology draws together the counting traditions – widely known for their practicality and policy appeal – and the widely used FGT class of axiomatic measures in order to assess multidimensional poverty, and stands on the shoulders of both traditions.

5.2.1 The Deprivation Cutoffs: Identifying Deprivations and Obtaining Deprivation Scores

Bourguignon and Chakravarty (2003) contend that ‘a multidimensional approach to poverty defines poverty as a shortfall from a threshold on each dimension of an individual’s wellbeing’.[4]  Following them and the plethora of counting methods reviewed in Chapter 4, the AF measures use a deprivation cutoff for each dimension, defined and applied as described in this section.

As introduced in section 2.2, the base information in multidimensional poverty measurement is typically represented by an  dimensional achievement matrix , where  is the achievement of person  in dimension . For simplicity, as done in section 2.2, it is assumed that achievements can be represented by non-negative real numbers (i.e. ) and that higher achievements are preferred to lower ones.

For each dimension , a threshold  is defined as the minimum achievement required in order to be non-deprived. This threshold is called a deprivation cutoff. Deprivation cutoffs are collected in the -dimensional vector . Given each person’s achievement in each dimension , if the ith person’s achievement level in a given dimension  falls short of the respective deprivation cutoff , the person is said to be deprived in that dimension (that is, if ). If the person’s level is at least as great as the deprivation cutoff, the person is not deprived in that dimension.

As Chapter 2 introduced, from the achievement matrix  and the vector of deprivation cutoffs , one can obtain a deprivation matrix  such that  whenever  and , otherwise, for all  and for all . In other words, if person  is deprived in dimension , then the person is assigned a deprivation status value of 1, and 0 otherwise. The matrix  summarizes the deprivation status value of all people in all dimensions of matrix . The vector  summarizes the deprivation status values of person  in all dimensions, and the vector  summarizes the deprivation status values of all persons in dimension .

The deprivation in each of the  dimensions may not have the same relative importance. Thus, a vector  of weights or deprivation values is used to indicate the relative importance of a deprivation in each dimension. The deprivation value attached to dimension  is denoted by . If each deprivation is viewed as having equal importance, then this is a benchmark ‘counting’ case. If deprivations are viewed as having different degrees of importance, general weights are applied using a weighting vector whose entries vary, with higher weights indicating greater relative value.

Intricate weighting systems create challenges in interpretation, so it can be useful to choose the dimensions such that the natural weights among them are roughly equal or else to group dimensions into categories that have roughly equal weights (Atkinson 2003). The deprivation values affect identification because they determine the minimum combinations of deprivations that will identify a person as being poor. They also affect aggregation by altering the relative contributions of deprivations to overall poverty (for more on weights see Chapter 6). Yet importantly the deprivation values do not function as weights that govern trade-offs between dimensions for every possible combination of ratio-scale achievement levels, as they do in a traditional composite index.  Because each deprivation status value is binary, the role of deprivation values differs from the role of weights in traditional composite indices.

Based on the deprivation profile, each person is assigned a deprivation score that reflects the breadth of each person’s deprivations across all dimensions. The deprivation score of each person is the sum of her weighted deprivations. Formally, the deprivation score is given by . The score increases as the number of deprivations a person experiences increases, and reaches its maximum when the person is deprived in all dimensions. A person who is not deprived in any dimension has a deprivation score equal to 0. We denote the deprivation score of person  by  and the column vector of deprivation scores for all persons by .

5.2.2    Alternative Notation and Presentation

Distinct notational presentations can be employed for the weights, deprivation scores, deprivation score vector, poverty cutoff, poverty measures, and partial indices. Substantively, alternative presentations are identical in that they each identify precisely the same persons as poor and generate the same poverty measure value and identical partial indices. What differ are the numerical values of weights, deprivation scores, and poverty cutoff. For didactic purposes we explain the main options so as to avoid confusion among researchers using different notational conventions.

Alternative notations arise from two decisions. The first decision is whether to define weights that sum to one, i.e. , or whether weights sum to the number of dimensions under consideration, . We refer to the first as normalized weights and to the second as non-normalized or numbered weights. The normalized weight of a dimension reflects the share (or percentage) of total weight given to a particular dimension. The deprivation score then shows the percentage of weighted dimensions in which a person is deprived and lies between 0 and 1. In the numbered case, deprivation scores range between 0 and . If person  is deprived in all dimensions, then . Depending on the weighting structure, one of these options may be more intuitive than the other. For example, if dimensions are equally weighted, the deprivation count vector shows the number of dimensions in which each person is deprived. Thus, while in the normalized case one may state that a person is deprived in 43% of the weighted dimensions, in the non-normalized case one states that a person is deprived in three out of seven dimensions, which is more intuitive. However, if dimensions are not equally weighted, as is common in practice, normalized weights may be more intuitive. Suppose there are seven dimensions and a person is deprived in two dimensions having weights of 25% and 10%, respectively. Their numbered deprivation score would be 2.45 = (0.25*7 + 0.10*7). This same situation could be communicated more intuitively by saying that this person is deprived in 35% of the weighted dimensions.

The second decision is whether to express the formulas using the deprivation matrix  and the (explicitly separate) weighting vector  in an explicit way, or whether to express them in terms of a weighted deprivation matrix denoted by  such that  if  and  if . These two decisions lead to four possible—but totally equivalent—notations, as detailed in Box 5.7. This chapter, and most of this book, uses normalized weights and expresses formulas using the deprivation matrix and the weight vector. We refer to this as Method I. Method II uses normalized weights with the weighted deprivation matrix. Method III uses non-normalized weights and expresses formulas using the deprivation matrix and the weight vector. Methods II and III are not further discussed in this chapter, but all the formulas are stated in Box 5.7. Finally, Method IV uses non-normalized weights and expresses the formulas using the weighted deprivation matrix, aligned with the notation used in Alkire and Foster (2011a), which is presented in Box 5.3, Box 5.6, and Box 5.7. What is particularly elegant about Method IV is that the AF measures can be expressed as the mean of the relevant censored deprivation matrix, as we shall elaborate subsequently.

5.2.3    The Second Cutoff: Identifying the Poor

In addition to the deprivation cutoffs , the AF methodology uses a second cutoff or threshold to identify the multidimensionally poor. This is called the poverty cutoff and is denoted by . The poverty cutoff is the minimum deprivation score a person needs to exhibit in order to be identified as poor. This poverty cutoff is implemented using an identification function , which depends upon each person’s achievement vector   the deprivation cutoff vector , the weight vector , and the poverty cutoff . If the person is poor, the identification function takes on a value of 1; if the person is not poor, the identification function has a value of 0. Notationally, the identification function is defined as  if and  otherwise. In other words,  identifies person  as poor when his or her deprivation score is at least ; if the deprivation score falls below the cutoff , then person  is not poor according to . Since  is dependent on both the set of within-dimension deprivation cutoffs  and the across-dimension cutoff ,  is referred to as the dual cutoff method of identification, or sometimes as the ‘intermediary’ method.

Within the counting approach to identification, the most commonly used multidimensional identification strategy is the union criterion.[5] Most of the poverty indices discussed in Chapter 3 use the union criterion, by which a person  is identified as multidimensionally poor if she is deprived in at least one dimension (). At the other extreme, another identification criterion is the intersection criterion, which identifies person  as being poor only if she is deprived in all dimensions (). Both these approaches have the advantage of identifying the same people as poor regardless of the relative weights set on the dimensions. But the identification of who is poor in each case is exceedingly sensitive to the choice of dimensions. Also these strategies can be too imprecise for policy: in many applications, a union identification identifies a very large proportion of the population as poor, whereas an intersection approach identifies a vanishingly small number of people as poor. A natural middle-ground alternative is to use an intermediate cutoff level for  that lies somewhere between the two extremes of union and intersection.

The AF dual-cutoff identification strategy provides an overarching framework that includes the two extremes of union and intersection criteria and also the range of intermediate possibilities.[6] Notice that  includes the union and intersection methods as special cases. In the case of union, the poverty cutoff is less than or equal to the dimension with the lowest weight:  Whereas in the case of intersection, the poverty cutoff takes its highest possible value of . In Box 5.1, we present different identification strategies using an example. 

Box 5.1 Different Identification Strategies: Union, Intersection, and Intermediate Cutoff

Suppose there is a hypothetical society containing four persons and multidimensional poverty is analysed using four dimensions: standard of living as measured by income, level of knowledge as measured by years of education, nutritional status as measured by Body Mass Index (BMI), and access to public services as measured by access to electricity. The  matrix  contains the achievements of four persons in four dimensions.


For example, the income of Person 3 is 400 Units; whereas Person 4’s is 800 Units. Person 1 has completed fourteen years of schooling; whereas Person 2 has completed thirteen years of schooling. Person 3 is the only person who is malnourished of all four persons. Two persons in our example have access to improved sanitation. Thus, each row of matrix  contains the achievements of each person in four dimensions, whereas each column of the matrix contains the achievements of four persons in each of the four dimensions. All dimensions are equally weighted and thus the weight vector is . The deprivation cutoff vector is denoted by  (500, 5, Not malnourished, Has access to improved sanitation), which is used to identify who is deprived in each dimension. The achievement matrix  has three persons who are deprived (see the underlined entries) in one or more dimensions. Person 1 has no deprivation at all.

Based on the deprivation status, we construct the deprivation matrix , where a deprivation status score of 1 is assigned if a person is deprived in a dimension and a status score of 0 is given otherwise.


The weighted sum of these status scores is the deprivation score () of each person. For example, the first person has no deprivation and so the deprivation score is 0, whereas the third person is deprived in all dimensions and thus has the highest deprivation score of 1. Similarly, the deprivation score of the second person is  (). The union identification strategy identifies a person as poor if the person is identified as deprived in any of the four dimensions. In that case, three of the four persons are identified as poor. On the other hand, an intersection identification strategy requires that a person is identified as poor if the person is deprived in all dimensions. In that case, only one of four persons is identified as poor in this case. An intermediate approach sets a cutoff between union and intersection, say, , which is equivalent to being deprived in two of four equally weighted dimensions. This strategy identifies a person as poor if the person is deprived in half or more of weighted dimensions, which in this case means that two of the four persons are identified as poor.

The dual-cutoff identification strategy has a number of characteristics that deserve mention. First, it is ‘poverty focused’ in that an increase in an achievement level  of a non-poor person leaves its value unchanged. Second, it is ‘deprivation focused’ in that an increase in any non-deprived achievement  leaves the value of the identification function unchanged; in other words, a person’s poverty status is not affected by changes in the levels of non-deprived achievements. This latter property separates  from the ‘aggregate achievement’ approach which allows a higher level of achievement to compensate for lower levels of achievement in other dimensions. Finally, the dual-cutoff identification method can be meaningfully used with ordinal data, since a person’s poverty status is unchanged when an admissible transformation is applied to an achievement level and its associated cutoff.

5.2.4    Dual-Cutoff Approach and Censoring

The transition between the identification step and the aggregation step is most easily understood by examining a progression of matrices. There are two kinds of censoring, each of which follows the application of the two kinds of cutoffs: deprivation and poverty. By applying the deprivation cutoffs to the achievement matrix , we constructed the deprivation matrix  replacing each entry in  that is below its respective deprivation cutoff  with 1 and each entry that is not below its deprivation cutoff with 0. This is the first censoring, because the achievements above their corresponding deprivation cutoff are converted into 0. The deprivation matrix provides a snapshot of who is deprived in which dimension.

Next, the poor are identified by applying the poverty cutoff  and thus a new matrix can be obtained from the deprivation matrix: the censored deprivation matrix, which is denoted by . Each element in  is obtained by multiplying the corresponding element in  by the identification function . Formally,  for all  and for all . What does this do?  If person  is poor and thus , then the person’s deprivation status in every dimension remains unchanged and so does the row containing the deprivation information of the person. If person  is not poor and thus , then their deprivation status in every dimension becomes 0, which is equivalent to censoring the deprivations of persons who are not poor. This second censoring step is key to the AF methodology. As we will see in subsequent sections, the censored deprivation matrices embody the identification step and are the basic constructs used in the aggregation step.

From the censored deprivation matrix, a censored deprivation score can be obtained. This applies the identification function to the original deprivation score vector used to identify the poor. The censored deprivation score of person  is denoted by , and can be obtained as . The censored deprivation score vector is denoted by . Note that by definition,  has been censored of all deprivations that are less than the value of . Thus, when , then  (deprivation score of the person), but if , then .[7]

Note that there is one case where the second censoring is not relevant: when the poverty cutoff  corresponds to the union approach, then any person who is deprived in any dimension is considered poor and the censored and original matrices are identical.

Although the censored matrices are used to construct multidimensional poverty measures, the original deprivation matrix still provides useful information, as we shall see later in constructing ‘raw’ or uncensored deprivation headcount ratios by dimension and analysing their change over time.

Before moving on to the aggregation step to create the Adjusted Headcount Ratio, let us provide an example of how to obtain the censored deprivation score vector from an achievement matrix in Box 5.2.

Box 5.2 Obtaining the Censored Deprivation Score Vector from an Achievement Matrix

Consider the  achievement matrix  and the deprivation cutoff vector  in Box 5.1. As earlier, each of the four dimensions receives a weight equal to 0.25 and weights sum to one. Assume in this case that a person is identified as poor if deprived in half or more of the four equally weighted dimensions, i.e. .

The achievement matrix  has three persons who are deprived in one or more dimensions. Based on the deprivation status, a deprivation matrix  is constructed in which a deprivation status score of 1 is assigned if a person is deprived in a dimension and a status score of 0 is given otherwise. The weighted sum of these status scores yields the deprivation score of each person .

Note that two persons (second and third) have deprivation scores that are greater than or equal to 0.5. They are considered to be poor (), and hence their entries in the censored deprivation matrix are the same as in the deprivation matrix. However, the fourth person has a single deprivation and hence is not poor. This single deprivation is censored in the censored deprivation matrix, which only displays the deprivations of the poor, as depicted below.[8]


5.3     Aggregation: The Adjusted Headcount Ratio

The aggregation step of our methodology builds upon the FGT class of unidimensional poverty measures and likewise generates a parametric class of measures. Just as each FGT measure can be viewed as the mean of an appropriate vector built from the original data and censored using the poverty line, the Adjusted Headcount Ratio is the mean of the censored deprivation score vector:



This section elaborates the Adjusted Headcount Ratio; the other measures in the AF class are presented in section 5.7.

A second way of viewing  is in terms of partial indices—measures that provide basic information on a single aspect of poverty. The Adjusted Headcount Ratio, denoted as , can also be written as the product of two partial indices. The first partial index  is the percentage of the population that is poor or the multidimensional headcount ratio or the incidence of poverty. The second index  is the intensity of poverty.




The headcount ratio or poverty incidence  is the proportion of the population that is poor. It is defined as , where  is number of persons identified as poor using the dual-cutoff approach.[9]

In turn, poverty intensity  is the average deprivation score across the poor. Notice that the censored deprivation score  represents the share of possible deprivations experienced by a poor person . So the average deprivation score across the poor is given by . Like the poverty gap information in income poverty, this partial index conveys relevant information about multidimensional poverty, in that persons who experience simultaneous deprivations in a higher  fraction of dimensions have a higher intensity of poverty and are poorer than others having a lower intensity.

Thus,  is given by



As a simple product of the two partial indices  and , the measure  is sensitive to the incidence and the intensity of multidimensional poverty. It clearly satisfies dimensional monotonicity, since if a poor person becomes deprived in an additional dimension, then  rises and so does . Another interpretation of  is that it provides the share of weighted deprivations experienced by the poor divided by the maximum possible deprivations that could possibly be experienced if all people were poor and were deprived in all dimensions.

Let us provide an example using the same censored deprivation matrix and the censored deprivation score vector as in Box 5.2.


The headcount ratio  is the proportion of people who are poor, which is two out of four persons in the above matrix. The intensity  is the average deprivation share among the poor, which in this example is the average of  and , i.e. equal to . It is easy to see that the multidimensional headcount ratio . In this example   and , so . It is straightforward to verify that  is the average of all elements in the censored deprivation score vector , i.e. . Analogously, it is equivalent to compute  as the weighted sum of deprivation status values divided by the total number of people: .

Box 5.3 An Alternative Presentation of the Adjusted Headcount Ratio Using Non-Normalized Weights

We have outlined the different expressions in terms of normalized weights (Method I in Box 5.7). Let us provide an alternative approach for computing the Adjusted Headcount Ratio when the weights are non-normalized such that  and , i.e. adding to the total number of dimensions, following the notation presented in Alkire and Foster (2011a). In order to do so, we need to introduce the weighted deprivation matrix. From the deprivation matrix, a weighted deprivation matrix can be constructed by replacing the deprivation status value of a deprived person with the value or weight assigned to the corresponding dimension. Formally, we denote the weighted deprivation matrix by  such that  if  and  if . Like the censored deprivation matrix, the censored weighted deprivation matrix  can be constructed such that  for all  and all . From the weighted deprivation matrix , the Adjusted Headcount Ratio can be defined as



That is,  is the mean of the weighted censored deprivation matrix. Thus, the Adjusted Headcount Ratio is the sum of the weighted censored deprivation status values of the poor or  divided by the highest possible sum of weighted deprivation status values, or .

Let us provide an example and show how the Adjusted Headcount Ratio is computed using this approach. Recall this deprivation matrix in Box 5.1. In this example, suppose the dimensions are unequally weighted and the weight vector is denoted by . Note that the weights sum to the number of dimensions. The weighted deprivation matrix  for this example can be denoted as follows:


The deprivation score of each person is obtained by summing the weighted deprivations. For example, the third person is deprived in all dimensions and so receives a deprivation score equal to four. Similarly, the fourth person is deprived only in the second dimension, which is assigned a weight of 1 and so her deprivation score is 1. If the poverty cutoff is , then only two persons are identified as poor. The censored weighted deprivation matrix can be obtained from the censored deprivation matrix as shown below.


The sum of the weighted deprivation status values of the poor is six. The highest possible sum of weighted deprivation status values is . Thus, .

Box 5.4 An Alternative Notation of the Identification Function

The application of the identification function can also be shown explicitly using another notation. An identification function  takes a value of 1 if the indicated condition   is true for the th person, and 0 otherwise, such that  if  and  otherwise.

In this notation, the identification function for the th person is multiplied by the weighted deprivation score  of the th person. This censors (replaces by 0) the deprivations of the non-poor. The sum of deprivation scores thus censored by the identification function, divided by , provides the value of .



The headcount ratio or incidence of multidimensional poverty  can also be expressed using this  alternative notation as



And the other partial indices such as intensity or the censored headcount ratios  introduced in section 5.5.3 can also be expressed using the identification function.

5.4     Distinctive Characteristics of the Adjusted Headcount Ratio

The  measure described in the previous section has several characteristics that merit special attention. First, it can be implemented with indicators of ordinal scale that commonly arise in multidimensional settings. In formal terms,  satisfies the ordinality property introduced in section 2.5. The ordinality property states that whenever variables (and thus their corresponding deprivation cutoffs) are modified in such a way that their scale is preserved—what has been defined in section 2.3 as an admissible transformation—the poverty value should not change. [10]

The satisfaction of this property is a consequence of the combination of the identification method and the aggregation method. Because identification is performed with the counting approach, which dichotomizes achievements into deprived and non-deprived, equivalent transformations of the scales of the variables will not affect the set of people who are identified as poor. Note that the weights attached to deprivations are independent of the indicators’ scale and implemented after the deprivation status has been determined. This is clearly relevant for consistent targeting within policies or programmes using ordinal indicators. In turn, aggregation to obtain the  measure is performed using the censored deprivation matrix, which represents the deprivation status of each poor person in every dimension and also uses the 0–1 dichotomy. In the aggregation procedure, the deprivations of the poor are weighted, but, again, the weights are independent of the indicators’ scale and implemented after the deprivation status of the poor has been determined. Thus, equivalent transformations of the scales of the variables will not affect the aggregation of the poor and thus will not affect the overall poverty value.

The fact that  satisfies the ordinality property is especially relevant when poverty is viewed from the capability perspective, since many key functionings are commonly measured using ordinal (or ordered categorical) variables. Virtually every other multidimensional methodology defined in the literature (including , , and, in general, the  measures with , which are defined in section 5.6) do not satisfy the ordinality property. In the case of the  measures with , while the set of people identified as poor does not change under equivalent representations of the variables, the aggregation procedure will be affected as it is no longer based on the censored deprivation matrix but on a matrix that considers the depth of deprivation in each dimension. In other measures, the violation of ordinality occurs at the identification step. Moreover, for most measures, the underlying ordering is not even preserved, i.e.  and  can both be true. Special care must be taken not to use measures whose poverty judgements are meaningless (i.e. reversible under equivalent representations) when variables are ordinal.

There is a methodology that combines the identification method used in the AF measures  with the headcount ratio as the aggregate measure: . which was used in previous counting measures surveyed in Chapter 4, satisfies the ordinality property. But it does so at the cost of violating dimensional monotonicity, among other properties. In contrast, the methodology that combines a counting approach to identification and  as the aggregate measure,  provides both meaningful comparisons and favourable axiomatic properties and is arguably a better choice when data are ordinal.

Second, while other measures have aggregate values whose meaning can only be found relative to other values,  conveys tangible information on the deprivations of the poor in a transparent way. As stated in section 5.3, it can either be interpreted as the incidence of poverty ‘adjusted’ by poverty intensity or as the aggregate deprivations experienced by the poor as a share of the maximum possible range of deprivations that would occur if all members of society were deprived in all dimensions. As we shall see in section 5.5.3, the additive structure of the  measure permits it to be broken down across dimensions and across population subgroups to obtain additional valuable information, especially for policy purposes.

Third, the adjusted headcount methodology is fundamentally related to the axiomatic literature on freedom. In a key paper, Pattanaik and Xu (1990) explore a counting approach to measuring freedom that ranks opportunity sets according to the number of (equally weighted) options they contain. Let us suppose that the achievement matrix  has been normatively constructed so that each dimension represents an equally valued functioning. Then deprivation in a given dimension is suggestive of capability deprivation, and since  counts these deprivations, it can be viewed as a measure of ‘unfreedom’ analogous to Pattanaik and Xu. Indeed, the link between  and unfreedom can be made precise, yielding a result that simultaneously characterizes  and  using axioms adapted from Pattanaik and Xu.[11] This general approach also has an appealing practicality: as suggested by Anand and Sen (1997), it may be more feasible to monitor a small set of deprivations than a large set of attainments.

5.5     The Set of Consistent Partial Indices of the Adjusted Headcount Ratio

The Adjusted Headcount Ratio condenses a lot of information. It can be unpacked to compare not only the levels of poverty but also the dimensional composition of poverty across countries, for example, as well as within countries by ethnic group, urban and rural location, and other key household and community characteristics. This is why we sometimes describe  as a high-resolution lens on poverty: it can be used as an analytical tool to identify precisely who is poor and how they are poor. This section presents the partial indices and consistent indices that serve to elucidate multidimensional poverty for policy purposes.

5.5.1     Incidence and Intensity of Poverty

We have already shown in section 5.3 that the  measure is the product of two very informative partial indices: the multidimensional headcount ratio—or incidence of poverty ()—and the average deprivation share across the poor—or the average intensity of poverty (). Both are relevant and informative, and it is useful to present them both in all tables. In Box 5.5, we present an example to show that two societies may have the same Adjusted Headcount Ratios but very different levels of incidence and intensity.

Box 5.5 Similar M0 but Different Composition of Incidence and Intensity

Suppose there are four persons in both societies  (as in Box 5.1) and  and multidimensional poverty is analysed using four dimensions, which are weighted equally. A person is identified as poor if deprived in more than half of all weighted indicators (). The  achievement matrices of two societies are


and the deprivation cutoff vector  (500, 5, Not malnourished, Has access to improved sanitation). The corresponding deprivation matrices are denoted as follows.


The deprivation score vectors are thus  and , respectively. Clearly, the second and the third person are identified as poor in  and the second, third, and fourth persons are identified as poor in . The corresponding censored deprivation matrices are as follows.


Then, using the formulation of , we obtain . The two societies have the same level of Adjusted Headcount Ratio. However, if we break down  into incidence and intensity, we find that  and , whereas  and . Clearly,  has a lower headcount ratio but the poor suffer larger deprivation on average.

The breakdown of  into  and  can provide useful policy insights. A policymaker who is interested in reducing overall poverty when poverty is assessed by the Adjusted Headcount Ratio may do so in different ways. If  is reduced by focusing on the poor who have a lower intensity of poverty, then there will be a large reduction in  But there may not be a large reduction in the average intensity . On the other hand, if the policies are directed towards the poorest of the poor, then an overall reduction in  may be accomplished by a larger reduction in  instead of . Thus, while monitoring poverty reduction, it is possible to see how overall poverty has been reduced.

It should be noted that  and  are also partial indices of the other  measures. Additionally, these other measures, such as  and , also have other informative partial indices, discussed in section 5.1.

5.5.2    Subgroup Decomposition

In developing multidimensional methods, we would not want to lose the useful properties that the unidimensional methods have successfully employed over the years. Prime among them is population subgroup decomposability, which, as stated in section 2.5.3, posits that overall poverty is a population-share weighted sum of subgroup poverty levels. This property has proved to be of great use in analysing poverty by regions, by ethnic groups, and by other subgroups defined in a variety of ways.[12] The  measure, as well as the other  measures, satisfies the population subgroup decomposability property, a property that is directly inherited from the FGT class of indices (Foster, Greer, and Thorbecke 1984).

Population subgroup decomposability allows us to understand and monitor the subgroup  levels and compare them with the aggregate . The population share and the achievement matrix of subgroup  are denoted by  and , respectively. We express the overall  as



Given the additive form of equation (5.7), it is also possible to compute the contribution of each subgroup to overall poverty. Let us denote the contribution of subgroup  to overall poverty by , which is formulated as



Note that the contribution of subgroup  to overall poverty depends both on the level of poverty in subgroup and on the population share of the subgroup. Whenever the contribution to poverty of a region or some other group greatly exceeds its population share, this suggests that there is a seriously unequal distribution of poverty in the country, with some regions or groups bearing a disproportionate share of poverty. Clearly, the sum of the contributions of all groups needs to be 100%.[13]    Subgroup Decompositions of the Adjusted Headcount Ratio ()

Let us consider the example of the hypothetical society presented in Box 5.1 and show how the contribution of subgroups to the overall Adjusted Headcount Ratio is computed. For this example, let us assume a certain weighting structure and a certain poverty cutoff to identify who among these four persons is poor. We assume that a 40% weight is attached to income, a 25% weight is attached to years of education, and 25% weight is attached to undernourishment and the remaining 10% weight is attached to the access to improved sanitation. Thus, the weight vector is . We identify a person as poor if the person is deprived in 40% or more of weighted indicators, that is, .

For subgroup decomposition, we divide the entire population in  into two subgroups. Subgroup 1 consists of three persons, whereas Subgroup 2 consists of only one person as presented in Table 5.1. Note that the person in Subgroup 2 is deprived in all dimensions. We denote the achievement matrix of Subgroup 1 by  and that of Subgroup 2 by .


The deprivation matrices and deprivation scores of the two subgroups are presented in Table 5.2. Person 1 is not deprived in any dimension and so has a deprivation score of 0. Person 2 is deprived in two dimensions: standard of living and access to public services, and so the deprivation score is 0.375. Similarly, the deprivation score of Person 4 is 0.5 and Person 3’s is 1. Now, for , Person 2 and Person 4 are poor in Subgroup 1 and Person 3 is poor in Subgroup 2. In both subgroups, those who are deprived are identified as poor, and so there is no scope for censoring. The censored deprivation matrices for both groups are, in this particular case, the corresponding deprivation matrices.


Thus,  and . The overall Adjusted Headcount Ratio . It is straightforward to verify that the sum of the population-weighted Adjusted Headcount Ratios of subgroups is equal to the overall Adjusted Headcount Ratio. The population share of Subgroup 1 is  and that of Subgroup 2 is . Therefore, .

Note that the Adjusted Headcount Ratio in Subgroup 2 is more than three times larger than the Adjusted Headcount Ratio of Subgroup 1. Does this mean that the contribution of Subgroup 2 is equally large? Not necessarily. It may not always be the case because of different population sizes across different subgroups. Recall that the contribution of a subgroup to overall poverty depends on the population share of that subgroup as well. For our example, the contribution of Subgroup 1 to the overall Adjusted Headcount Ratio is  or 42.8%. The contribution of Subgroup  to the overall headcount ratio is  or 57.1%. It is worth noting that, in this case, the population Subgroup 2 does bear a disproportionate load of poverty since, despite being only 25% of the total population, it contributes nearly 60% of overall poverty. Because population shares affect interpretation, tables showing subgroup decompositions must include population shares for each subgroup, as well as poverty figures.

5.5.3  Dimensional Breakdown

As discussed in section 2.5, a multidimensional poverty measure that satisfies the dimensional breakdown property can be expressed as a weighted sum of the dimensional deprivations after identification. The  satisfies the dimensional breakdown property and thus can also be expressed as a weighted sum of post-identification dimensional deprivation, which in the particular case of  we refer to as the censored headcount ratio.

Why is this property useful? This property allows one to analyse the composition of multidimensional poverty. For example, Alkire and Foster (2011a), after decomposing overall poverty in the United States by ethnic group, break the poverty within those groups down by dimensions and examine how different ethnic groups have different dimensional deprivations, i.e. different poverty compositions.

The censored headcount ratio of a dimension is defined as the percentage of the population who are both multidimensionally poor and simultaneously deprived in that dimension. Formally, we denote the th column of the censored deprivation matrix  as  and the mean of the column for that chosen dimension  as . We define  as the censored headcount ratio of dimension . What is the interpretation of ? The censored headcount ratio  is the proportion of the population that are identified as poor () and are deprived in dimension .

The additive structure of the  measure allows it to be expressed as a weighted sum of the censored headcount ratios, where the weight on dimension  is the relative weight assigned to that dimension. We have already seen in expression (5.3) that . This expression can be reformulated as



Analyses based on the censored headcount ratios can be complemented in an interesting way by considering the percentage contribution of each dimension to overall poverty. The censored headcount ratio shows the extent of deprivations among the poor but not the relative value of the dimensions. Two dimensions may have the same censored headcount ratios but very different contributions to overall poverty. This is because the contribution not only depends on the censored headcount ratio but also on the weight or value assigned to each dimension. Let us denote the contribution of dimension  to the  by . Then, the contribution of dimension  for poverty cutoff  is given by



for each . Whenever the contribution to poverty of a certain indicator greatly exceeds its weight, there is a relatively high censored headcount in this indicator. The poor are more deprived in this indicator than in others. Clearly, the sum of the contributions of all indicators is 100%.[14] The censored headcount ratios and the percentage contributions have policy relevance for understanding the composition of poverty in different regions. Chapter 9 describes how they may be used to analyse intertemporal changes in multidimensional poverty and percentage contributions.

The uncensored (raw) headcount ratio of a dimension is defined as the proportion of the population that are deprived in that dimension. It aggregates deprivations pertaining to the poor (censored headcount) with deprivations among the non-poor.  The uncensored headcount ratio of dimension  is computed from the th column  of the (uncensored) deprivation matrix . We denote the mean of the column vector  by . Therefore,  is the uncensored (raw) headcount ratio of dimension .

The censored headcount ratio may differ from the uncensored headcount ratio except when the identification criterion used is union. In this case, a person is identified as poor if the person is deprived in any dimension, so no deprivations are censored. Thus, the censored and uncensored headcount ratios are identical.  The Censored and Uncensored Headcount Ratios and Percentage Contributions

Using a hypothetical illustration, we now show how the uncensored headcount ratios and the censored headcount ratios are computed and then show how the contribution of each dimension to the Adjusted Headcount Ratio is calculated. Let us consider the same achievement matrix and weight vector as was in the previous subsection, which consists of four persons and four dimensions.

First, we show how to compute the uncensored headcount ratio. The achievement matrix  and the deprivation cutoff vector  are used obtain the deprivation matrix , presented in Table 5.3. The uncensored headcount ratio of any dimension  is . The uncensored headcount ratio of the standard of living dimension is . In other words, 50% of the population is deprived in the standard of living dimension. Similarly, the uncensored headcount ratio of the knowledge dimension is 50%, of the nutritional status dimension is 25% and of the access to services dimension is 50%. The uncensored headcount ratios (summarized by vector ) are reported in the bottom-most row of the table.


Next, we show how to compute the censored headcount ratio. We identify a person as poor if the person is deprived in 40% of weighted indicators, i.e. . Using the identification function we construct the censored deprivation matrix, presented in Table 5.4. Note that we censor the deprivations of Person 4 and replace them by 0 even when Person 4 is deprived in the education dimension. Why do we do this? We do so because the deprivation score of Person 4 is only 0.25, which is less than the poverty cutoff of . It can be easily verified that the  measure obtained from the censored deprivation matrix is 0.350.


Let us finish this example by showing how to compute the contribution of a dimension to the Adjusted Headcount Ratio. We already know that the Adjusted Headcount Ratio is 0.350 and that the percentage contribution is . Let us consider the income dimension, which has a censored headcount ratio of 0.50 and the weight attached to it is 0.40. Then the contribution of the dimension to the Adjusted Headcount Ratio is  or 53.3%.  Similarly, the contribution of the education dimension is  or 16.7%. An interesting aspect to note is that the censored headcount ratio of the access to sanitation dimension is the same as that of the income dimension, but its contribution to the Adjusted Headcount Ratio is only 13.3%, which is lower than the contribution of the income dimension. The reason is that the weight attached to the standard of living dimension is twice the weight of these two dimensions.

5.6       A Case Study: The Global Multidimensional Poverty Index (MPI)

Now that we have learned how to compute the Adjusted Headcount Ratio and its partial indices, we provide an example showing one prominent implementation of the  measure: the global Multidimensional Poverty Index (MPI). The global MPI was introduced by Alkire and Santos (2010) and has been reported annually in the Human Development Report since 2010.[15] The index consists of ten indicators grouped into three dimensions as outlined in Table 5.5.


Note that the index uses nested weights. The weights are distributed such that each dimension reported in the first column receives an equal weight of 1/3 and the weight is equally divided among indicators within each dimension (note the distinction in terms here between indicator and dimension). Thus, each education and health indicator receives larger weights than the standard of living indicators. The weights for each indicator are reported in the third column. The deprivation cutoffs are outlined in the final column. Any person living in a household that fails to meet the deprivation cutoff is identified as deprived in that indicator. An abbreviation has been assigned to each indicator in the second column that will be useful for the presentations in next table.

Table 5.6 presents a hypothetical example of people living in four households, which will help explain how the MPI is constructed. The first two households live in urban areas and the third and the fourth households live in rural areas. In this illustration, the households are not of equal size. The household sizes are reported in the third column of the table. The deprivation matrix () is presented in columns 4 through column 13. Following the standard notation, a 1 indicates that a household is deprived in the corresponding indicator and 0 indicates that the household is not deprived in that indicator. For example, the first household is only deprived in mortality (M) and cooking fuel (C), whereas the fourth household is deprived in five indicators: schooling (Sc), mortality (M), electricity (E), cooking fuel (C), and asset ownership (A).


Let us first show how the deprivation score () of each person is computed. Note that in this example, all persons within a household are assigned the same deprivation score, which is the weighted sum of deprivations that the household faces. For example, the deprivation score of each person in the first household is

The deprivation scores are reported in column 14. The deprivation scores of the second, third, and fourth households are 0.72, 0.39, and 0.50, respectively. Thus, the second household has the largest deprivation score and the first household has the lowest deprivation score.

In the computation of the global MPI, a person is identified as poor if the person’s deprivation score is equal to 1/3 or higher. It is evident from column 14 that the first household’s deprivation score is less than 1/3, whereas the three other households’ deprivation scores are larger than 1/3. Thus, all persons in the first household are identified as non-poor, whereas all other persons in the last three households are identified as multidimensionally poor. Column 15 classifies the households as multidimensionally poor or not. The multidimensional headcount ratio or the incidence of poverty () is (hint: use the household size)

So 80% of the population are poor. Note that we have already discussed that the multidimensional headcount ratio (H) does not satisfy the dimensional monotonicity property, and so it does not change if any of the three poor households become deprived in an additional dimension. This limitation is overcome by the Adjusted Headcount Ratio (), which is called the MPI in this example. The censored deprivation scores are reported in column 16, where the deprivation score of the first household has been censored by replacing the score by 0. The MPI is the mean of the censored deprivation score vector and can be computed using expression (5.3) as (hint: use the household size)

One may be also interested in knowing how poor the poor people are or the intensity of multidimensional poverty. The intensity of poverty can be computed as

So, on average, poor people are deprived in 56.3% of the weighted indicators. It can be easily verified that the MPI is the product of the incidence of poverty and the intensity of poverty, i.e.

Let us now show how the subgroup decomposition property may be used to understand the subgroups’ multidimensional poverty and the contribution of the subgroup to the overall poverty. Using the same process as above, the , , and  can be computed for each population subgroup. The MPI of the two urban households is 0.46, which can be obtained either by summing the censored deprivation scores weighted by the population share of each household or as a product of  and . The MPI of the two rural households is 0.44, whereas  and . Indeed, the incidence of poverty in the rural households is higher because all persons are identified as multidimensionally poor; whereas in the urban households this is not the case. However, when comparing the MPIs, we find the urban households have higher poverty because the intensity is higher. The urban households contribute 55% of the total population, and the rural ones contribute 45%. Thus, following the decomposition formula in equation (5.7), it can be verified that the overall MPI is . Again, using equation (5.8), it can be verified that the urban contribution to the overall MPI is 56%, whereas the rural contribution to the overall MPI is only 44%.

Next, using the last rows of Table 5.6, we show how the dimensional breakdown property is used. We have seen in expression (5.9) that the overall  can be expressed as a weighted average of censored headcount ratios. How are the censored headcount ratios in Table 5.6 computed? The censored headcount ratio for the years of education indicator is equal to (7+4)/20 = 55%. Similarly, the censored headcount ratio of the cooking fuel indicator is equal to (7+4+5)/20 = 80%. Note that the first household is not identified as poor and thus censored. This is why the censored headcount ratios are different from the uncensored headcount ratios reported in the row above. Looking at them, we can see that the poor in this society exhibit the highest deprivation levels in access to electricity and cooking fuel, followed (though with much lower headcount ratios) by sanitation, years of education, mortality, and assets. The percentage contributions of the indicators, which are computed using expression (5.10), are reported in the final column of the table. It is evident that neither electricity nor sanitation nor assets have the highest contribution to the overall MPI. Why? Because the weights assigned to these dimensions are lower than those assigned to schooling and mortality.

We now provide the following example to show how the censored headcount ratio and the percentage contribution of dimensions are used in practice. Borrowing from Alkire, Roche, and Seth (2011), the example provides information on two subnational regions for a cross-country implementation of the MPI. These two regions have roughly the same  levels reported in the final row of Table 5.7. Breaking  down by dimension reveals how the underlying structure of deprivations differs across the two countries for the ten indicators.[19] In Ziguinchor (a region in Senegal), mortality deprivations contribute the most to multidimensional poverty, whereas in Barisal (a division in Bangladesh), the relative contribution of nutritional deprivations is much larger than, say, deprivations in school attendance. Although the overall poverty levels as measured by  are very similar, dimensional breakdown reveals a very different underlying structure of poverty, which in turn could suggest different policy responses.


5.7     AF Class Measures Used with Cardinal Variables

In this chapter our discussion has focused on the Adjusted Headcount Ratio as many poverty indicators in practice are of ordinal scale. However, if all indicators are cardinal, we can go beyond the Adjusted Headcount Ratio to measures that additionally reflect the depth of deprivations poor people experience below the deprivation cutoff in each dimension. The identification step proceeds in exactly the same way as with . The difference is in the aggregation step. This section introduces the normalized gap matrix, which is used for the aggregation step for all the  class of measures with . The section also presents the two most common members of the  class with :  and .

5.7.1     The Normalized Gap Matrix

The basic matrix for measures drawing upon cardinal data is the normalized gap matrix, which, like the deprivation matrix, is constructed from the achievement matrix and the vector of deprivation cutoffs. The entries in the normalized gap matrix are the shortfall or gap in deprived people’s achievements, expressed as a proportion of the respective dimensional deprivation cutoff.

In the normalized gap matrix  the typical element is defined by . In other words, the normalized gap matrix replaces each deprived entry in  with the respective normalized gap and replaces each entry that is not below its deprivation cutoff with 0. The normalized gap matrix provides a snapshot of the depth of deprivation of each person in each dimension. The squared gap matrix  replaces each deprived entry in  with the square of the normalized gap and replaces each entry that is not below its deprivation cutoff with 0. Thus the typical element of the squared gap matrix is . Squaring the normalized gaps puts relatively more emphasis on larger deprivations. Generalizing the above, we may define the normalized gap matrix of order  by raising the normalized gaps to the power  and denote this by , whose typical element is . Clearly, if the normalized gap matrix is raised to the power 0, we would return to the deprivation matrix , in which all entries take the value of 0 or 1. Similarly, if the normalized gap matrix is raised to the power one and two, we obtain  and , respectively.

From the normalized gap matrices, we apply the same identification function  to obtain the censored normalized gap matrix of order  as  such that . Recall that the identification function  is based on the vector of weighted deprivation counts  (generated, as before, from the matrix and the vector of weights) and the poverty cutoff . The identification function replaces all deprived entries of the non-poor with 0 and leaves the deprived entries of the poor unchanged. We define  as the censored normalized gap matrix and  as the censored squared gap matrix.

5.7.2     The Adjusted Poverty Gap, Adjusted FGT, and Ma Measures

The Adjusted Poverty Gap measure  can be defined as



In other words, the Adjusted Poverty Gap is the sum of the weighted normalized gaps of the poor or  divided by the population (). Another way of viewing  is in terms of partial indices:  is the product of  (incidence) and  (intensity) (which in turn is ) and the average deprivation gap among the poor . That is,



In words,  is the average value of the normalized gap among all instances in which any poor person is deprived (and hence where the censored normalized gap is positive). Thus,  provides information on the average depth of deprivations across all poor and deprived states.

As in the case of , the partial indices greatly aid in comparing multidimensional poverty across time and space. Suppose for example that  is higher in one region than in another. It could be useful to examine the extent to which the difference is due to a higher ,or to higher values of or  It is also possible to examine the average gaps for each dimension to identify in which dimension normalized gaps tend to be higher.

Under methodology (), if the deprivation of a poor person deepens in any dimension, then the respective  will rise and hence so will . Consequently, () satisfies the property of monotonicity.

To incorporate sensitivity to one form of inequality among the poor, which satisfies the transfer property defined in section 2.5, we turn to the censored matrix  of squared normalized shortfalls. The Adjusted Squared Gap measure or Adjusted FGT Measure   can be defined as



The adjusted squared gap is the sum of the weighted normalized squared gaps of the poor, or  divided by the population ().  can also be expressed in terms of partial indices as the product of  (incidence) and  (intensity) and the average severity index . That is



The average severity index  denotes the average severity among all instances where a poor person is deprived (and hence where the censored squared gap is positive). By summarizing the square of the normalized gaps,  places relatively greater emphasis on the larger gaps. Therefore, under , a given-sized increase in a deprivation of a poor person will have a greater impact the larger the initial level of deprivation. Consequently, the methodology satisfies the weak transfer property and is sensitive to the inequality with which deprivations are distributed among the poor.

We generalize , , and  to the class  of multidimensional poverty measures associated with the unidimensional FGT class. The adjusted FGT class of multidimensional poverty measures can be defined as



In other words,  is the sum of the powers of the normalized gaps of the poor, or  divided by the population (). In the notation (Method IV in Box 5.7) used in Alkire and Foster (2011a), the ,  and  indices are each the means of their respective matrices.

The general methodology employing the dual-cutoff function  and an associated FGT measure  is denoted by It is important to define the AF methodology fully, both the dual-cutoff identification strategy and the poverty measures, because it is this combined methodology which assures that the resulting measures satisfy the principles here specified.

As stated in section 3.6.1 and as a way to wrap up this chapter, it is worth recalling that all measures in the AF family satisfy symmetry, replication invariance, scale invariance, poverty focus, deprivation focus, dimensional monotonicity, population subgroup decomposability, and dimensional breakdown. For , the measure satisfies the ordinality property, making it suitable for implementation when at least some of the indicators used are of ordinal scale. For , the measures require all indicators to be cardinal. When, the measures satisfy strong monotonicity. When, the measures satisfy transfer and weak deprivation rearrangement. When the union criterion is used for identification and , the measures satisfy continuity.

Box 5.6 α An Alternative Presentation of  Measures Using Non-normalized Weights

In this chapter we have stated the formulas in terms of normalized weights (Method I in Box 5.7), but they can also be expressed using non-normalized weights such that  for all  and , so that they add to the total number of dimensions (Method IV in Box 5.7). In order to do so, we introduce the weighted normalized gap matrices. Like the weighted deprivation matrix  that we defined earlier in Box 5., we may also define the weighted normalized gap matrix of order  as  such that . In other words, in weighted normalized gap matrices, each deprived entry in  is replaced with its respective normalized gap of order  multiplied by its relative weight and each entry that is not below its deprivation cutoff is replaced with 0. For ,  is the weighted normalized gap matrix with the typical element being . Similarly, for ,  is the weighted squared gap matrix with .

The censored weighted normalized gap matrix of order  can be obtained as  such that . Thus,  is the censored weighted normalized gap matrix and  is the censored weighted normalized squared gap matrix. As with any censored matrix, these matrices are obtained by multiplying the (weighted) deprivation matrix by the identification function .

The adjusted FGT class of multidimensional poverty measures can be defined as



In this case,  is the weighted sum of the α powers of the normalized gaps of the poor, or  divided by the highest possible value for this sum, or .

Based on expression (5.16), the Adjusted Poverty Gap measure  is the mean of the censored weighted normalized gap matrix and can be defined as



Thus, the Adjusted Poverty Gap is the sum of the weighted normalized gaps of the poor, or  divided by the highest possible sum of normalized gaps, or .

Similarly, the Adjusted Squared Gap or the Adjusted FGT Measure is given by



Thus,  is the sum of the squared normalized gaps of the poor, or , divided by the highest possible sum of the squared normalized gaps, or .

 5.8    Some Implementations of the AF Methodology

As mentioned in Chapter 1, since its development, the Alkire-Foster approach to multidimensional poverty has generated some practical interest. These include a global Multidimensional Poverty Index (MPI) estimated over 100 developing countries[20] as well as official national multidimensional poverty measures in Mexico, Colombia, Bhutan, and the Philippines with many other regional, national and subnational measures in progress.[21] Adaptations of the methodology include the Gross National Happiness Index of the Royal Government of Bhutan (Ura et al. 2012) and the Women’s Empowerment in Agriculture Index (Alkire, Meinzen-Dick et al. 2013). Several academic studies have implemented the AF approach for different poverty measurement purposes and in different parts of the world. These are summarized in Table 5.8.


Table 5.8. Summary of Research Studies That Have Implemented the AF Methodology



Paper Title

Implements AF
method to...

Region of the world for which it was

Alkire Apablaza and Jung


Multidimensional Poverty Measurement for EU-SILC countries

Constructs trial measures using EU-SILC data 2006–2012 and analyses by country, gender, age, and dimensional composition


Alkire and Seth


Identifying BPL Households: A Comparison of Methods

Compares a simple targeting-based method to some of the proposed mthods for targeting the poor for the BPL programme in India.

India, South Asia

Alkire and Seth


Multidimensional Poverty Reduction in India between 1999 and 2006: Where and How?

Monitors and studies multidimensional poverty reduction in India

 India, South Asia

Alkire and Seth


Selecting a Targeting Method to Identify BPL Households in India

Propose a counting-based targeting methodology for the BPL programme in India

India, South Asia

Arndt et al


Ordinal Welfare Comparisons with Multiple Discrete Indicators: A First-Order Dominance Approach and Application to Child Poverty

Performs child poverty comparison over time and between regions

Vietnam, Mozambique

Azevedo and Robles


Multidimensional Targeting: Identifying Beneficiaries of Conditional Cash Transfer Programmes

Implements the AF methodology to propose a targeting method

Latin America



Multidimensional Measurement of Poverty Among Women in Sub-Saharan Africa

Measure multidimensional poverty among women in fourteen sub-Saharan African countries

Sub-Saharan Africa

Battiston et al


Income and Beyond: Multidimensional Poverty in Six Latin American Countries

Measure multidimensional poverty in six Latin American countries and track its reduction between 1992–2006

Latin America

Beja and Yap 


Counting Happiness from the Individual Level to the Group Level

Uses the counting measure to assess group-level happieness


Castro, Baca, Ocampo


(Re)counting the Poor in Peru: A Multidimensional Approach

Uses the AF methodology to compare headcount ratios of monetary povery and multidimensional poverty between 2004 and 2008 in regions of Peru.

Peru, Latin America

Foster, Horowitz and Méndez


An Axiomatic Approach to the Measurement of Corruption: Theory and Applications

Develops a measure of corruption

No regional application



Race, Poverty and Deprivation in South Africa

Measures poverty and material deprivation and the racial gap among South Africans after apartheid

South Africa



Towards a Multidimensional Measure of Governance

Develops a governance index for sub-Saharan African countries

Sub-Saharan Africa

Mitra, Posarac and Vick


Disability and Poverty in Developing Countries: A Multidimensional Study

Obtains the economic profile of persons (aged 18–65) with disabilities. Multidimensional poverty analysis is performed for persons with and without disability

Burkina Faso, Ghana, Kenya, Malawi, Mauritius, Zambia, and Zimbabwe; Bangladesh, Lao PDR, Pakistan, and the Philippines; Brazil, Dominican Republic, Mexico, and Paraguay

Mitra, Jones et al.


Implementing a Multidimensional Poverty Measure Using Mixed Methods and a Participatory Framework

Measures multidimensional poverty among people with psychiatric diagnoses.


Nicholas and Ray 


Duration and Persistence in Multidimensional Deprivation: Methodology and Australian Application

Constructs dynamic deprivation measures and assesses the duration of deprivation across multiple dimensions


Notten and Roelen


A New Tool for Monitoring (Child) Poverty: Measures of Cumulative Deprivation

Measures material deprivation, cumulative deprivation, child poverty.

UK, Germany, France, the Netherlands

Nussbaumer et al.


Measuring Energy Poverty: Focusing on What Matters

Derives the Multidimensional Energy Poverty Index (MEPI)

Angola, Benin, Burkina Faso, Cameroon, Congo Brazzaville, Congo Democratic Republic, Egypt, Ethiopia, Ghana, Guinea, Kenya, Lesotho, Liberia, Madagascar, Malawi, Mali, Morocco, Mozambique, Namibia, Niger, Nigeria, Rwanda, Senegal, Sierra Leone, Swaziland, Tanzania, Uganda, Zambia, Zimbabwe

Peichl and Pestel


Multidimensional Affluence: Theory and Applications to Germany and the US

Constructs an index of affluence instead of poverty to study affluence in Germany and the US

Germany, USA

Peichl and Pestel


Multidimensional Well-Being at the Top: Evidence for Germany

Constructs an index of well-being to study well-being in Germany




Monitoring Progress in Child Poverty Reduction: Methodological Insights and Illustration to the Case Study of Bangladesh

Measures multidimensional poverty among children in Bangladesh and analyze the patterns of poverty reduction.

Bangladesh, South Asia



Has Poverty Decreased in Cameroon between 2001 and 2007? An Analysis Based on Multidimensional Poverty Measures

Analyses changes in multidimensional poverty in Cameroon between 2001 and 2007.

Cameroon, sub-Saharan Africa



Tracking Poverty Reduction in Bhutan: Income Deprivation Alongside Deprivation in Other Sources of Happiness

Measure multidimensional poverty in Bhutan and track its trend between 2003 and 2007

Bhutan, South Asia

Siegel and Waidler


Migration and Multi-Dimensional Poverty in Moldovan Communities

Examines multidimnesional poverty in 180 Moldovian communities in 2011

Moldova, Eastern Europe

Trani and Cannings


Child Poverty in an Emergency and Conflict Context: A Multidimensional Profile and an Identification of the Poorest Children in Western Darfur

Measure child poverty

Darfur, Sudan



An Exercise to Evaluate an Anti-Poverty Program with Multiple Outcomes Using Program Evaluation

Evaluates a programme using multidimensional poverty measures with difference-in-difference matching estimators


Trani et al


The Multidimensionality of Child Poverty: Evidence from Afghanistan

Measures poverty among children in Afghanistan




Multidimensional Poverty in China: Findings based on the CHNS

Measures multidimensional poverty in China and track its progress between 2000 and 2009.




The Counting-Based Measurement of Multidimensional Poverty: The Focus on Economic Resources, Inner Capabilities, and Relational Resources in the United States

Comparing a two-step process of the dimensional approach to AF method



Other papers do not directly implement the AF methodology but engage with it in various ways. These include Ferreira (2011), Ravallion (2011b) and others in the Journal of Economic Inequality, vol. 9 (2011), Ferreira and Lugo (2013), Ravallion (2012), Foster et al. (2010), Betti et al. (2012), Cardenas and Carpenter (2013), Larochelle (2014), Berenger et al. (2013), Siminski and Yerokhin (2012), and Smith (2012).

Chapter 6 which follows explains the normative decisions required to apply the AF framework of multidimensional poverty measurement empirically. It identifies the different decisions required, delineate their normative content and key considerations, and presents alternative courses of action.


Note: This figure uses the normalized notation in which weights   sum to 1 and , with the deprivation matrix.

Adjusted Headcount Ratio ) – Interpretation
The proportion of deprivations that poor people in a society experience, as a share of the deprivations that would be experienced if all persons were poor and deprived in all dimensions of poverty. It is the product of two intuitive partial indices, the Incidence and Intensity of Poverty ().

Alkire–Foster methodology
The AF methodology uses dual cutoffs to identify who is poor according to the count of ‘joint’ deprivations a person experiences and measures poverty using an extension of the FGT measures. AF measures are consistent with sub-indices that show the incidence and intensity and dimensional composition of poverty and, for cardinal variables, the depth and severity of deprivations in each dimension. The AF methodology can be used with different indicators, weights, and cutoffs to create measures for different societies and situations.

Censored headcount ratios
The proportion of people who are multidimensionally poor and deprived in each of the indicators.

This is the process of removing from consideration deprivations belonging to people who do not reach the poverty cutoff and focusing in on those who are multidimensionally poor.

The process of breaking down the Adjusted Headcount Ratio to show the composition of poverty for different groups. Groups might include countries, regions, ethnic groups, urban versus rural location, gender, age or occupational categories, or other groups.

Deprivation cutoffs ()
The achievement levels for a given dimension below which a person is considered to be deprived in a dimension.

A person is deprived if their achievement is strictly less than the deprivation cutoff in any dimension.

Functionings are ‘the various things a person may value doing or being’ (Sen 1999: 75). In other words, functionings are valuable activities and states that make up people’s wellbeing—such as being healthy and well nourished, being safe, being educated, having a good job, and being able to visit loved ones. They are related to resources and income but describe what a person is able to do or be with these, given their particular ability to convert those resources into functionings.

Incidence ()
The proportion of people (within a given population) who experience multidimensional poverty. This is also called the ‘multidimensional headcount ratio’ or simply the ‘headcount ratio’. Sometimes it is called the ‘rate’ or ‘incidence’ of poverty. It is the number of poor people  over the total population .

Intensity ()
The average proportion of deprivations experienced by poor people (within a given population) or the average deprivation score among the poor. The intensity is the sum of the deprivation scores, divided by the number of poor people.

Percentage contribution of each indicator
The extent to which each weighted indicator contributes to overall poverty.

A person is identified as poor if their deprivation score (the sum of their weighted deprivations) is at least as high as the poverty cutoff.

Poverty cutoff ()
This is the cutoff or cross-dimensional threshold used to identify the multidimensionally poor. It reflects the proportion of weighted dimensions a person must be deprived in to be considered poor. Because more deprivations (a higher deprivation score) signifies worse poverty, the deprivation score of all poor people meets or exceeds the poverty cutoff.

Uncensored or raw headcount ratios
The deprivation rates in each indicator, which includes all people who are deprived, regardless of whether they are multidimensionally poor or not.



Alkire et al. (2013): Alkire, S., Meinzen-Dick, R., Peterman, A., Quisumbing, A. R., Seymour, G., and Vaz, A. (2013). ‘The Women’s Empowerment in Agriculture Index’. World Development, 53: 71–91.

Alkire et al. 2014a: Alkire, S., Conconi, A., and Seth, S. (2014). ‘Multidimensional Poverty Index 2014: Brief Methodological Note and Results’. OPHI MPI Methodological Note: MPI 2014.

Alkire, S. and Foster, J. (2007). ‘Counting and Multidimensional Poverty Measurement’. OPHI Working Paper 7, Oxford University.

Alkire, S. and Foster, J. (2011a). ‘Counting and Multidimensional Poverty Measurement’. Journal of Public Economics, 95(7–8): 476–487.

Alkire, S. and Santos, M. E. (2010). ‘Acute Multidimensional Poverty: A New Index for Developing Countries’. OPHI Working Paper 38, Oxford University; also published as Human Development Research Paper 2010/11.

Alkire, S. and Santos, M. E. (2013). ‘A Multidimensional Approach: Poverty Measurement and Beyond’. Social Indicators Research, 112(2): 239–257.

Alkire, S. and Santos, M. E. (2014). ‘Measuring Acute Poverty in the Developing World: Robustness and Scope of the Multidimensional Poverty Index’. World Development, 59: 251–274.

Alkire, S. and Seth, S. (2013a). ‘Identifying BPL Households’. Economic & Political Weekly, 48(2): 49–67.

Alkire, S. and Seth, S. (2013b). ‘Multidimensional Poverty Reduction in India between 1999 and 2006: Where and How?’ OPHI Working Paper 60, Oxford University.

Alkire, S. and Seth, S. (2013c). ‘Selecting a Targeting Method to Identify BPL Households in India’. Social Indicators Research 112 (2): 417–446.

Alkire, S., Apablaza, M., and Jung, E. (2014). ‘Multidimensional Poverty Measurement for EU-SILC countries’. OPHI Research in Progress 36c.

Alkire, S., Conconi, A., and Roche, J. M. (2013). ‘Multidimensional Poverty Index 2013: Brief Methodological Note and Results’. OPHI MPI Methodological Note: MPI 2013.

Alkire, S., Roche, J. M., and Seth S. (2011). ‘Sub-National Disparities and Inter-Temporal Evolution of Multidimensional Poverty across Developing Countries’. OPHI Research in Progress 32a, Oxford University.

Alkire, S., Roche, J. M., Santos, M. E., and Seth, S. (2011). ‘Multidimensional Poverty Index 2011: Brief Methodological Note’. OPHI Briefing 07.

Anand, S. and Sen, A. (1997). ‘Concepts of Human Development and Poverty: A Multidimensional Perspective’, in Human Development Papers 1997: Poverty and Human Development. UNDP, 1–20.

Arndt et al.: Arndt, C., Distante, R., Hussain, M. A., et al. (2012). ‘Ordinal Welfare Comparisons with Multiple Discrete Indicators: A First Order Dominance Approach and Application to Child Poverty’. World Development, 40(11): 2290–2301.

Atkinson, A. B. (2003). ‘Multidimensional Deprivation: Contrasting Social Welfare and Counting Approaches’. Journal of Economic Inequality, 1(1): 51–65.

Azevedo, V. and Robles, M. (2013). ‘Multidimensional Targeting: Identifying Beneficiaries of Conditional Cash Transfer Programs’. Social Indicators Research, 112(2): 447–475.

Batana, Y. M. (2013). ‘Multidimensional Measurement of Poverty Among Women in Sub-Saharan Africa’. Social Indicators Research, 112(2): 337–362.

Battiston et al. (2013): Battiston, D., Cruces, G., Lopez-Calva, L. F., Lugo, M. A., and Santos, M. E. (2013). ‘Income and Beyond: Multidimensional Poverty in Six Latin American Countries’. Social Indicators Research, 112(2): 291–314.

Beja, E. L., Jr. and Yap D. B. (2013). ‘Counting Happiness from the Individual Level to the Group Level’. Social Indicators Research, 114(2): 621–637.

Berenger et al. (2013): Berenger, V., Deutsch, J., and Silber, J. (2013). ‘Durable Goods, Access to Services and the Derivation of an Asset Index: Comparing Two Methodologies and Three Countries’. Economics Modeling, 35: 881–891.

Betti et al. (2012): Betti, G., Gagliardi, F., Lemmi, A., Verma, V.. (2012). ‘Subnational Indicators of Poverty and Deprivation in Europe: Methodology and Applications’. Cambridge Journal of Regions Economy and Society, 5(1): 129–147.

Bourguignon, F., and Chakravarty, S. R. (2003). ‘The Measurement of Multidimensional Poverty’. Journal of Economic Inequality, 1(1): 25–49.

Cardenas, J. C. and Carpenter, J. (2013). ‘Risk Attitudes and Economic Well-Being in Latin America’. Journal of Development Economics, 103(C): 52-61.

Castro et al. (2012): Castro, J. F., Baca, J., and Ocampo, J. P. (2012). ‘(Re)Counting the Poor in Peru: A Multidimensional Approach’. Latin American Journal of Economics-formerly Cuadernos de Economía, 49(1): 37-65, Instituto de Economía. Pontificia Universidad Católica de Chile.

Chakravarty et al. (1998): Chakravarty, S. R., Mukherjee, D., and Ranade, R. R. (1998). ‘On the Family of Subgroup and Factor Decomposability Measures of Multidimensional Poverty’, in D. J. Slottje (ed.), Research on Economic Inequality 8. JAI Press, 175–194.

DHS Bangladesh (2007): National Institute of Population Research and Training (NIPORT), Mitra and Associates, and Macro International. (2009). Bangladesh Demographic and Health Survey 2007. Dhaka, Bangladesh and Calverton, Maryland, U.S.A.

DHS Senegal (2005): Ndiaye, Salif, et Mohamed Ayad. (2006). Enquête Démographique et de Santé au Sénégal 2005. Calverton, Maryland, U.S.A., Centre de Recherche pour le Développement Humain [Sénégal] et ORC Macro.

Ferreira, F. H. G. (2011). ‘Poverty Is Multidimensional. But What Are We Going To Do About It?’ Journal of Economic Inequality, 9(3): 493–495.

Ferreira, F. H. G. and Lugo, M. A. (2013). ‘Multidimensional Poverty Analysis: Looking for a Middle Ground’. World Bank Research Observer, 28(2): 220–235.

Foster et al. (2010): Foster, J., Greer, J., and Thorbecke, E. (2010). ‘The Foster–Greer–Thorbecke (FGT) Poverty Measures: 25 Years Later’. Journal of Economic Inequality, 8(4): 491–524.

Foster et al. (2012): Foster, J. E., Horowitz A. W., and Méndez, F. (2012). ‘An Axiomatic Approach to the Measurement of Corruption: Theory and Applications’. World Bank Economic Review, 26(2): 217–235.

Foster J., Greer, J., and Thorbecke, E. (1984). ‘A Class of Decomposable Poverty Measures’.  Econometrica, 52(3): 761–766.

Gradín, C. (2013). ‘Race, Poverty and Deprivation in South Africa’. Journal of African Economies, 22(2): 187–238.

Larochelle et al. (2014): Larochelle, C, Alwang, J., and Taruvinga, N. (2014). ‘Inter-temporal Changes in Well-being During Conditions of Hyperinflation: Evidence from Zimbabwe’. Journal of African Economics, 23(2): 225–256.

Mitra, S. (2013). ‘Towards a Multidimensional Measure of Governance’ Social Indicators Research. 112(2): 477–496

Mitra et al. (2013): Mitra, S., Jones, K., Vick, B., et al. (2013). ‘Implementing a Multidimensional Poverty Measure Using Mixed Methods and a Participatory Framework’. Social Indicators Research, 110(3): 1061–1081.

Mitra, S, Posarac, A., and Vick, B. (2013). ‘Disability and Poverty in Developing Countries: A Multidimensional Study’. World Development, 41: 1–18.

Nicholas, A. and Ray, R. (2012). ‘Duration and Persistence in Multidimensional Deprivation: Methodology and Australian Application’. Economic Record 88(280): 106–126.

Notten, G. and Roelen, K. (2012). ‘A New Tool for Monitoring (Child) Poverty: Measures of Cumulative Deprivation’. Child Indicators Research, 5(2): 335–355.

Nussbaumer et al. (2012): Nussbaumer, P., Bazilian, M., and Modi, V. (2012). ‘Measuring Energy Poverty: Focusing on what Matters’. Renewable & Sustainable Energy Reviews, 16(1): 231–243.

Pattanaik, P. K. and Xu, Y. (1990). ‘On Ranking Opportunity Sets in Terms of Freedom of Choice’. Recherches Economiques de Louvain. 56, 383-390.

Peichl, A. and Pestel, N. (2013a). ‘Multidimensional Affluence: Theory and Applications to Germany and the US.’ Applied Economics, 45(32): 4591–4601.

Peichl, A. and Pestel, N. (2013b). ‘Multidimensional Well-Being at the Top: Evidence for Germany’. Fiscal Studies, 34(3): 355–371.

Ravallion, M. (2011b). 'On Multidimensional Indices of Poverty'. Journal of Economic Inequality, 9(2): 235–248.

Roberts, F. S. (1979). Measurement Theory with Applications to Decisionmaking, Utility, and the Social Sciences, Vol. 7. CUP.

Roche, J. M. (2013). ‘Monitoring Progress in Child Poverty Reduction: Methodological Insights and Illustration to the Case Study of Bangladesh’. Social Indicators Research 112(2): 363–390.

Ruggeri Laderchi, C., Saith, R., and Stewart, F. (2003). ‘Does It Matter that We Don’t Agree on the Definition of Poverty? A Comparison of Four Approaches’. Oxford Development Studies 31(3): 243–274.

Santos, M.E. (2013). ‘Tracking Poverty Reduction in Bhutan: Income Deprivation Alongside Deprivation in Other Sources of Happiness’. Social Indicators Research, 112(2): 259–290.

Sen, A. K. (1976). ‘Poverty: An Ordinal Approach to Measurement’. Econometrica, 44(2): 219–231.

Sen, A. K. (1997). On Economic Inequality. Expanded Edition with a Substantial Annex by James E Foster and Amartya Sen. OUP.

Sen, A. K. (1999). Development as Freedom. OUP.

Siani Tchouametieu, J. R. (2013). ‘Has Poverty Decreased in Cameroon between 2001 and 2007? An Analysis Based on Multidimensional Poverty Measures’. Economics Bulletin, 33(4): 3059-3069.

Siegel, M. and J. Waidler. (2012). ‘Migration and Multi-dimensional Poverty in Moldovan Communities’. Eastern Journal of European Studies, 3: 105-119. Centre for European Studies, Alexandru Ioan Cuza University.

Siminski, P. and Yerokhin, O. (2012). ‘Is the Age Gradient in Self-Reported Material Hardship Explained By Resources, Needs, Behaviors, or Reporting Bias?’ Review of Income and Wealth, 58(4): 715–741.

Smith, S. C. (2012). ‘The Scope of NGOs and Development Programme Design: Application to Problems of Multidimensional Poverty’. Public Administration and Development, 32(4–5): 357–370.

Tonmoy Islam, T. M. (2014). ‘An Exercise to Evaluate an Anti-Poverty Program with Multiple Outcomes Using Program Evaluation’. Economics Letters, 122(2): 365–369.

Trani et al. (2013): Trani, J.-F., Biggeri, M., and Mauro, V. (2013), ‘The Multidimensionality of Child Poverty: Evidence from Afghanistan’. Social Indicators Research, 112(2): 391-416.

Trani, J.-F. and Cannings, T. I. (2013). ‘Child Poverty in an Emergency and Conflict Context: A Multidimensional Profile and an Identification of the Poorest Children in Western Darfur’. World Development, 48(C): 48–70.

Tsui, K. (2002). ‘Multidimensional Poverty Indices’. Social Choice and Welfare 19(1): 69–93.

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Wagle, U. R. (2014). ‘The Counting-Based Measurement of Multidimensional Poverty: The Focus on Economic Resources, Inner Capabilities, and Relational Resources in the United States’. Social Indicators Research, 115(1): 223–240.

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[1] We are following the ‘normalized’ notation here; for other notations see section 5.2.2  and Box 5.7.

[2] One common assumption is that prices exist and are adequate normative weights for the dimensions; however, as noted by Tsui (2002), this assumption is questionable. Prices may be adjusted to reflect externalities, but exchange values do not and ‘indeed cannot give…interpersonal comparisons of welfare or advantage’ (Sen 1997: 208). Subjective poverty lines cannot replace prices for all attributes, and markets may be missing or imperfect (Bourguignon and Chakravarty 2003; Tsui 2002). In practice, income may not be translated into basic needs (Ruggeri Laderchi, Saith, and Stewart 2003; Sen 1979). Finally, aggregating across dimensions entails strong assumptions regarding cardinality and comparability, which are impractical when data are ordinal (Sen 1997).

[3] The use of the word methodology when referring to AF indicates that it comprises both an identification and an aggregation method.

[4] See also Chakravarty et al. (1998) and Tsui (2002) on this point.

[5] Atkinson (2003) applied the terms ‘union’ and ‘intersection’ in the context of multidimensional poverty.

[6] See Chapter 6 on the choice of  (and ).

[7] In the case of deprivation scores, the poverty cutoff fixes a minimum level of deprivations that identify poverty. This is in contrast to the unidimensional context, where a person is identified as poor if her achievement is below the poverty line.

[8] This example has an identical relative weight across dimensions; the general case admits a wide variety of identification approaches. For example, if one dimension had overriding importance and its relative weight was set above or equal to , then any person deprived in that dimension would be considered poor.

[9] While informative, this measure is insufficient as a standalone index for two reasons. First, if a poor person becomes deprived in a new dimension,  remains unchanged, violating the property of dimensional monotonicity. Second,  cannot be further broken down to show how much each dimension contributes to poverty.

[10] The set of the poor and the measured value of poverty are therefore meaningful in the sense of Roberts (1979). Note that M0 can also be applied to categorical variables (which do not necessarily admit a unique ordering across categories), so long as achievements can be separated into deprived and non-deprived sets.

[11] For a fuller discussion see Alkire and Foster (2007).

[12] Additive decomposable measures satisfy subgroup consistency, but the converse does not hold. See section 5.2 for further details on these and other properties.

[13] Note that other measures in the AF class discussed in section 5.7 satisfy the population subgroup decomposability property as well, and expressions (5.7) and (5.8) are equally applicable to these measures.

[14] Note that if poverty as measured by  is very low, the censored headcount ratios are also low, and contributions require care in interpretation. One indicator can have an 80% contribution, not because there is a massive deprivation in that indicator but because it is one of the few indicators that have a non-zero censored headcount, explaining most of the (very low) poverty.

[15] See also Alkire and Santos (2014), where the MPI is presented and scrutinized with a host of robustness tests.

[16] If a household has no school-aged children, the household is treated as non-deprived.

[17] An adult with a Body Mass Index below 18.5 m/kg2 is considered undernourished. A child is considered undernourished if his or her body weight, adjusted for age, is more than two standard deviations below the median of the reference population.

[18] If no person in a household has been asked this information, the household is treated as non-deprived.

[19] Data are drawn from the Demographic and Health Surveys (DHS) for Bangladesh (2007) and Senegal (2005), which are nationally representative household surveys.

[20] UNDP (2010a); Alkire and Santos (2010, 2014); Alkire Roche Santos and Seth (2011); Alkire Conconi and Roche (2013); Alkire Conconi and Seth (2014a).

[21] These experiences are documented on the often-updated site

4 Counting Approaches: Definitions, Origins, and Implementions

An assessment of measurement methodologies based on their properties and normative characteristics is illuminating and draws our attention to many interesting distinctions among measure, as we saw in Chapter 3. Yet, as Tony Atkinson observed in the landmark 2003 paper that catalysed many responses, including this book, ‘Empirical studies of multiple deprivation to date have not typically adopted a social welfare function approach. Rather they have tended to concentrate on counting the number of dimensions in which people suffer deprivation’. To catalyse policy-relevant measurement methodologies, it may be useful to analyse some measures which have served to guide policy, to see why they were implemented and how they have been used, as well as the criticisms and difficulties they faced. Our task in this chapter is to begin such an exploration of counting-based measures.

4.1 Definition and Origins

The measurement of multidimensional poverty, as discussed in Chapter 1, involves three fundamental steps: selecting the space, deciding who is poor, and aggregating the information of the poor. The fundamental step of deciding who is poor is identification (Sen 1976). A ‘counting approach’ is one way to identify the poor in multidimensional poverty measurement. It entails, as Atkinson (2003: 51) notes, ‘counting the number of dimensions in which people suffer deprivation, (…) the number of dimensions in which they fall below the threshold’.[1]

As mentioned in section 2.2.2 and section 3.6.1, a counting approach to identifying the poor can be broken down into the following steps:

1. Defining a set of relevant indicators;

2. Defining a threshold of satisfaction (deprivation cutoff) for each indicator such that if the person does not reach it, she is considered deprived;

3. Creating binary deprivation scores for each person in each indicator, where 1 is being deprived and 0 is being non-deprived;

4. Assigning a weight or deprivation value to each considered indicator;

5. Producing a deprivation score by taking the weighted sum of deprivations (or counting the number of deprivations, if they are equally weighted);

6. Setting a threshold score of poverty (or poverty cutoff) such that if the person has a deprivation score at or above the threshold, she is considered poor.

Most steps involve normative judgements, which are largely discussed in Chapter 6. Step (4) entails deciding whether all deprivations should be given the same weight. Step (6) specifies the extent of deprivations which must be experienced by a person in order to be considered poor which, as outlined in section 2.2.2, can range from experiencing at least one deprivation (union) to experiencing all deprivations (intersection). In practice, either the union or intermediate criteria have been most commonly used; the intersection criterion has rarely been used. The need to define a ‘poverty cutoff’ in step (6) is what led Alkire and Foster to name their identification methodology as ‘dual cutoff’, as it involves defining a set of indicator cutoffs in step (2) and the poverty cutoff in step (6). The dual-cutoff strategy is clearly applicable to any approach following a counting method to identify the poor.

Counting approaches have been widely used in empirical studies, with one developed and one developing region being particularly pioneering in this work: Europe and Latin America. Interestingly, applications of the counting approach have been inspired and motivated by different conceptual approaches, and have developed relatively independently of each other.

One such influential approach was the basic needs approach, which emerged in the mid-1970s as a reaction to the prevailing economic growth-centred approach to development of the time.[2] The Cocoyoc Declaration, adopted in 1974 by participants in the UNEP/UNCTAD symposium on ‘Patterns of Resource Use, Environment and Development Strategies’ articulated this approach as follows: ‘Human beings have basic needs: food, shelter, clothing, health, education …We are still in a stage where the most important concern of development is the level of satisfaction of basic needs for the poorest sections in each society … Development should not be limited to the satisfaction of basic needs … Development includes freedom of expression and impression, the right to give and to receive ideas and stimulus …, the right to work’ (UNEP/UNCTAD 1975 896–7). The Cocoyoc Declaration was echoed by several subsequent studies and reports released in 1976.[3],[4]

The basic needs approach had a policy focus, but in practice it influenced poverty measurement, especially in Latin America. Until the 1970s, the prevailing approach to measuring poverty used an income poverty line for identifying the poor, which Sen (1981) called the income method.[5] The first European use of an (implicit) poverty line was by the London School Board during the 1880s in order to exempt destitute families from paying school fees (Gillie 1996).[6] The poverty line was then used in the seminal surveys of Booth (1894, 1903), Rowntree (1901), and Bowley and Burnett-Hurst (1915), which were conducted in specific cities in the UK. As expressed by Rowntree, the poverty line represented the ‘minimum necessaries for the maintenance of merely physical efficiency’ (i.e. nutritional requirements, clothing, fuel, and household sundries) in monetary terms (Townsend 1954: 131).[7] The poor were those whose household income was below the poverty line corresponding to their family size. In the 1950s, the income method of poverty measurement appeared to be consistent with growth emphasis of development (Sen 1960). Clearly, a commodity-focused concept of basic needs underlay the income method of poverty measurement, as the poverty line indicated the minimum amount of resources to cover such needs. Subsequently the basic needs approach, alongside other approaches we will mention, such as social exclusion, drew attention to the importance of looking at the actual satisfaction of basic needs (or at least access to key commodities), thus fostering the so-called direct method of poverty measurement (Sen 1981). A list of needs considered to be basic alongside minimum levels of satisfaction (cutoffs) would be specified. It is in such a context that counting the number of deprivations naturally emerged as a method of identifying the poor and of monitoring progress towards meeting basic needs.

As the Cocoyoc Declaration quote shows, the basic needs approach was originally quite comprehensive in the goals it regarded as intrinsically important, including, for example, freedom of expression and the right to have decent work. Later, as the approach was intended to have a direct policy impact, empirical studies were conducted in order to determine which goods and services, incomes, and resources were needed for everyone to enjoy a ‘full life’ (Streeten et al. 1981). Resources were understood to be of secondary importance and merely as means to ends by most basic needs advocates (Stewart 1985). Unfortunately, when the idea caught on, some operational programmes designed by the International Labour Organization (ILO) and the World Bank, under Robert MacNamara, were ‘focused on commodity inputs to health, education, clothing, shelter, sanitation and hygiene …. The problem was that the overemphasis on commodities misinterpreted the basic needs approach, and in so doing redefined and subverted it’ (Alkire 2005: 116, cf. 2006). The policy urgency was defended as being appropriate and necessary, but in fact it implemented only a subset of priorities of the basic human needs approach (Stewart 1985).

Some years before the emergence of the basic needs approach, Europe started to develop social indicators, which enabled empirical studies of non-monetary aspects of social welfare (Delors 1971). Erikson (1993) describes how criticisms of GNP per capita as a measure of welfare in the 1950s led to a 1954 UN expert group, which proposed to measure well-being using ‘level of living’. In the late 1960s, interest was renewed in constructing ‘a parsimonious set of specific indices covering a broad range of social concerns’ (Vogel 1997: 105). In 1968 Sweden implemented a Level of Living Survey that was repeated and spread in other Scandinavian countries, and this, together with parallel work on social indicators such as Delors (1971), catalysed discussions of poverty measurement: ‘Johansson [(1973)], in his first discussion of the level of living concept, suggested a concentration on “bad conditions”’ (Erikson 1993: 80).[8]

While basic needs was one concept informing measures of deprivation in Europe (Galtung 1980), this was supplemented by other conceptual motivations.[9] Atkinson and Marlier (2010) observe that the multidimensional concept of ‘social exclusion’ (Lenoir 1974) has most widely motivated European approaches to measurement for public policy. In 1974, the Council adopted a ‘resolution concerning a social action programme’ which prompted responses to poverty and social exclusion (Atkinson et al. 2005: 29). The Council defined the poor (in 1975) as ‘individuals or families whose resources are so small as to exclude them from the minimum acceptable way of life of the Member State in which they live’, with ‘resources’ being defined as ‘goods, cash income plus services from public and private sources’ (Atkinson et al. 2005: 18).[10] Social exclusion became seen as going ‘beyond the elimination of poverty’ to focus on ‘the mechanisms whereby individuals and groups are excluded from taking part in the social exchanges, from the component practices and rights of social integration’ (European Commission 1992, cited in Atkinson and Marlier 2010: 18).

Although the social inclusion approach was (and often still is) widely described as ‘relative’, this depends upon the evaluative space. Amartya Sen wrote, ‘[t]he characteristic feature of “absoluteness” is neither constancy over time, nor invariance between different societies, nor concentration merely on food and nutrition. It is an approach of judging a person’s deprivation in absolute terms (in the case of poverty study, in terms of certain specified minimum absolute levels), rather than in purely relative terms vis-à-vis the levels enjoyed by others in the society’ (1985: 673).

A landmark moment in the mainstreaming of social inclusion into European Union (EU) policies occurred at the Lisbon Summit of March 2000, where ‘EU Heads of State and Government decided that the Union should adopt the strategic goal for the next decade of becoming “the most competitive and dynamic knowledge-based economy ... with more and better jobs and greater social cohesion”. Importantly, the phrase “social cohesion” appeared in the same sentence as “most competitive economy”’ (Atkinson et al. 2002: 17). Another inflection point in Europe was the very explicit political processes for engaging member states in the ‘open method of coordination’ for social measures and policies. ‘The open method of coordination, which is designed to help member states progressively to develop their own policies, involves fixing guidelines for the Union, establishing quantitative and qualitative indicators to be applied in each member state, and periodic monitoring’ (Atkinson et al. 2002: 1–5).

A third influential conceptual framework for developing counting-based poverty measures has been Amartya Sen’s capability approach, outlined in section 1.1. It gained increasing recognition as providing an appropriate space for evaluating poverty: the space of capabilities and functionings rather than the space of resources upon which basic needs programmes had come to concentrate.[11] Applications of counting approaches intending to operationalize the capability approach sought to look at failures such as the ability to meet nutritional requirements, be clothed and sheltered, enjoy functional literacy and numeracy, or the power to participate in the social life of the community, which are some of the basic functionings mentioned by Sen from the very start (1979: 218). Yet even in work inspired by the capability framework, the indicators considered in counting approaches are data-constrained, hence often include resource-based indicators that are linked to key functionings, much as in the basic needs approach.

This chapter briefly reviews key empirical implementations of counting approaches to identifying the poor that are motivated by any of the aforementioned conceptual approaches.[12] Before proceeding to the salient applications of the counting approach, let us clarify that their emphasis is on identifying the poor. Most measurement applications of the counting approach have used the proportion of people identified as poor—the so-called headcount ratio defined in equation (3.23) in section 3.6.—for the third fundamental step of poverty measurement: aggregation. By using only the headcount ratio, the poverty measure is not able to discriminate according to the number or extent of deprivations among the poor, what we call intensity.[13] The focus of this chapter is on the identification step; Chapters 3 and 5 address forms of aggregation that provide more informative poverty measures than the multidimensional headcount ratio.

4.2 Measures of Deprivation in Europe and their Influence

Townsend (1979) conducted an early seminal study using a counting approach to poverty in the United Kingdom, analysing a 1968–9 survey covering about 2000 households in Britain. To assess the magnitude of ‘relative deprivation’, Townsend defined sixty indicators covering twelve dimensions: diet, clothing, fuel and light, home amenities, housing conditions and facilities, the immediate environment of the home, conditions at work, family support, recreation, education, health, and social relations. Each indicator was equally weighted, although the number of indicators within each dimension varied greatly. For ‘illustrative purposes’, he then focused on a shorter list of twelve items covering major aspects of dietary, household, familial, recreational, and social deprivation. Townsend used a minimum score of five (out of the twelve) ‘as suggestive of deprivation’ (p. 252). In other words, a poverty cut-off of five out of twelve was chosen to identify the poor.[14] He did not use a union criterion because he recognized the potential problems: ‘No single item by itself, or pair of items by themselves, can be regarded as symptomatic of general deprivation. People are idiosyncratic and will indulge in certain luxuries and apply certain prohibitions for religious, moral, educational or other reasons, whether they are rich or poor’ (p. 252).[15] However, he actually did not use this counting approach to analyse poverty. Rather, he explored the correlation between deprivation scores and household income (adjusted for household size) in order to derive an income threshold below which people are ‘disproportionately deprived’ (p. 255). In other words, he used a direct approach to ‘validate’ the poverty line to be used in the indirect income poverty measure.

Townsend’s study inspired much subsequent work on poverty and social exclusion in Europe and, in particular, another benchmark study on poverty: Mack and Lansley’s Poor Britain (1985). This study was also influenced by Sen’s writings on the direct approach to poverty measurement (Sen 1981). A novelty of this study was that the list of items considered as necessities was, for the first time, constructed using a survey of the public’s perceptions of minimum needs (PSE [1983] Breadline Britain). That is why their method has been called the ‘consensual or perceived deprivation approach to measuring poverty’. Of the original thirty-five items, they retained the twenty-six that were considered necessities by strictly more than 50% of the population. The survey usefully distinguished people who lacked an item because they could not afford it from those for whom it was a voluntary choice.[16] The authors identified as poor those who could not afford three or more items from the equally weighted items (p. 178).[17] This poverty cutoff was selected after analysing the association between the number of deprivations, income levels, and spending patterns. Mack and Lansley proposed that a lack of three or more necessities was a matter of force rather than choice.[18] ‘Very few of the better-off lack this level of necessities. And nearly all those who lack this level of necessities cut back on non-necessities, a majority cutting back substantially’ (p. 176). In addition to their benchmark cutoff, the authors reported degrees of deprivation using two additional poverty cutoffs: ‘Broadly speaking, those who cannot afford five or more necessities are sinking deeper into poverty; and those who cannot afford seven or more necessities are in intense poverty’ (p. 184). British authors continue working along these lines, with new surveys in 1990, 1999, and 2012. Breadline-Britain-type surveys were also replicated elsewhere in Europe.[19] It also inspired the structure of two much-used datasets: the European Community Household Panel survey (ECHP) and the European Union Statistics on Income and Living Conditions (EU-SILC).[20]

Gordon et al. (2000) compare the 1983, 1990, and 1999 Breadline Britain surveys in terms of the items considered as necessities and assess the evolution in poverty levels. Using an updated list of thirty-five items to evaluate poverty, they identified a household as poor if they could not afford two or more items and, additionally, had relatively low incomes.[21] The report also constructed a measure of child poverty using a list of twenty-seven socially perceived necessities for children. They used a poverty cut-off of one or more and another more restrictive cut-off of two or more. In both cases the poverty cut-off was set using discriminant function analysis.[22] Note that the poverty cut-off selected using discriminant analysis, which is a data-driven approach, may provide different conclusions when applied to different datasets, making comparison across time difficult (section 3.4.4). Because the poverty cut-off is not normatively considered or justified, in contrast to other measures reviewed, there is no link to ethical assessments of poverty.

Building upon the work of Mack and Lansley (1985) and Ringen (1987, 1988), Callan, Nolan, and Whelan (1993) also proposed to identify the poor by combining both resource and deprivation measures.[23] They used data from a household survey conducted in Ireland by the Economic and Social Research Institute (ESRI) in 1987, which used Mack and Lansley’s (1985) format. Starting from a list of twenty-four items, the authors used factor analysis to observe possible indicator clusterings and accordingly used three dimensions: (1) basic lifestyle (eight items such as food and clothes), (2) housing and durables (seven items related to housing quality and facilities), and (3) ‘other’ aspects of lifestyle (nine items such as social participation, leisure activities, and having a car or telephone). People’s perceptions regarding the necessity of indicators restricted their material deprivation index to the eight-item basic lifestyle dimension.[24] They identified as poor anyone who both lacked one or more of the eight items and fell below the relative income poverty line, set at 60% of the average equivalent disposable income in the sample.[25] This work sparked a series of surveys and studies to monitor poverty in Ireland using variations on this combined method of resources and material deprivation. These were used to build a ‘consistent measure of poverty’[26] which identifies a person as poor if she is both income poor and deprived in some minimum set of deprivations.

Muffels et al. (1992) built upon Muffels and Vriens (1991) and designed an index of relative and subjective deprivation using a Dutch socioeconomic panel survey inspired by Mack and Lansley (1985). Their innovation was to use household weights and poverty cutoffs. They first constructed an (objective) deprivation score for every head of household as the weighted sum of deprivation in each of a large set of items related to living conditions. The weight for each item varied across households and represented the respondent’s perceived importance of the item, compared to the perceived importance of the item by the household head (p. 195). They selected a subjective poverty cutoff, termed the ‘subjective deprivation poverty line’, using an econometric model. In the model, the dependent variable was the respondent’s subjective assessment of whether he/she was poor or not on a scale of one to ten. The independent variables were their deprivation score plus control variables such as income, age of the household head, family status, and financial stress factors. Using the estimated coefficients, the ‘subjective deprivation poverty line’ was calculated for each household separately, as the weighted deprivation score that would produce a subjective assessment score of 5.5.[27] Then they explored the degree of overlap between those identified as poor using the subjective deprivation poverty line and those identified as ‘insecure’ using three alternative income poverty lines (a subjective line, equal to the minimum income reported by households as ‘the minimum income they need to acquire a certain minimum standard of welfare’; a ‘national social minimum income standard’; and a ‘European statistical minimum income standard’). They examined the relationship between both measures in terms of bivariate distributions using contingency tables and regression analysis and found significant mismatches. They concluded that a multi-method approach combining income and (direct) deprivation measures was needed to assess poverty.

Halleröd (1994, 1995) used data from the Swedish standard of living survey in 1992, which also followed Mack and Lansley (1985). A key difference was that they retained all thirty-six originally included items but weighted them by the proportion of the population that regarded each as necessary. Weights were adjusted by certain groups to reflect significant differences in preferences.[28] The index was labelled the Proportional Deprivation Index (PDI). The author selected a poverty cutoff that produced the same headcount ratio as the Consensual Poverty Line (CPL). The CPL was an interpersonally comparable income level at which, on average, respondents in different circumstances would subjectively indicate that their current income was just sufficient for them to make ends meet. While both methods identified nearly 21.3% of the population as poor, only 8.8% of the population were identified poor by both. Acknowledging that both the income and direct methods may be subject to substantial measurement problems, the author advocated the use of a combination of both methods and defined those 8.8% of the population who were poor by both CPL and PDI as the ‘truly poor’. Subsequently, Halleröd et al. (2006) used a variant of the PDI to compare poverty levels in Britain, Finland, and Sweden.

Using ECHP data, Layte et al. (2001) constructed a material deprivation index from thirteen items to assess the relationship between (relative) income poverty and material deprivation. For each country, they weighted each item by the proportion of households possessing that item, and they defined the poverty cutoff of the deprivation index endogenously as the threshold which generated a headcount ratio equal to that of the (relative) income poverty line of the country. They performed this exercise for different relative poverty lines: at 40%, 50%, and 60% of the median income in each country. Their results showed that the overlap between the two poverty measures was very limited and thus supported a method that combines both measures. Whelan, Layte, and Maître (2004), using the ECHP to identify persistently poor persons, found a similar mismatch as mentioned in section 1.2.1.

Eurostat (2002) constructed an index of non-monetary poverty (pauvreté d’existence) for European countries. Following the analysis of Whelan et al. (2001b) of the first ECHP survey, a list of twenty-four dichotomous items (‘having’/‘not having’) available in that survey were grouped into five dimensions using factor analysis.[29] For each individual, a deprivation score per dimension was obtained as the weighted sum of deprivations in the indicators of that dimension, where the weight attached to an indicator was inversely related to the deprivation rate in that indicator in the corresponding country. Then, the dimensional deprivation scores were also aggregated by taking a weighted sum, where the dimensional weight attached to a dimension was proportional to the weighted average of the coefficients of variation among that dimension’s indicators (pp. 155–6). People with a deprivation score of 60% or more were considered poor.[30]

Additional implementations of the counting approach to identifying the poor in Europe included studies of poverty in Sweden (Erikson 1993), the reports on poverty in Belgium by Vranken and other authors (Vranken 2002), and recent work on the search for a relative deprivation index for Europe (Guio 2005, 2009; Guio and Maquet 2006; Decancq et al. 2013). In 2011, the European Commission implemented an ‘EU-2020’ multidimensional poverty measure using union identification across three indicators: relative income poverty, severe material deprivation, and quasi-joblessness. This landmark measure identified those ‘at risk of poverty and social exclusion’ in order to set and monitor a poverty reduction target for 2020. It represents the most high-profile policy application to date—hence, perhaps, the most closely scrutinized.

Nolan and Whelan’s book Poverty and Deprivation in Europe offers a systematic conceptual and empirical study of ‘why and how non-monetary indicators of deprivation can play a significant role in complementing (not replacing) income in order to capture the reality of poverty in Europe’ (2011: 1). It is thus relevant to this book at many points, as they too survey research on mismatches in identification between different indicators of poverty—by social group and in one period and across time—scrutinize indicator design, apply robustness tests, consider the poverty cut-off, and propose ways of strengthening the EU-2020 Poverty Target. Maître, Nolan, and Whelan (2013) offer a critical evaluation of the EU-2020 Target and Whelan, Nolan, and Maître (2014) explored the use of the AF method for the case of the European Union using EU-SILC data; they advocate the replacement of the current approach by the AF approach as it is more structured, less ad hoc, and more transparent, as well as being flexible in terms of the poverty cutoff and the axiomatic properties of its measures (see section 2.5 and Chapter 5). Alkire, Apablaza and Jung (2014) also apply an AF measure to EU-SILC data 2006–12, and explore the inclusion of social indicators.

Townsend (1979) and Mack and Lansley (1985) also influenced work outside Europe. For example, Mayer and Jencks (1989) severely criticized the income approach to poverty measurement in the United States based on a survey in Chicago on material hardship. They collected information on ten indicators covering dimensions of food, housing, and medical care. The number of hardships (equally weighted) were analysed alongside income and subjective satisfaction with living standard. They found that the family’s income-to-needs ratio explained less than a quarter of the variation in the total number of hardships that families report.[31]

The consensual approach or socially perceived necessities to poverty measurement initiated by Mack and Lansley (1985) and its survey structure were replicated elsewhere. In particular, it served as a model for a Basic Necessities Survey (BNS) (Davies 1997; Davies and Smith 1998). The BNS method weights each item by the proportion of people who said it is a basic necessity. It suggests defining a poverty cutoff across the BSN score such that it identifies as poor the same proportion of people as those who have subjectively identified themselves as poor. Davies’ BNS method was implemented in Vietnam and Mali (Nteziyaremye and MkNelly 2001), Bangladesh (Ahmed 2007), Ireland (Nolan and Whelan 1996), Japan (Abe 2006), Europe (Eurobarometer 2007), and South Africa (Wright 2008), among other countries.

4.3 Measures of Unsatisfied Basic Needs in Latin America and Beyond

Latin America is the other region where the direct method to measure poverty alongside a counting approach to identifying the poor has been widely implemented. Rather than a focus on social exclusion or ‘relative deprivation’ as in Europe, in Latin America it was operationalized under the unsatisfied basic needs (UBN) approach. The first implementation was in Chile in 1975 when the first ‘Map of Extreme Poverty’ was produced (Kast and Molina 1975). However, the method became known and generalized in the region with a seminal study conducted by the Institute of Statistics and Census of Argentina (INDEC) and the United Nations Economic Commission for Latin America and the Caribbean (ECLAC or Comisión Económica para América Latina y el Caribe/CEPAL in Spanish) (INDEC 1984). INDEC recognized the multidimensionality of poverty and sought to assess disadvantage across a wide set of basic needs or—alternatively—with information on income (p. 10). Thus, initially the UBN method was presented as an imperfect proxy for income poverty measurement.

The selection of census indicators was first performed by ECLAC with an empirical study using data from the 1980 census of Argentina. The study acknowledges that the census did not provide data on income or consumption nor on key health variables such as nutrition. However, the census provided data from all areas in the country and, importantly, with a useful level of disaggregation at smaller geographical entities. Within these constraints, three criteria guided the selection of indicators (INDEC 1984: 11):

1. The indicators represented the degree of failure to satisfy some specific group of basic needs;

2. These indicators were significantly associated with [income] poverty;

3. They were comparable across regions of the country so that poverty maps could be constructed.

In order to fulfil the second criterion, as part of the project, CEPAL undertook an empirical study using data from a 1980 survey in two urban areas of Argentina: the Greater Buenos Aires area and Goya (taken as representative of other urban areas).[32] The aim was ‘to select the characteristics that not only represented some intrinsically important deprivation but were also sufficiently associated with situations of [income] poverty so as to represent the other [unmeasured] deprivations that constitute such situation’ (INDEC 1984: 500). Both absolute and relative poverty lines were used; the former followed Altimir (1979) and the latter was set at half of the mean private per capita consumption according to national accounts. Census indicators were selected if they were empirically assessed to be strong predictors of income poverty in regression analysis—thus not using normative criteria. Step two, thus, in practice, dominated the three criteria mentioned above as well as the three-step selection of (a) the basic need, (b) the specific indicator, and (c) the deprivation cutoff (Feres and Mancero 2001). The census indicators chosen by INDEC and CEPAL were:

1. Households with more than three people per room (overcrowding);

2. Households with precarious housing;

3. Households with no kind of toilet;

4. Households with children of school age (6–12 years old) not attending school;

5. Households with four or more dependents per occupied member (high dependency ratio) and whose household head’s education was at most second grade of primary education.

The union criterion was used: all members of any household with at least one unsatisfied basic need were considered poor. The intuition was that because very low deprivation cutoffs were used for each indicator, one sole deprivation seemed sufficient to signal poverty (Rio Group 2006: 110). However, the information was reported in different ways and using different cutoffs: alternative (1) the proportion of households and people experiencing each UBN, (2) the proportion of households and people with one or more UBN, and also (3) the proportion of people withtwo or more and three or more UBNs.

The set of census indicators outlined by INDEC and CEPAL for Argentina was replicated by official statistical institutes in many Latin American countries: Bolivia, Chile, Colombia, Ecuador, Guatemala, Honduras, Nicaragua, Paraguay, Peru, Uruguay, and Venezuela. While there was some variation in indicators, the dimensions considered remained essentially the same, as they were limited by the information contained in the countries’ censuses. Feres and Mancero (2001: 67) noted that they belonged to four broad categories:

1. Access to housing that met minimum housing standards;

2. Access to basic services that guarantee minimum sanitary conditions;

3. Access to basic education;

4. Economic capacity to achieve minimum consumption levels.

In all these countries, the UBN methodology was used to construct detailed and disaggregated poverty maps using census data. Poverty maps became a valuable tool for policy design (Kaztman 1996: 24). Coady, Grosh, and Hoddinott (2004) observe that poverty mapping has been widely used for geographical targeting purposes—and not only in Latin America. ‘Much of the history of poverty mapping has used a “basic needs” approach with poverty defined in terms of access to basic services. The simplest form of geographic targeting involves the use of a single variable such as nutritional status. … the choice of variables is largely guided by a combination of philosophy and data availability’ (Coady, Grosh, and Hoddinott 2004: 63). In other cases, such as in Argentina or Chile, maps were constructed using the proportion of people with different numbers of UBNs. Poverty maps have guided investments in infrastructure, implementation of public works programmes and social funds, subsidized services, and the allocation of conditional cash transfer (CCT) programmes (usually alongside a complementary targeting method).[33] While some countries have built consumption-based poverty maps (Elbers, Lanjouw, and Lanjouw 2002), these have been less common than the basic needs maps, and their policy interpretation is more challenging.[34]

Beyond the policy impact that basic needs poverty maps have, it is worth noting that, while in Europe the direct method of measuring poverty was implemented alongside an effort to collect new data that would (a) reveal socially perceived necessities and (b) distinguish whether the lack of items was enforced or by choice, in Latin America the direct method was restricted to the data available at the time (census data). Thus, the range of indicators that could be included was severely constrained. The direct method in Latin America did not seek to reflect people’s views of their own necessities, and it did not deliberately permit ‘choice’. Relatedly, it must also be noted that in Latin America the definition of the UBN indicators was done from an absolute poverty perspective, whereas in Europe it was justified with respect to a relative or perceptual concept of poverty. Despite these differences, a strong common feature emerged: the interest in crossing the direct method with the indirect one. This gave rise in Latin America to the ‘integrated method’ to measure poverty proposed by Beccaria and Minujin (1985) and Kaztman (1989). The indirect or income method was being applied using data from household surveys, which started to be progressively implemented in the late 1980s and 1990s in the region. An absolute income poverty line approach was applied using the cost of basic needs method (Altimir 1979). The idea of the integrated method was to identify four sets of people: (1) the income and UBN poor, (2) the UBN poor but income non-poor, (3) the income poor but UBN non-poor, and (4) the non-poor by any method, as expressed in Table 4.1. This could be done using data from household surveys, which collected information on the UBN indicators as well as on income.

Kaztman (1989) terms the first group ‘chronically poor’, not because of information on poverty over time but because he assumes that insufficient income coupled with at least one UBN (most had more than one) would reproduce poverty over time. This group would be equivalent to the ‘consistently poor’ in the European literature. Other names belie other assumptions, but in any case empirical mismatches proved widespread.[35]


As Boltvinik (1991) argued, these studies showed that the two methods, income poverty and UBN poverty, were (unintentionally) complementary, identifying to a great extent different slices of the population. Evidence from Montevideo (Uruguay) in 1984 and from Greater Buenos Aires (Argentina) in 1976 suggested that 10–15% of the households were in the ‘recently poor’ category, 4–9% of the households were in the ‘with inertial deprivations’ category, and only 7% were poor by both methods, i.e. in the chronically poor category. In Peru nearly 40% of the population were identified as both income and UBN poor (chronically poor). Yet 30% of the population was identified as either income poor or UBN poor but not both, which shows the ‘mismatches’ between the two methods, covering nearly the entire population.

Boltvinik (1992) proposed an ‘Improved Integrated Method to Measure Poverty’, which involved changes in each method separately, as well as in their combination. His method was applied in Mexico (Boltvinik 1995, 1996). He proposed first that UBN indicators be those associated with public investment, which could not be purchased individually, and dimensions that could be purchased using private resources should be considered in the income component.[36] This generated, second, a higher poverty line based on a more comprehensive basket of goods and services. Third, he incorporated gaps in the measurement of income poverty and also of UBN. Thus, rather than dichotomizing achievements in each of the UBN indicators, he proposed (controversially) computing normalized deprivation gaps for each indicator as if they had cardinal data. Fourth, he allowed deprivation gaps to take negative values (reflecting achievements above the deprivation cut-off) thus permitting substitutability across deprived and non-deprived items. Finally he normalized the gaps to vary between minus one and one. Boltvinik’s proposal entails cardinalizing ordinal variables, which imposes multiple value judgements for which there is no clear agreement (see section 2.3 for a detailed explanation of the problems involved). The problem is that measures thus constructed are very unlikely to be robust to different value judgements used in their construction.[37]

To identify the poor, Boltvinik suggested using three alternative poverty cutoffs. Boltvinik also discussed alternative methods for weighting the UBN indicators: (a) equal weights, (b) the complement of indicators’ deprivation rates (Desai and Shah 1988), and (c) a combination of monetary and time valuations[38] of each need. As in the case of the UBN index, negative gaps were allowed for income. These were normalized to range between minus one and one by dividing them by (the absolute value of) a normative maximum negative gap, and replacing them by minus one whenever the absolute value of the negative gap was higher than the maximum normative gap. On the other hand, each person would have an individual UBN score which would be the weighted sum of the normalized deprivation gaps (ranging from minus one to one). In the combined method, each UBN indicator would be weighted by the proportion of the total cost required to fulfil each set of needs, and an individual’s UBN score would be added to her income poverty score.

Boltvinik’s revised integrated method to measure poverty was altogether different from the integrated method outlined in Table 4.1 and is no longer a counting measure. It ceased to consider mismatches between the UBN poor and the income poor. His identification method is not a counting approach but relied on a score obtained as the weighted sum of the aggregated gaps of cardinalized ordinal data (permitting substitutability). The reasons for the ‘Improved Integrated Method to Measure Poverty’ not acquiring popularity seem to be (a) that it required a number of controversial estimations, such as those related to time use and monetary valuations of UBN indicators, (b) that it attached a cardinal meaning to categories of response in ordinal variables and thus the intensity in the UBN index was dependent on the particular cardinalization used (which again could be contested), (c) some steps such as the cardinalization of ordinal data and the consideration of negative gaps prevented the resulting measure from satisfying many desirable axiomatic properties outlined in section 2.5.[39] Overall, in trying to accomplish too much, the method lost the public intuition and policy relevance that characterizes the counting approach and direct method of poverty measurement. That being said, many important distinctions were considered in this development, including the importance of time poverty.

The UBN approach has also been used in other parts of the world. In the Arab Region, Lebanon pioneered the UBN approach in 1997, using eleven indicators in four dimensions from the 1994–5 population and HH survey (a mini census covering 10% of the population). The document Mapping Living Conditions in Lebanon was published in 1998 by the Lebanese Ministry of Social Affairs (MoSA) and UNDP, and was used to define poverty after the civil war in the absence of other data. This report, which mapped Lebanon’s six governorates and twenty-six districts, was updated in 2007 using 2004 survey data (UNDP and MoSA 2007), and an expanded index was published together with monetary poverty measures from the same survey in 2009. Iraq’s Ministry of Planning together with UNDP also completed a significant three-volume study Mapping of Deprivation and Living Conditions in Iraq using 2003 data (UNDP and MPDC 2006). The study was used for budget allocation and policy priorities. In 2011 the same partners published a second study using 2007 data. A seven-country study was also produced using PAPFAM data which covered seventeen indicators grouped into five dimensions: education, health, housing, home necessities, and economic conditions (League of Arab States et al. 2009). Jordan published a two-volume study using the same methodology using the 2010 data (first volume) and a comparative study 2002–10 (second volume) (MOPIC, DOS, UNDP, ESCWA). Other studies (ESCWA-AUDI) have covered particular topics such as urban deprivation index in Tripoli, Lebanon (Nehmeh 2013).

The counting approach to identifying the poor has also been used in a new generation of poverty measures with renewed interest being shown in the direct method that uses solid aggregation methodologies based on axiomatic frameworks analogous to those which gave rise to the advances in income poverty measurement in the 1970s and 1980s (Alkire and Santos 2014). The following sections review some of these.

4.3.1 Counting Approaches in Measures of Child Poverty

A counting approach to identifying the poor is a natural approach for various policy-oriented measures of child poverty. Understanding child poverty is widely agreed to require a multidimensional approach in both European and developing contexts (Trani et al. 2013, Boyden and Bourdillon 2012; Minujin and Nandy 2012; Gardiner and Evans, 2011). A pioneering and internationally comparable measure of child poverty in developing countries was computed in 2003 (Gordon et al. 2003; Gordon et al. 2001; UNICEF 2004), whose indicators and cutoffs reflect the Convention of the Rights of the Child. A number of studies have more recently measured and analysed child poverty using the AF method, including Alkire and Roche (2012), Apablaza and Yalonetzky (2011), Roche (2013), Trani et al. (2013), de Neubourg et al. (2012), and Dickerson and Popli (2013). In particular, it is worth highlighting that de Neubourg et al. (2012) is a step-by-step guide to implementing the Multiple Overlapping Deprivation Analysis (MODA) tool, developed at UNICEF’s Research Office for global child comparisons using the measure of the AF method.[40]

4.4 Counting Approaches in Targeting

The implementations of the counting approach observed in Europe, the US, Latin America, and elsewhere were originally developed mostly within universities, and later became a tool for policy design and even targeting, although usually complemented by some other methodology. However, other implementations of the counting approach have stemmed immediately from a much more pragmatic motivation: targeting beneficiaries in programmes run by the national or regional governments and non-governmental organizations.

One good illustration is the case of India, where a series of different methodologies have been used to identify rural households as ‘below the poverty line’ (BPL). BPL households are eligible for certain benefits, such as subsidized food or electricity, and programmes to construct housing and encourage self-employment. Poverty measurement in India has largely been based on consumption and expenditure poverty. Since 1992, the Indian government’s census-based targeting methods have gradually evolved towards a counting approach (GOI 2009; Alkire and Seth 2013c). For example, in 2002, the BPL census collected information on thirteen dimensions covering topics such as food, housing, work, land ownership, assets, and education, and an aggregate achievement approach was implemented. This methodology was criticized on a number of grounds, including the cardinalization of ordinal variables and the substitutability of achievements among others.[41] Alkire and Seth (2008) compare the 2002 BPL method with a method based on a counting approach and show the possible mismatches that may occur between the two methods.

In 2008, the Indian government appointed an Expert Group Committee, under the chairmanship of N. C. Saxena, to provide a critical review of the 2002 BPL methodology and data contents, and to propose a new method for identification.[42] Their three-stage proposal implicitly used a counting method with a union approach in the first two stages leading to a counting-based identification in the third. It sparked informative empirical studies and ongoing methodological debates (Drèze and Khera 2010; Roy 2011; Sharan 2011; Alkire and Seth 2013).

Other subnational initiatives in South Asia use counting approaches for targeting. Two cases might illustrate this. The first concerns the Indian state of Kerala, an emblematic case of development and poverty reduction, whose government has been using a counting approach for targeting poor households since the late 1990s. The method was originally developed by non-governmental organizations (NGOs) and subsequently used for a women-based participatory poverty eradication programme named ‘Kudumbashree’ (Thomas et al. 2009). Kudumbashree uses nine equally weighted indicators related to housing, water, sanitation, literacy, income sources, food, presence of infants, presence of mentally or physically challenged or chronically ill persons, and caste/tribe. If the household presents deprivations in four or more indicators, it is considered poor; if it presents eight or nine, it is destitute. The identification of poor households is verified by neighbourhood groups comprising households that live in proximity. The identified households are eligible for a number of programmes, including microcredit.[43]

In our second case, a counting approach to celebrating ‘graduation from poverty’ is used by two acclaimed Bangladeshi NGOs, the Grameen Bank and BRAC. The Grameen Bank, the ‘bank for the poor’, was founded by Muhammad Yunus in 1976 in Bangladesh, originally as a local microcredit project. The project evolved into a nationwide bank with over eight million borrowers of whom 96% were women, and has spread elsewhere. Grameen uses a set of ten indicators to identify participants. When a household has zero deprivations (intersection approach), it is considered to have ‘graduated’ from poverty. A counting approach to identifying the poor is also implicitly used by BRAC (Bangladesh Rural Advancement Committee), another prominent microfinance NGO, initiated by Fazle Hasan Abed in 1972 in Bangladesh, which has spread widely. The BRAC programme, ‘Target the Ultra-Poor Programme’ (TUP), uses a counting-based method to target asset grants, skills training, community support, and healthcare services.

Moving further east, in Indonesia the measurement of poverty has primarily used the indirect income approach. However, multidimensional perspectives using counting approaches are emerging (CBS 2008). A ‘family welfare approach’ was initially proposed by the Family Planning Coordination Board in 1999 (CBS 2008: 10). This approach identified a family as poor if it was deprived in one of five indicators (a union approach): religious freedom, meals per day, clothing, size of house, and access to modern medicine, but the approach was not implemented because the five indicators were not relevant to all families. The Central Bureau of Statistics (CBS) then proposed a ‘poverty criteria approach’, which identified people as poor if they were deprived in five out of eight indicators. The eight indicators were floor area; type of floor; water access; type of water; asset ownership; income per month; expenditure spent on food; and consumption of meat, fish, eggs, and chicken. A census instrument conducted in three provinces—South Kalimantan, DKI Jakarta, and East Java—in the years 1999, 2000, and 2011, respectively (CBS 2008: 18), used this method to determine whether households had the right to receive basic necessity subsidies (CBS 2008: 19).[44]

A distinct yet related methodology for identifying the poor is the poverty scorecard developed by Mark Schreiner (Schreiner 2002, 2006, 2010). Schreiner proposed the method both for measuring poverty as well as for targeting beneficiaries. The poverty scorecard uses an individual or household card, and grades five to ten achievements to produce a score. Indicators are sought that are strongly correlated with income poverty and have the following characteristics: ease of acceptance, inexpensive to observe and verify, already commonly collected, objective, liable to change over time as poverty status changes, variety vis-à-vis other selected indicators, and applicable across countries and across regions within a country. The indicators proposed for poverty scorecards for seven countries include housing quality; drinking water and toilet facilities; cooking arrangements; school attendance; ownership of land; and ownership of televisions, radios, or telephones (Schreiner 2010).

The indicators are fielded in nationally representative household surveys that also collect information on income or expenditure. Indicator weights are set through a logit regression as follows. The individuals are categorized into two groups: income poor and income non-poor, and this categorization is used as the dependent variable in the regression, with the selected indicators as explanatory variables. The logit weights are transformed such that all weights are non-negative integers and the minimum score is 0 and the maximum is 100.[45] For example, in his proposed scorecard for Pakistan (Schreiner 2010), if the household does not have a flush toilet (most deprived), it receives a score of 0; if it has a flush toilet to pit (less deprived), it receives fourteen points; and if it has a flush toilet to public sewer (not deprived at all), it receives nineteen points. The total poverty score for each household is obtained as the sum of the household’s scores obtained in all indicators. A person is identified as poor if that person’s poverty score lies below a poverty threshold which, as Schreiner indicates, can be determined according to the aim and scope of the particular programme.

As in the case of Boltvinik’s method, Schreiner’s poverty scorecard method departs from a counting approach. Furthermore, it cardinalizes ordinal data, based on logit regressions, which does not seem legitimate, as section 2.3 argued. Scores are then standardized and aggregated to obtain an overall score, which is compared to an overall threshold. This step, like the aggregate achievement approach (section 2.2.2), allows substitutability between non-deprived and deprived achievements. If all the variables had been cardinal, the score would be the (weighted) sum of achievements. But given that usually most variables are ordinal, such a score actually has no direct interpretation. This procedure has been followed in the Unsatisfied Basic Needs Index in Lebanon, Iraq, and other Arab states.[46] With a particular normalization of the variables, it has also been the method used by the Scottish Area Deprivation Index (Kearns, Gibb and Mackay 2000), as well as by the Multidimensional Poverty Assessment Tool (MPAT) (Cohen 2010; Saisana and Saltelli 2010), among others.

With the method described above, Schreiner developed poverty scorecards for various microfinance institutions, and also developed adaptations such as the Progress out of Poverty Index (PPI).[47] A related method was used by the Benazir Income Support Programme in Pakistan, which targets benefit recipients using a scorecard of twelve observable indicators, each of which receives a weight based on an Ordinary Least Squares (OLS) model of household expenditure per adult equivalent regressed on various sets of predictors (proxy-means test) (Khan and Qutub 2010).[48] The approach thus has spread widely yet without clarifying fundamental methodological concerns.

In the area of targeting, the AF methodology is also underway both via academic studies and in policy programmes. For example, Robano and Smith (2014) examine the TUP programme of BRAC, developing measures for the existing targeting methods as well as for a proposed alternative, and present and implement an impact evaluation methodology using rather than any single outcome as the dependent variable. Azevedo and Robles (2013) propose an multidimensional targeting approach to identifying beneficiaries that explicitly takes into consideration the multiple objectives of conditional cash transfer programmes and the multiple deprivations of the poor household. Using data from Mexico’s prominent Oportunidades programme, they find multidimensional targeting to be significantly better than either the current targeting method or an alternative income proxy-means test at identifying households with deprivations that matter for the programme objectives. An ex ante evaluation suggests that programme transfers could have a greater impact if potential beneficiaries were selected by the AF method. Alkire and Seth (2013c) set out the powerful benefits of linking multidimensional targeting methods to national multidimensional poverty measures, such as policy coherence, monitoring and evaluation synergies, and the ability to update the targeting methodology and the targeting census instrument consistently across time. They suggest how an targeting method can be developed, justified, and linked with a national multidimensional poverty measure. This kind of approach is being implemented with increasing frequency: for example, Angulo et al. (2013) describe the geographical targeting that is used in Colombia.

In sum, the necessity of defining a target population for poverty reduction programmes has motivated the use of counting methods with a variety of specificities and prompted the development of related new identification methods. However, the measurement properties and features of the alternative targeting instruments are rarely discussed (or, one suspects, clearly communicated), which makes it difficult for policymakers to make an informed decision.

4.5 Final Comments on Counting Approaches

Counting approaches emerged as a natural procedure for identifying the poor with the basic needs and the social exclusion approaches, giving form to various direct methods to measure poverty. Counting the number of observable deprivations in core indicators has an appealing intuition and simplicity that has attracted not only academics but also policymakers and practitioners. Over time, counting methods have been implemented in a variety of useful formats in terms of poverty measurement—namely, the European Measures of Relative Deprivation, the Consensual Approach to Poverty Measurement, the Consistent Poverty Approach, the Latin American Basic Needs Approach—and it has been incorporated into solid axiomatic poverty measures in the academic literature. Moreover, the counting approach has also been used to measure child poverty and to construct targeting tools for poverty reduction programmes. The counting approach has motivated the collection of new data in some cases, and the construction of powerful policy tools such as poverty maps, in others.

It is also worth observing some prominent approaches that look similar to the counting approach yet differ in fundamental ways—such as assigning (by a normative or a statistical procedure) cardinal values to categories of ordinal variables, or using an aggregate line approach—and thus, in the end, are altogether different. This is the case in Boltvinik’s improved integrated method and Schreiner’s poverty scorecard method, among others.

The AF methodology uses a counting approach to identify the poor, and, as a consequence, it inherits its simplicity and intuition and stands on the shoulders of this venerable tradition in both academic and policy circles. Additionally, it introduces axiomatic rigour by (a) scrutinizing the counting approach as an identification method of the multidimensionally poor in a formal framework and (b) combining it with sound aggregation methodologies also within a formal axiomatic framework. Chapter 5 will present the AF methodology in depth.


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[1] Note that a counting approach to identifying the poor can be implemented only with multidimensional poverty measures that use unit-level data to consider the joint distribution of achievements across dimensions.

[2] The study of how economic growth occurs and how it advances basic needs has evolved significantly. See Commission on Growth and Development (CGD) (2008); Stiglitz, Sen, and Fitoussi (2009); Drèze and Sen (2013).

[3] Dag Hammarskjöld Foundation (1976); Herrera et al. (1976); ILO (1976).

[4] Philosophically, the basic needs approach seeks to elaborate some minimal material requirements of human well-being and justice. See Rawls (1971), Stewart (1985), Braybrooke (1987), Hamilton (2003), and Reader (2006).

[5] Alkire and Santos (2014) further elaborate the income method vs direct methods of poverty measurement.

[6] Earlier the Poor Laws in England and Wales provided a nascent welfare system. They started in mid-1350s in response to the Black Death in England, and an increase in the number of beggars and people looking for better pay as feudalism started to decay. The goal was to induce every able-bodied person to work (Townsend 1786; Quigley 1998; Hollen Lees 1998). Targeting was accomplished using a mix of ‘visual’ verification and ‘self-targeting’ rather than income poverty or counting-based measures.

[7] Cf. Tout (1938), Pagani (1960), Dubois (1899), and Townsend (1952).

[8] Johansson (1973) had already raised the need for measures to employ indicators having dichotomous and ordinal scales; cf. section 2.3, section 3.6, and Chapter 5.

[9] For example, in 1989 the European Commission proposed a ‘Community Charter of Fundamental Social Rights’, which was adopted by eleven of the twelve then-member states. See discussions in Room (1995), Silver (1995), and Nolan and Whelan (1996, 2011).

[10] In order to define the ‘minimally acceptable way of life’ for empirical measurement, different processes were explored including the socially perceived necessities approach commented on later.

[11] Sen’s capability approach built upon the basic needs approach: ‘The focus on basic capabilities can be seen as a natural extension of Rawls’s concern with primary goods, shifting attention from goods to what goods do to human beings’ (Sen 1979: 218–19).

[12] See Nolan and Whelan (1996, 2011) for a more thorough review of counting approaches to identify the poor as well as for a review of empirical evidence of the mismatches between income poverty and non-monetary deprivation.

[13] As we shall see, some implementations of counting approaches use multiple poverty cutoffs (i.e. required alternative numbers of deprivations to identify the poor). While informative, this stops short of incorporating intensity into one summary measure, which, like the Adjusted Headcount Ratio (), can be broken down by dimension.

[14] Townsend used the terms deprived and poor interchangeably, whereas we define these terms differently.

[15] Note however that he did not use the term ‘union’ to refer to this criterion.

[16] This pioneering study analysed patterns of what people considered necessary, correlations with income, and the free choice of voluntary deprivations. It includes vivid testimonies from interviews and a fascinating discussion of contemporary policies.

[17] The authors acknowledged that they could have discerned the ‘seriousness’ of deprivations in different indicators by assigning more weight to items considered by more people as a necessity (i.e. by the rank order of the necessities). However, they dismissed this possibility by arguing that people in poverty with an equal number but different combinations of indicators should, by definition, face an equally ‘serious’ situation.

[18] ‘Two criteria have been applied: first, those who lack this level of necessities [three out of twenty-two] should have low incomes, falling in the bottom half of the income range; second, their overall spending patterns should reflect financial difficulty rather than high spending on other goods’ (Mack and Lansley 1985: 175–6).

[19] Gordon et al. (2000: 72, Appendix 1) lists other studies using the Breadline Britain survey.

[20] These surveys do not collect information on socially perceived necessities, but do ascertain whether the lack of an item is voluntary.

[21] The thirty-five items selected in 1999 were such that 50% of people or more considered them as socially perceived necessities.

[22] A discriminant function analysis (DFA) divides the population into poor and non-poor by predicting whether each person belongs to one group or the other based on a set of characteristics of the unit of analysis, taken as ‘explanatory variables’. In these studies, alternative numbers of deprivations (poverty cut-offs) were tested, and the explanatory variables included family income, the employment status of the household, the number of children, ethnicity, and region of residence, among many others.

[23] Ringen (1987) stated, ‘we need to establish not only that people live as if they were poor but that they do so because they do not have the means to avoid it’ (p. 162, cited in Callan, Nolan, and Whelan 1993).

[24] The eight items are: going into arrears/debt to meet ordinary living expenses such as food and rent, not having a substantial meal all day, having to go without heating because of lack of money, involuntary lack of new clothes, lack of two pairs of shoes, not being able to afford a roast or equivalent once a week, not being able to afford a meal with meat or fish every second day, and not being able to afford a warm coat.

[25] Those fulfilling both conditions were identified as experiencing ‘generalised deprivation’ due to a lack of resources (Callan, Nolan, and Whelan 1993; Nolan and Whelan 1996). They also identified subgroups who were income poor but not materially deprived and vice versa.

[26] Callan et al. (1999), Whelan et al. (2001a), Layte et al. (2000), and Whelan, Nolan, and Maître (2006), among others.

[27] The reason for using the subjective assessment score of 5.5 is that in the Netherlands schooling system, a score of 5.5 in a 1–10 scale is considered to be a dividing line between a ‘satisfactory’ and an ‘unsatisfactory’ score.

[28] The groups were: men, women, age groups, household types, and geographic regions.

[29] Boarini and d’Ercole (2006: 33) and Eurostat (2002: 25) present these.

[30] This threshold was selected so that the average rate of non-monetary poverty across the fifteen countries equalled the average income poverty rate.

[31] Bauman (1998, 1999) critically evaluated their work and instead advocated the use of measures of hardship to complement, not substitute for, income poverty measures.

[32] The Encuesta Permanente de Hogares was already being conducted regularly by INDEC but it was restricted to the Greater Buenos Aires area.

[33] Progresa (later renamed Oportunidades) in Mexico and Bolsa de Familia in Brazil started in 1997 and were pioneer programmes in Latin America and in the world (Fiszbein and Schady 2009).

[34] The construction of income poverty maps typically matches a census with household survey information to predict income poverty, which is not directly measured in the census. Tarozzi and Deaton (2009) criticize such methods as they require a degree of spatial homogeneity, which is not guaranteed by the matching methods. See also Elbers et al. (2007) and Bedi et al. (2007).

[35] See Kaztman (1989: 130) and Stewart et al. (2007).

[36] Boltvinik (1992, 2012) proposed that the UBN indicators should be as follows: sanitation, electricity, and services (such as phone and garbage collection); housing and overcrowding; education; furniture and appliances; access to health care and social security. He considered the following items to be covered by income poverty: food, petrol, personal and household hygiene, clothing, footwear and personal care, transport, basic communications, recreation and culture, basic services expenditure in health care and education, and other expenditures.

[37] Boltvinik (1992) also proposes independent and revised UBN and income poverty measures.

[38] Boltvinik (1992: 360–1) has a detailed description of how this alternative was implemented.

[39] Boltvinik’s aggregation method resembles the aggregate achievement approach described in sections 2.2.2 and 3.6.2.

[40] <>, accessed 24 November 2014.

[41] For a list of criticisms, see Sundaram (2003), Hirway (2003), Jain (2004), Mukherjee (2005), Jalan and Murgai (2007), Alkire and Seth (2008), GOI (2009), Thomas et al. (2009), Drèze and Khera (2010), and Roy (2011).

[42] Alkire and Seth (2013c) provide further details.

[43] The Kolkata-based NGO Bandhan also uses a counting approach to identify participants and ‘graduates’ from poverty.

[44] For a few other attempts and proposals for multidimensional non-counting approaches to poverty measurement, see Pradhan et al. (2000) and Gönner et al. (2007).

[45] Schreiner (2010) divides the sample into components for ‘construction’ (50%), ‘calibration’ (25%), and ‘validation’.

[46] UNDP and MoSA (1998, 2007); UNDP and MDPC (2006); League of Arab States (2009); Nehmeh (2013).

[47] <>, accessed April 2013.

[48] <>, accessed April 2013.

 2 The framework

This chapter introduces the notation and basic concepts that will be used throughout the book. The chapter is organized into four sections. Section 2.1 starts with a review of unidimensional poverty measurement with particular attention to the well-known FGT measures (Foster, Greer, and Thorbecke 1984) because many methods presented in Chapter 3, as well as the Alkire and Foster (2007, 2011a) measures presented in Chapter 5, are based on FGT indices. Section 2.2 introduces the notation and basic concepts for multidimensional poverty measurement that will be used in subsequent chapters. Section 2.3 delves into the issue of indicators’ scales of measurement, an aspect often overlooked when discussing methods for multidimensional analysis and which is central to this book. Section 2.4 addresses comparability across people and dimensions. Finally Section 2.5 presents in a detailed form the different properties that have been proposed in axiomatic approaches to multidimensional poverty measurement. Such properties enable the analyst to understand the ethical principles embodied in a measure and to be aware of the direction of change they will exhibit under certain transformations.

2.1 Review of Unidimensional Measurement and FGT Measures

The measurement of multidimensional poverty builds upon a long tradition of unidimensional poverty measurement. Because both approaches are technically closely linked, the measurement of poverty in a unidimensional way can be seen as a special case of multidimensional poverty measurement. This section introduces the basic concepts of unidimensional poverty measurement using the lens of the multidimensional framework, so serves as a springboard for the later work. The measurement of poverty requires a reference population, such as all people in a country. We refer to the reference population under study as a society. We assume that any society consists of at least one observation or unit of analysis. This unit varies depending on the measurement exercise. For example, the unit of analysis is a child if one is measuring child poverty, it is an elderly person if one is measuring poverty among the elderly, and it is a person or—sometimes due to data constraints—the household for measures covering the whole population. For simplicity, unless otherwise indicated, we refer to the unit of analysis within a society as a person (Chapter 6 and Chapter 7) We denote the number of person(s) within a society by , such that is in or , where is the set of positive integers. Note that unless otherwise specified, refers to the total population of a society and not a sample of it. Assume that poverty is to be assessed using number of dimensions, such that . We refer to the performance of a person in a dimension as an achievement in a very general way, and we assume that achievements in each dimension can be represented by a non-negative real valued indicator. We denote the achievement of person in dimension by for all and , where is the set of non-negative real numbers, which is a proper subset of the set of real numbers .[30] Subsequently, we denote the set of strictly positive real numbers by Throughout this book, we allow the population size of a society to vary, which allows comparisons of societies with different populations. When we seek to permit comparability of poverty estimates across different populations, we assume to denote a fixed set (and number) of dimensions.[31] The achievements of all persons within a society are denoted by an -dimensional achievement matrix which looks as follows: formula1 We denote the set of all possible matrices of size by and the set of all possible achievement matrices by , such that . If , then matrix contains achievements for persons in dimensions. Unless specified otherwise, whenever we refer to matrix , we assume . The achievements of any person in all dimensions, which is row of matrix , are represented by the -dimensional vector for all . The achievements in any dimension for all persons, which is column of matrix , are represented by the -dimensional vector for all . In the unidimensional context, the dimensions considered in matrix —which are typically assumed to be cardinal—can be meaningfully combined into a well-defined overall achievement or resource variable for each person , which is denoted by . One possibility, from a welfarist approach, would be to construct each person’s welfare from her vector of achievements using a utility function .[32] Another possibility is that each dimension refers to a different source of income (labour income, rents, family allowances, etc.). Then, one can construct the total income level for each person as the sum of the income level obtained from each source, that is . Alternatively, each dimension can be measured in the quantity of a good or service that can be acquired in a market. Then, one can construct the total consumption expenditure level for each person as the sum of the quantities acquired at market price, that is , where , the price of commodity is used as its weight. In any of these three cases, the achievement matrix is reduced to a vector containing the welfare level or the resource variables of all persons. In other words, the distinctive feature of the unidimensional approach is not that it necessarily considers only one dimension, but rather that it maps multiple dimensions of poverty assessment into a single dimension using a common unit of account.[33]

2.1.1 Identification of the Income Poor

Since Sen (1976), the measurement of poverty has been conceptualised as following two main steps: identification of who the poor are and aggregation of the information about poverty across society. In unidimensional space, the identification of who is poor is relatively straightforward: the poor are those whose overall achievement or resource variable falls below the poverty line , where the subscript simply signals that this is a poverty line used in the unidimensional space. Analogous to the construction of the resource variable, the poverty line can be obtained aggregating the minimum quantities or achievements considered necessary in each dimension. It is assumed that such quantities or levels are positive values, that is .[34] These minimum levels are collected in the -dimensional vector . If the overall achievement is the level of utility, a utility poverty line needs to be set as .[35] On the other hand, when the overall achievement is total income or total consumption expenditure, the poverty line is given by the estimated cost of the basic consumption basket —or some increment thereof.[36] Then, given the person’s overall resource value or utility value and the poverty line, we can define the identification function as follows: identifies person as poor if , that is, whenever the resource or utility variable is below the poverty line, and identifies person as non-poor if . We denote the number of unidimensionally poor persons in a society by and the set of poor persons in a society by , such that .

2.1.2 Aggregation of the Income Poor

In terms of aggregation, a variety of indices have been proposed.[37] Among them, the Foster, Greer and Thorbecke or FGT (1984) family of indices has been the most widely used measures of poverty by international organizations such as the World Bank and UN agencies, national governments, researchers, and practitioners.[38] For simplicity, we assume the unidimensional variable to be income. Building on previous poverty indices including Sen (1976) and Thon (1979), the FGT family of indices is based on the normalized income gap—called the ‘poverty gap’ in the unidimensional poverty literature—which is defined as follows: Given the income distribution , one can obtain a censored income distribution by replacing the values above the poverty line by the poverty line value itself and leaving the other values unchanged. Formally, if and if . Then, the normalized income gap is given by: formula-2.1   The normalized income gap of person is her income shortfall expressed as a share of the poverty line. The income gap of those who are non-poor is equal to 0.[39] The individual income gaps can be collected in an -dimensional vector . Each element is the normalized poverty gap raised to the power and it can be interpreted as a measure of individual poverty where is a ‘poverty aversion’ parameter. The class of FGT measures is defined as , thus can be interpreted as the average poverty in the population. The FGT measures can also be expressed in a more synthetic way as , where is the mean operator and thus denotes the average or mean of the elements of vector . This presentation of the FGT indices is useful in understanding the AF class (Alkire and Foster 2011a). Within the FGT measures, three measures, associated with three different values of the parameter , have been used most frequently. The deprivation vector , for , replaces each income below the poverty line with 1 and replaces non-poor incomes with 0. Its associated poverty measure is called the headcount ratio, or the mean of the deprivation vector. It indicates the proportion of people who are poor, also frequently called the incidence of poverty.[40] The normalized gap vector , for , replaces each poor person’s income with the normalized income gap and assigns 0 to the rest. Its associated measure , the poverty gap measure, reflects the average depth of poverty across the society. The squared gap vector for , replaces each poor person’s income with the squared normalized income gap and assigns 0 to the rest. Its associated measure—the squared gap or distribution sensitive FGT—is ; it emphasizes the conditions of the poorest of the poor as Box 2.1 explains. The FGT measures satisfy a number of properties, including a subgroup decomposability property that views overall poverty as a population-share weighted average of poverty levels in the different population subgroups.[41] As noted by Sen (1976), the headcount ratio violates two intuitive principles: (1) monotonicity: if a poor person’s resource level falls, poverty should rise and yet the headcount ratio remains unchanged; (2) transfer: poverty should fall if two poor persons’ resource levels are brought closer together by a progressive transfer between them, and yet the headcount ratio may either remain unchanged or it can even go down. The poverty gap measure satisfies monotonicity, but not the transfer principle; the measure satisfies both monotonicity and the transfer principle.[42]

Box 2.1. A numerical example of the FGT measures

A simple example [43] can clarify the method and these axioms, and will also prove useful in linking the Alkire and Foster methodology (fully described in Chapter 5) to its roots in the FGT class of poverty measures. Consider four persons whose incomes are summarized by vector and the poverty line is . The headcount ratio : Consider first the case of . Each gap is replaced by a value of 1 if the person is poor and by a value of 0 if non-poor. The deprivation vector is given by: , indicating that the second and third persons in this distribution are poor. The mean of this vector—the measure—is one half: , indicating that 50% of the population in this distribution is poor. Undoubtedly, it provides very useful information. However, as noted by Watts (1968) and Sen (1976), the headcount ratio does not provide information on the depth of poverty nor on its distribution among the poor. For example, if the third person became poorer, experiencing a decrease in her income so that the income distribution became , the measure would still be one half; that is, it violates monotonicity. Also, if there was a progressive transfer between the two poor persons, so that the distribution was , the measure would not change, violating the transfer principle. This has policy implications. If this was the official poverty measure, a government interested in maximising the impact of resources on poverty reduction would have an incentive to allocate resources to the least poor, that is, those who were closest to the poverty line, leaving the lives of the poorest of the poor unchanged. The poverty gap (or FGT-1): Here Each gap is raised to the power , giving the proportion in which each poor person falls short of the poverty line and if the person is non-poor. The normalized gap vector is given by . The measure is the mean of this vector. , indicates that the society would require an average of 20% of the poverty line for each person in the society to remove poverty. In fact, $4 is the overall amount needed in this case to lift both poor persons above the poverty line. Unlike the headcount ratio , the measure is sensitive to the depth of poverty and satisfies monotonicity. If the income of the third person decreased so we had the corresponding normalized gap vector would be , so . Clearly, . Indeed, all measures with satisfy monotonicity. However, a transfer to an extremely destitute person from a less poor person would not change , since the decrease in one gap would be exactly compensated by the increase in the other. By being sensitive to the depth of poverty (i.e. satisfying monotonicity), the measure does make policy makers want to decrease the average depth of poverty as well as reduce the headcount. But because of its insensitivity to the distribution among the poor, does not provide incentives to target the very poorest, whereas the FGT-2 measure does. The Squared Poverty Gap (or FGT-2): When we set , each normalized gap is squared or raised to the power . The squared gap vector in this case is given by: . By squaring the gaps, bigger gaps receive higher weight. Note for example that while the gap of the second person () is three times bigger than the gap of the third person (), the squared gap of the second person () is nine times bigger than the gap of the third person (). The mean of the vector—the measure—is . The measure is sensitive to the depth of poverty: if the income of the third person decreases one unit such that , the squared gap vector becomes , increasing the aggregate poverty level to ). It is also sensitive to the distribution among the poor: if there is a transfer of $1 from the third person to the second one, so becomes , the squared gap vector becomes , decreasing the aggregate poverty level to . Squaring the gaps has the effect of emphasising the poorest poor and providing incentives to policy makers to address their situation urgently. All measures with satisfy the transfer property

2.2 Notation and Preliminaries for Multidimensional Poverty Measurement

We now extend the notation to the multidimensional context. We represent achievements as dimensional achievement matrix , as in the unidimensional framework described in section 2.1. We make two practical assumptions for convenience. We assume that the achievement of person in dimension can be represented by a non-negative real number, such that for all and . Also, we assume that higher achievements are preferred to lower ones.[44] In a multidimensional setting, in contrast to a unidimensional context, the considered achievements may not be combinable in a meaningful way into some overall variable. In fact, each dimension can be of a different nature. For example, one may consider a person’s income, level of schooling, health status, and occupation, which do not have any common unit of account. As in the unidimensional case, we allow the population size of a society to vary, and we assume to denote a fixed set (and number) of dimensions.[45] We denote the set of all possible matrices of size by and the set of all possible achievement matrices by , such that . If , then matrix contains achievements for persons and a fixed set of dimensions. Unless specified otherwise, whenever we refer to matrix , we assume . The achievements of any person in all dimensions, which is row of matrix , are represented by the -dimensional vector for all . The achievements in any dimension for all persons, which is column of matrix , are represented by the -dimensional vector for all . In multidimensional analysis, each dimension may be assigned a weight or deprivation value based on its relative importance or priority. We denote the relative weight attached to dimension by , such that for all . The weights attached to all dimensions are collected in a vector . For convenience we may restrict the weights such that they sum to the total number of considered dimensions, that is: Alternatively, weights may be normalized; in other words, the weights sum to one: .[46]

2.2.1 Identifying Deprivations

A common first step in multidimensional poverty assessment in several of the methodologies reviewed in Chapter 3, as well as in the Alkire and Foster (2007, 2011a) methodology, requires defining a threshold in each dimension. Such a threshold is the minimum level someone needs to achieve in that dimension in order to be non-deprived. It is called the dimensional deprivation cutoff. When a person’s achievement is strictly below the cutoff, she is considered deprived. We denote the deprivation cutoff in dimension by ; the deprivation cutoffs for all dimensions are collected in the -dimensional vector . We denote all possible -dimensional deprivation cutoff vectors by . Any person is considered deprived in dimension if and only if . For several measures reviewed in Chapter 3, and for the AF method, it will prove useful to express the data in terms of deprivations rather than achievements. From the achievement matrix and the vector of deprivation cutoffs , we obtain a deprivation matrix (analogous to the deprivation vector in the unidimensional context) whose typical element whenever and , otherwise, for all and for all . In other words, if person is deprived in dimension , then the person is assigned a deprivation status value of 1 and 0, otherwise. Thus, matrix represents the deprivation status of all persons in all dimensions in matrix . Vector represents the deprivation status of person in all dimensions and vector represents the deprivation status of all persons in dimension . From the matrix one can construct a deprivation score for each person such that . In words, denotes the sum of weighted deprivations suffered by person .[47] In the particular case in which weights are equal and sum to the number of dimensions, the score is simply the number of deprivations or deprivation counts that the person experiences. Whenever weights are unequal but sum to the number of dimensions, person deprivation score is defined as the sum of her weighted deprivation counts. The deprivation scores are collected in an -dimensional column vector . On certain occasions, it will be useful to use the deprivation-cutoff-censored achievement matrix which is obtained from the corresponding achievement matrix in , replacing the non-deprived achievements by the corresponding deprivation cutoff and leaving the rest unchanged. We denote the th element of by . Then, formally, if and , otherwise. In this way, all achievements greater than or equal to the deprivation cutoffs are ignored in the censored achievement matrix. When data are cardinally meaningful for all and all , and , in other words, when all the achievements take non-negative values and the deprivation cutoffs take strictly positive values, one can construct dimensional gaps or shortfalls from the censored achievement matrix as:[48] formula-2-2 Each or normalized gap, expresses the shortfall of person in dimension as a share of its deprivation cutoff. Naturally, the gaps of those whose achievement is above the corresponding dimensional deprivation cutoff are equal to 0. Generalizing the above, the individual normalized gaps can be collected in an dimensional matrix where each element is the normalized gap defined in (2.2) raised to the power ; such normalized gaps can be interpreted as a measure of individual deprivation in dimension . When , we have the deprivation matrix already defined. When , we have the matrix of normalized gaps, and when , we have the matrix of squared gaps. Analogous to the FGT measures, is a deprivation aversion parameter.

2.2.2 Identification and Aggregation in the Multidimensional Case

Sen’s (1976) steps of identification of the poor and aggregation also apply to the multidimensional case. It is clear that the identification of who is poor in the unidimensional case is relatively straightforward. The poverty line dichotomizes the population into the sets of poor and non-poor. In other words, in the unidimensional case, a person is poor if she is deprived. However, in the multidimensional context, the identification of the poor is more complex: the terms ‘deprived’ and ‘poor’ are no longer synonymous. A person who is deprived in any particular dimension may not necessarily be considered poor. An identification method, with an associated identification function, is used to define who is poor. We denote the identification function by , such that identifies person as poor and identifies person as non-poor. Analogous to the unidimensional case, we denote the number of multidimensionally poor people in a society by and the set of poor persons in a society by , such that . It could be the case that the identification method is based on some ‘exogenous’ variable, in that it is a variable not included in achievement matrix For example, the exogenous variable could be being beneficiary of some government programme or living in a specific geographic area. One may also define an identification method based on one particular dimension of matrix . One may consider the corresponding normative cutoff to identify the person as poor, in which case the function is , or one may consider a relative cutoff identifying as poor anyone who is below the median or mean value of the distribution, in which case the function is . Alternatively, identification may be based on the whole set of achievements not necessarily considering dimensional deprivation cutoffs but rather the relative position of each person on the aggregate distribution . There are many different ways of identifying the poor in the multidimensional context. A particularly prevalent set of methods consider the person’s vector of achievements and corresponding deprivation cutoffs, such that identifies person as poor and identifies person as non-poor.[49] Within this specification of the identification function, at least two approaches can be followed. An approach closely approximating unidimensional poverty is the ‘aggregate achievement approach’, which consists of applying an aggregation function to the achievements across dimensions for each person to obtain an overall achievement value. The same aggregation function is also applied to the dimensional deprivation cutoffs to obtain an aggregate poverty line. As in the unidimensional case, a person is identified as poor when her overall achievement is below the aggregate poverty line. Another method, which we refer to as ‘censored achievement approach’, first applies deprivation cutoffs to identify whether a person is deprived or not in each dimension and then identifies a person considering only the deprived achievements. The ‘counting approach’ is one possible censored achievement approach, which identifies the poor  according to the number (count) of deprivations they experience. Note that ‘number’ here has a broad meaning as dimensions may be weighted differently. Chapter 4 and the AF method (Ch 5-10) use a counting approach. When the scale of the variables allows, other identification methods could be developed using the information on the deprivation gaps.
In counting identification methods, the criterion for identifying the poor can range from ‘union’ to the ‘intersection’. The union criterion identifies a person as poor if the person is deprived in any dimension, whereas the intersection criterion identifies a person as poor only if she is deprived in all considered dimensions. In between these two extreme criteria there is room for intermediate criteria. Many counting-based measurement exercises since the mid-1970s have used intermediate criteria (see Chapter 4). The AF methodology formally incorporated it into an axiomatic framework.[50] Once the identification method has been selected, the aggregation step requires selecting a poverty index, which summarizes the information about poverty across society. A poverty index is a function that converts the information contained in the achievement matrix and the deprivation cutoff vector into a real number. We denote a poverty index as . An identification and an aggregation method that are used together constitute what we call a multidimensional poverty methodology, and we denote it as . It will prove useful to introduce notation for two further partial indices related to the overall poverty index . Each of them offers information on different ‘slices’ of the achievement matrix as analysed by the corresponding multidimensional poverty methodology ; that is, the partial indices are dependent on the selected identification and aggregation methods. One of these partial indices is the poverty level of a particular subgroup within the total population, denoted as , where is the achievement matrix of this particular subgroup (which could be people of a particular ethnicity, for example) contained in matrix . Visually, this partial index is based on a horizontal slice of the achievement matrix (i.e. a set of rows). The other partial index is a function of the post-identification dimensional deprivations, denoted as , where, as stated above, vector represents the achievements in dimension for all persons and is the deprivation cutoffs vector.[51] This is precisely why the full vector of deprivation cutoffs is an argument of the index and not just the particular deprivation cutoff . Recall that under some identification methods, it is possible to have some people who experience deprivations but are not identified as poor (for example, when a counting approach is used with an identification strategy that it is not union). In such cases, their deprivations will not be considered in the poverty measure and therefore will not be considered in this partial index either. In a visual way, this partial index is based on a vertical slice of the achievement matrix (i.e. a column). These two partial indices will be used when introducing the different principles of multidimensional poverty measures in section 2.5.3. As we shall see, although identification and aggregation have, since Sen (1976), usually been recognized as key steps in poverty measurement, some methods in the multidimensional context do not follow these steps. We will clarify this in Chapter 3.

2.2.3 The Joint Distribution

Throughout this book we will frequently refer to the joint distribution in contrast to the marginal distribution and we will also use the expression joint deprivations. The concept of a joint distribution comes from statistics where it can be represented using a joint cumulative distribution function.[52] The relevance of the joint distribution in multidimensional analysis was articulated by Atkinson and Bourguignon (1982), who observed that multidimensional analysis was intrinsically different because there could be identical dimensional marginal distributions but differing degrees of interdependence between dimensions.[53] In this book we treat the achievement matrix as a representation of the joint distribution of achievements. Each row contains the (vector of) achievements of a given person in the different dimensions, and each column contains the (vector of) achievements in a given dimension across the population. From that matrix, considered with deprivation cutoffs, it is possible to obtain the proportion of the population who are simultaneously deprived in different subsets of dimensions. In other words, it is possible to obtain the proportion of people who experience each possible profile of deprivations. This is visually clear in the deprivation matrix , which represents the joint distribution of deprivations. The higher order matrices and obviously offer further information regarding the joint distribution of the depths of deprivations. The importance of considering the joint distribution of achievements, which in turn enables us to look at joint deprivations, is best understood in contrast with the alternative of looking at the marginal distribution of achievements, and, thus, the marginal deprivations. The marginal distribution is the distribution in one specific dimension without reference to any other dimension.[54] The marginal distribution of dimension is represented by the column vector . From the marginal distribution of each dimension, it is possible to obtain the proportion of the population deprived with respect to a particular deprivation cutoff. However, by looking at only the marginal distribution, one does not know who is simultaneously deprived in other dimensions.[55] Table 2.1 illustrates the relevance of the joint distribution in the basic case of persons and dimensions using a contingency table.


Table 2.1. Joint distribution of deprivation in two dimensions

Table-2.1-v2We denote the number of people deprived and non-deprived in the first dimension by and , respectively; whereas, the number of people deprived and non-deprived in the second dimension are denoted by and , respectively. These values correspond to the marginal distributions of both dimensions as depicted in the final row and final column of the table. They could equivalently be expressed as proportions of the total, in which case, for example, () would represent the proportion of people deprived (or the headcount ratio) in Dimension 1. The marginal distributions, however, do not provide information about the joint distribution of deprivations, which is described in the four internal cells of the table. In particular, the number of people deprived in both dimensions is denoted by , the number of people deprived in the first but not the second dimension is denoted by , and the number of people deprived in the second and not in the first dimension is denoted by . We know that people are deprived in both dimensions and the sum of is the number of people deprived in at least one dimension. These values correspond to the joint distribution of deprivations. Consider now the case of four dimensions and four people, to see how valuable information can be added by the joint distribution. Table 2.2[56] presents the deprivation matrix of two hypothetical distributions, and . Such a matrix presents joint distributions of deprivations in a compact way and is used regularly throughout this book.


Table 2.2. Comparison of two joint distributions of deprivations in four dimensions


In the table, the marginal distributions of each dimension’s are identical in deprivation matrices and . Thus, the proportions of people deprived in each dimension are the same in the two distributions (25%). Yet, while, in distribution one person is deprived in all dimensions and three people experience zero deprivations, in distribution , each of the four persons is deprived in exactly one dimension. In other words, although the marginal distributions are identical, the two joint distributions and are very different. We understand that multiple deprivations that are simultaneously experienced are at the core of the concept of multidimensional poverty, and this is the reason why the consideration of the joint distribution is important. However, as we shall see, not all methodologies consider the joint distribution. In the next section, we introduce the notation for two methodologies of this type.

2.2.4 Marginal Methods

Some of the methods for multidimensional poverty assessment introduced in Chapter 3 can be called marginal methods because they do not use information contained in the joint distribution of achievements. In other words, they ignore all information on links across dimensions. Following Alkire and Foster (2011b), a marginal method assigns the same level of poverty to any two matrices that generate the same marginal distributions. In Table 2.2, a marginal method would assign the same poverty level to distribution (four deprivations are experienced by one person) and distribution (each person experiences exactly one deprivation). That is, it would not be able to show whether the deprivations are spread evenly across the population or whether they are concentrated in an underclass of multiply deprived persons. Such marginal methods can also be linked to the order of aggregation while constructing poverty indices (Pattanaik et al 2012). Specifically, a measure can be obtained by first aggregating achievements or deprivations across people (column-first) within each dimension and then aggregating across dimensions, or it can be obtained by first aggregating achievements or deprivations for each person (row-first) and then aggregating across people. Only measures that follow the second order of aggregation (i.e., first across dimensions for each person and then across persons) reflect the joint distribution of deprivations (Alkire 2011: 61, Figure 7). Measures that follow the first order of aggregation fall under marginal methods of poverty measurement.
Marginal methods also include cases where achievements for different dimensions are drawn from different data sources and/or from different reference groups within a population—as occurred, for example, in many indicators associated with the Millennium Development Goals (MDGs). In this case, rather than having an dimensional matrix of achievements, one may have a ‘collection’ of vectors , representing the achievements of people in dimension for all . Note that each vector may refer to different sets of people such as children, adults, workers, or females, to mention a few. Suppose as before, that a deprivation cutoff is defined. Then, we define a dimensional deprivation index for dimension by , which assesses the deprivation profile of people in dimension . The deprivation index might be simply the percentage of people who are deprived in this indicator or some other statistic such as child mortality rates. Note that deprivations are clearly identified in each dimension; however, because the underlying columns of dimensional achievements are not linked, no decision on who is to be considered multidimensionally poor can be made. This is a key difference between the dimensional deprivation index and the post-identification dimensional deprivation index which depends on the entire deprivation cutoff vector (and not just ) and thus captures the joint distribution of deprivations (see Section 2.2.2). Different dimensional deprivation indices can be considered in a set, constituting the ‘dashboard approach’, or combined by some aggregation function, what is often called a ‘composite index’ (Chapter 3).

2.2.5. Useful Matrix and Vector Operations

Throughout the book, we use specific vector and matrix operations. This section introduces the technical notation covering vectors and matrices. We denote the transpose of any matrix by where has the rows of matrix converted into columns. Formally, if and the th element of is written , then , where is the th element of for all and . The same notation applies to a vector, with being the transpose of . Thus, if is a row vector, is a column vector of the same dimension. As stated in section 2.1 the average or mean of the elements of any vector is denoted by , where . Similarly, the average or mean of the elements of any matrix is denoted by , where . Later in the book we use a related expression, the so-called ‘generalized mean of order’ . Given any vector of achievements , where , the expression of the weighted generalized mean of order is given by formula-2-3 where and . When weights are equal, for all . Each generalized mean summarizes distribution into a single number and can be interpreted as a ‘summary’ measure of well- or ill-being, depending on the meaning of the arguments . When for all , we write simply as . When , reduces to the arithmetic mean and is simply denoted by When , more weight is placed on higher entries and is higher than the arithmetic mean, approaching the maximum entry as tends to . For more weight is placed on lower entries, and is lower than the arithmetic mean, approaching the minimum entry as tends to . The case of is known as the geometric mean and as the harmonic mean. Expression (2.3) is also known as a constant elasticity of substitution function, frequently used as a utility function in economics. When generalized means are computed over achievements, it is natural to restrict the parameter to the range of giving a higher weight to lower achievements and penalizing for inequality (Atkinson 1970). Likewise when generalized means are computed over deprivations, it is natural to restrict the parameter to the range of giving a higher weight to higher deprivations and also penalizing for inequality. Box 2.2 contains an example of generalized means.

Box 2.2. Example of generalized Means[57]

Consider two distributions and with the following distribution of achievements in a particular dimension: and . We first show how to calculate for certain values of and then compare two distributions with a graph where ranges from to . In this example, we assume that all dimensions are equally weighted: . Arithmetic Mean:The arithmetic mean () of distribution is: . Geometric Mean:If , then is the ‘geometric mean’ and by the formula presented in (2.3) can be calculated as: . Harmonic Mean: If , then is the harmonic mean and can be calculated as: . The following graph depicts the values of the of and for different values of . Note that when , given that the two distributions have the same arithmetic mean. In both cases, when , the generalized means are strictly lower than the arithmetic mean, because the incomes are unequally distributed. Note moreover that for this range, because has a more unequal distribution. On the other hand, for , as the higher incomes receive a higher weight.

Another matrix transformation that we use is replication. A matrix is a replication of another matrix if it can be obtained by duplicating the rows of the original matrix a finite number of times. Suppose the rows of matrix are replicated number of times, where . Then the corresponding replication matrix is denoted by . notation may be used for replication of any vector : . We do not consider column replication, as we consider a fixed set of dimensions. Analogous We also use three types of matrices associated with particular operations: a permutation matrix, a diagonal matrix, and a bistochastic matrix. A permutation matrix, denoted by , is a square matrix with one element in each row and each column equal to 1 and the rest of the elements equal to 0. Thus the elements in every row and every column sum to one. We eliminate the special case when a permutation matrix is an ‘identity matrix’ with the diagonal elements equal to 1 and the rest equal to 0. What does a permutation matrix do? If any matrix is pre-multiplied by a permutation matrix, then the rows of matrix are shuffled without their elements being altered. Similarly, if any matrix is post-multiplied by a permutation matrix, then the columns of are shuffled without their elements being altered.
Example of Permutation Matrix: Let . Consider . Then . Thus, the rows of are merely swapped. Similarly, consider . Then . Note that the second and the third columns have swapped their positions. The first column did not change its position because the first diagonal element in is equal to one.
A diagonal matrix, denoted by , is a square matrix whose diagonal elements are not necessarily equal to 0 but all off-diagonal elements are equal to 0. Let us denote the th element of by . Then, for all . For our purposes, we require the diagonal elements of a diagonal matrix to be strictly positive or . What is the use of a diagonal matrix? If any matrix is post-multiplied by a diagonal matrix, then the elements in each column are changed in the same proportion. Note that different columns may be multiplied by different factors.
Example of Diagonal Matrix: Let . Consider . Then . Each element in the first column has been halved and each element in the third column has been doubled. However, the second column did not change because the corresponding element of is equal to one.
A bistochastic matrix, denoted by , is a square matrix in which the elements in each row and each column sum to one. If the th element of is denoted by , then for all and for all . Why do we require a bistochastic matrix? If a matrix is pre-multiplied by a bistochastic matrix, then the variability across the elements of each column is reduced while their average or mean is preserved. Note that if a diagonal element in a bistochastic matrix is equal to one, the achievement vector of the corresponding person remains unaffected. If the bistochastic matrix is a permutation matrix or an identity matrix, then the variability remains unchanged.
Example of Bistochastic Matrix: Let . Consider . Then . The bistochastic matrix equalizes the achievements across its elements.

2.3 Scales of Measurement: Ordinal and Cardinal Data

An important element of the framework in multidimensional poverty measurement relates to the scales of measurement of the indicators used. Scales of measurement are key because they affect the kind of meaningful operations that can be performed with indicators. In fact, as we will observe, certain types of indicators may not allow a number of operations and thus cannot be used to generate certain poverty measures. What does scale of measurement refer to exactly? Following Roberts (1979) and Sarle (1995), we define a scale of measurement to be a particular way of assigning numbers or symbols to assess certain aspects of the empirical world, such that the relationships of these numbers or symbols replicate or represent certain observed relations between the aspects being measured. There are different classifications of scales of measurement. In this book, we follow the classification introduced by Stevens (1946) and discussed in Roberts (1979). Stevens’ classification is consistent with Sen (1970, 1973), which analysed the implications of scales of measurement for welfare economics, distributional analysis, and poverty measurement, and it has largely stood the test of time.[58] Stevens’ (1946) classification relies on four key concepts: assignment rules, admissible transformations, permissible statistics, and meaningful statements. First, the defining feature of a scale is the rule or basic empirical operation that is followed for assigning numerals, as elaborated below. Second, each scale has an associated set of admissible mathematical transformations such that the scale is preserved. That is, if a scale is obtained from another under an admissible transformation, the rule under the transformed scale is the same as under the original one. Third, a permissible statistic refers to a statistical operation that when applied to a scale, produces the same result as when it is applied to the (admissibly) transformed scale. While the word ‘permissible’ may sound rather strong, it is justifiable under the premise that ‘one should only make assertions that are invariant under admissible transformations of scale’ (Marcus-Roberts and Roberts 1987, 384).[59] Fourth, a statement is called meaningful if it remains unchanged when all scales in the statement are transformed by admissible transformations (Marcus-Roberts and Roberts 1987: 384).[60] Stevens (1946) considered four basic empirical operations or rules that define four types of scales: equality, rank-order, equality of intervals, and equality of ratios. Following them, he defined four main types of scales: nominal, ordinal, interval, and ratio. Stevens' classification is not exhaustive. For example, it only applies to scales that take real values and which are regular.[61] Also, note that alternative terms are sometimes used for some of Stevens’ types. For example, nominal scales are sometimes referred to as categorical scales. Table 2.3 lists the scale types mentioned above from ‘weakest’ to ‘strongest’ in the sense that interval and ratio scales contain much more information than ordinal or nominal scales. The column that presents the rule defining each scale type is cumulative in the sense that a rule listed for a particular scale must be applicable to the scales in rows preceding it. The column that lists the permissible statistics is also cumulative in the same sense. In contrast, the column that lists the admissible transformations goes from general to particular: the particular operation listed in a row is included in the operation listed above. We now introduce each scale ‘type’. The scale pertains to an indicator used to measure dimension . The term ‘indicator ’ denotes the indicator of dimension . Achievements in indicator across the population are represented by vector , where is the achievement of person in the indicator. Indicator is said to be nominal or categorical if the scale is based on mutually exclusive categories, not necessarily ordered. Nominal variables are frequently called categorical variables. The rule or basic empirical operation behind this type of scale is the determination of equality among observations. A nominal scale is ‘the most unrestricted assignment of numerals. The numerals are used only as labels or type numbers, and words or letters would serve as well’ (Stevens 1946: 678). That is, numbers assigned to the various achievement levels in this domain are simply placeholders. Stevens introduces two common types of nominal variables. One uses ‘numbering’ for identification, such as the identification number of each household in a survey or the line number of individuals living within a household. The other uses numbering for a classification, such that all members of a social group (ethnic, caste, religion, gender, or age) or geographical regions (rural/urban areas, states, or provinces) are assigned the same number. The first type of nominal variable is simply a particular case of the second. There is a wide range of admissible transformations for this type of scale. In fact, any transformation that substitutes or permutes values between groups, that is, any one-to-one substitution function such that for all , will leave the scale form invariant. Given that in a nominal variable, the different categories do not have an order, neither arithmetic operations nor logical operations (aside from equality) are applicable. In terms of relevant statistics, if the nominal variable is simply an identifier, then only the number of categories is a relevant statistic; if the nominal variable contains several cases in each category, then the mode and contingency methods can be implemented, as can hypotheses tests regarding the distribution of cases among the classes (Stevens 1946: 678–9). Indicator is said to be ordinal if the order matters but not the differences between values. The rule or basic empirical operation behind this type of scale is the determination of a rank order. Categories can be ordered in terms of ‘greater’, ‘less’, or ‘equal’ (or ‘better’, ‘worse’, ‘preferred’, ‘not preferred’). Admissible transformations consist of any order-preserving transformation, that is, any strictly monotonic increasing function such that for all , as these will leave the scale form invariant. Thus, admissible transformations include logarithmic operation, square root of the values (nonnegative), linear transformations, and adding a constant or multiplying by another (positive) constant. Examples of ordinal scales are preference orderings over various categories, or subjective rankings. Given that the true intervals between the scale points are unknown, arithmetic operations are meaningless (because results will change with a change of scale), but logical operations are possible. For example, we can assert that someone reporting a health level of four feels ‘better’ than someone reporting a health level of ‘three’, who in turn feels better than a ‘two’, but we cannot assert whether the difference between level three and four is the same as the difference between level two and three. Nevertheless, some statistics are applicable to ordinal variables, namely, the number of cases, contingency tables, the mode, median, and percentiles.[62] Statistics such as mean and standard deviation cannot be used. Clearly, an ordinal variable is a nominal variable but the converse is not true. Ordinal and nominal (or categorical) variables are also sometimes referred to as qualitative variables.
Unordered categorical variables—such as eye colour—are not relevant for the construction of poverty measures. Relevant categorical variables are those that can be exhaustively and non-trivially partitioned into at least two sets according to some exogenous condition, and in which those sets can be arranged in a complete ordering. There will be fewer sets than there are categorical responses, or else the original variable would already have been ordinal. If a set contains multiple elements, it may not be possible to rank those elements against one another. Hence the resulting construction would be a ‘semi-order’ (Luce 1956) or ‘quasi-order’ (Sen 1973). Additionally, it may be possible to distinguish set(s) that are considered to be adequate achievements from those that are inadequate, forming a ‘weak order’, that is, some pairs of responses can be ranked as ‘preferred to’ and some others cannot be ranked.[63] For example, because it is difficult to assess whether it is better to have access to a public tap than to a borehole or a protected well as sources of drinkable water, the Millennium Development Goal indicator considers all three of them to be adequate sources of drinkable water (unrankable). Similarly, while one cannot rank access to an unprotected spring versus access to rainwater, both sources are considered inadequate by MDG standards. A variable thus constructed is often called an ‘ordered categorical’; we might also call the variables obtained as a weak order of categories in a nominal variable, a ‘weak-ordinal’ variable. Admissible transformations of weak ordinal variables include any transformations that partition the categorical variables into the relevant sets (safe water sources) in the same order; any apparent ordering of elements within the relevant sets can vary freely. Indicator 1 is said to be of interval scale if the rule or basic empirical operation behind its scale is the determination of equality of intervals or differences. Importantly, interval scales do not have a predefined zero point. The admissible transformations of interval scale consist of the linear transformation 1 (with 1), as this preserves the differences between categories. While the difference between two values of an interval-scale variable is meaningful, the ratios are not. The most cited example in this literature refers to two temperature scales: Celsius (oC) and Fahrenheit (oF). While the difference between 15oC and 20oC is the same as the difference between 20oC and 25oC, one cannot say that 20oC is twice as hot as 10oC because 0oC does not mean ‘no temperature’. That is, the Celsius scale (and Fahrenheit scale) lack a natural zero. Also, the difference between 15oC and 20oC and between 20oC and 25oC is also precisely the same if measured in Fahrenheit (59oF and 68oF vs. 68oF and 77oF) although the value of the difference is nine rather than five degrees. An interval scale allows addition and subtraction and the computation of most statistics, namely, number of cases, mode, contingency correlations, median, percentiles, mean, standard deviation, rank-order correlation, and product-moment correlation, but it is not meaningful to compute the coefficient of variation or any other ‘relative’ measure. In multidimensional poverty measurement, one indicator that is usually of interest is the z-score of under 5-year-old children’s nutritional achievement. We consider a nutritional z-score to be of interval-scale type. Box 2.3 provides a more detailed explanation of how to compute z-scores. Z-scores range from negative to positive values, spaced in (the reference population’s) standard deviation units, and the zero value means that the child’s nutritional achievement is at the median of what is considered a healthy population. Indicator 1 is said to be of ratio scale if the rule or basic empirical operation behind its scale is the determination of equality of ratios. Such a rule requires the scale to have a ‘natural zero’, namely the value 0 means ‘no quantity’ of that indicator. In other words, the value 0 is the absolute lowest value of the variable. Admissible transformations of interval-scale variables consist of functions such as 1 (with 1), as this preserves the ratio differences. Examples of ratio-scale variables are age, height, weight, and temperature in Kelvin, as 0o Kelvin means ‘no temperature’; 200 pounds is twice as much as 100 pounds, sixty years as thirty years and so on. Ratio-scale variables allow statements such as ‘a value is twice as large as another’, and they allow any type of mathematical operation, as well as the computation of any statistic (number of cases, mode, contingency correlations, median, percentiles, mean, standard deviation, rank-order correlation, product-moment correlation, and coefficient of variation). Interval- and ratio-scale variables constitute what are commonly referred to as cardinal variables. It is interesting to observe that in the order presented, from nominal to ratio scales, the admissible transformations become more restricted but the meaningful statistics become more unrestricted, ‘suggesting that in some sense the data values carry more information’ (Velleman and Wilkinson 1993: 66). Stevens (1959, 24) provided an insightful example of how measurement can progress from weaker to stronger scales. Early humans probably could only distinguished between cold and warm and thus used a nominal scale. Later, degrees of warmer and colder had been introduced and so the use of an ordinal scale gained prominence. The introduction of thermometers led to the use of an interval scale. Finally, the development of thermodynamics led to the ratio scale of temperature by introducing the Kelvin scale.

Box 2.3. Children’s nutritional z-scores

The nutritional status of children under 5 years old is assessed with three anthropometric indicators: weight-for-age, also called ‘underweight’; weight-for-height, also called ‘wasting’; and height-for-age, also called ‘stunting’. The indicators are constructed as follows. The World Health Organization has measured the height and weight of a reference population of children from different ethnicities, which is considered to constitute a standard of well nourishment (WHO 2006). From that population, a distribution of weights according to each age, a distribution of heights according to each age, and a distribution of height according to each weight are obtained. Each of these are discriminated by gender. How is a child’s nutrition assessed? Let us consider the case of the weight-for-age () indicator. Once the weight and age of the child have been documented, this information can be expressed in her weight-for-age z-score. This is computed as the child’s observed weight minus the median weight of children of the same sex and age in the reference population, divided by the standard deviation of the reference population. That is:

where is the z-score of weight-for-age, is the observed weight of child , is the median weight of children of the same sex and age as child in the reference population (healthy children), and is the standard deviation of the weight of children of that age in the reference population. The z-scores for weight-for-height () and height-for-age () are computed in an analogous way. Thus, for all and all , and . Thus, for example, suppose 14-month-old Anna weighs 8.3 kilograms. The median weight in the reference population of children of that age is 9.4 and the standard deviation is 1.* Thus, the z-score of Anna is , meaning that Anna is about one standard deviation below the median weight of healthy children. It is considered that children with z-scores that are more than two standard deviations below the median of the reference population suffer moderate undernutrition, and, if their z-score is more than three standard deviations below, they suffer severe undernutrition (underweight, wasting, or stunting, correspondingly). Children with a z-score of weight-for-height above standard deviations above the median are considered to be overweight (WHO 1997). An alternative way to assess the nutritional status of children is to use percentiles rather than z-scores, but z-scores present a number of advantages. Most importantly, they can be used to compute summary statistics such as a mean and standard deviation, which cannot be meaningfully done with percentiles (O’Donnell et al. 2008). Note that if we take a linear transformation of the z-score for weight-for-age such that , where and , then . Note that the difference (or ) has the same implication as the difference . This equivalence would hold for any linear transformation, exhibiting the characteristics of an interval-scale indicator.

*These values were taken from WHO’s reference tables:

Having introduced the scales of measurement, it is worth making a few clarifications regarding other frequently-mentioned types of indicators. First, Stevens' classification makes no reference to continuous versus discrete variables, for example. Continuous variables can take any value on the real line within a range. Discrete variables, in contrast, can only take a finite or countably infinite number of values.[64] Ordinal variables are discrete variables, but cardinal ones (interval and ratio scale) can be either discrete or continuous.[65] Second, note that count variables such as counts of publications, number of children in a household, or number of chickens, are particular cases of ratio-scale variables (Stevens 1946), such that the only admissible transformation is the identity function, i.e. 1. Roberts (1979) refers to the counting scale type as an absolute scale. Third, dichotomous (also called binary) variables can be of different scales, depending on the meaning of their categories. When the two values simply refer to unordered, mutually exclusive categories, such as being male or female, the variable is of nominal scale. When there is an order between the categories, such as being deprived or not in a specific dimension, the variable is cardinal. If the two values refer to having or lacking the same thing, such as a fully functional method for wood smoke ventilation, the variable may be interpreted as of ratio scale.[66] Fourth, the reader may wonder where do Likert scales—introduced by Likert (1932) and often used in social sciences—fit? Likert scales are obtained from responses to a set of (carefully phrased) statements to which each respondent expresses her level of agreement on a scale such as one to five: strongly disagree, disagree, neutral, agree, or strongly agree. Each statement and its responses are known as a Likert item. A Likert scale is obtained by summing or averaging the responses to each item so that a score is acquired for each person. Likert scales are frequently treated as interval scales, under the assumption (at times empirically verified) that there is an equal distance between categories (Brown 2011; Norman 2010). Thus, descriptive statistics (like means and standard deviations) and inferential statistics (like correlation coefficients, factor analysis, and analysis of variance) are regularly implemented with Likert scales. However, this has been criticized as being ‘illegitimate to infer that the intensity of feeling between ‘strongly disagree’ and ‘disagree’ is equivalent to the intensity of feeling between other consecutive categories on the Likert scale’ (Cohen et al. 2000, cited in Jamieson 2004: 1217). Thus, there is ongoing disagreement about whether Likert scales should be treated as ordinal scales (Pett 1997; Hansen 2003; Jamieson 2004). Often empirical psychometric tests are performed to ‘verify’ whether the assumption that the scale can be treated as cardinal holds for a particular dataset.
Stevens’ (1946) landmark work sparked a body of literature on measurement theory, which raised strong warnings regarding the applicability of statistics to different scales, including Stevens (1951, 1959), Luce (1959), and Andrews et al. (1981). This engendered an extensive and on-going debate across literatures from psychology to statistics. Some social scientists and statisticians consider that, although, in theory, certain statistics such as the mean and standard deviations are inappropriate for ordinal data, they can still be ‘fruitfully’ applied if the problem in question and data structure (and comparability, if relevant) have been tested and seem to warrant assumptions that they can be treated as interval scale. The arguments in favour of this position are interestingly articulated by Velleman and Wilkinson (1993). For example, it is stated that ‘the meaningfulness of a statistical analysis depends on the questions it is designed to answer’ (Lord 1953, Guttman 1977), that in the end ‘every knowledge is based on some approximation’ (Tukey 1961), and that ‘…if science was restricted to probably meaningful statements it could not proceed’ (Velleman and Wilkinson 1993). Another argument is that parametric methods are highly robust to violation of assumptions and thus can be implemented with ordinal data (Norman 2010). Although we acknowledge this on-going debate, in what follows we do endorse the requirement of meaningfulness—as defined above—in order to compute statistics or mathematical measures in each scale type; hence, we limit the operations with ordinal data to those that cohere with its definition and point out deviations from this. As this section suggests, the indicators’ scale and the analysts’ considered response to scales of measurement must be articulated before selecting a methodology to measure and assess multidimensional poverty. In Chapter 3, we clarify whether each method surveyed can be used with ordinal data. In general, when an indicator is of ratio scale, such as income, consumption, or expenditure (all expressed in monetary units), arithmetic operations with elements of the scale are permissible. Thus, it is meaningful to compute the normalized deprivation gaps as defined in Expression (2.2). When all considered indicators are of ratio scale, multidimensional poverty measures based on normalized deprivation gaps can be used. If the indicator is of interval scale, gaps may be redefined appropriately. When indicators are ordinal, measures based on normalized deprivation gaps are meaningless because results would vary under different admissible transformations of the indicators’ scales. The Adjusted Headcount Ratio presented in Chapters 5–10 can be meaningfully implemented using ordinal indicators, provided that issues of comparability are addressed as we will now clarify.

2.4 Comparability across People and Dimensions

The last section established the scales of measurement by which we can rigorously compare achievement levels in one variable, and the mathematical and statistical operations that can be performed on that variable. The discussion enabled us to identify the scale of measurement of each single indicator. Yet multidimensional measures seek to compare people’s achievements or deprivations across indicators, in ways that respect the scale of measurement of each indicator. This is by no means elementary. As Sen (1970) pointed out, cardinally meaningful variables may not necessarily be cardinally comparable—across people or, in multidimensional measurement, across dimensions. This section scrutinizes how these comparisons can legitimately proceed. That is, it takes a step back from the material presented thus far, to make explicit assumptions that have usually been implicit in work on multidimensional poverty measurement.
The issue of comparability across dimensions raised in this section has potentially significant empirical implications for quantitative methods beyond measurement. For example, as Chapter 3 will show, dichotomous data are regularly used in techniques that implicitly attribute cardinal meaning and comparability to the 0-1 values from several dimensions. If, in fact, the associated deprivation values of these data differ significantly from one another, then results based on techniques that treat each variable as cardinally equivalent may be affected; such exercises should, strictly, employ the 0-w variables because it is their relative weights or deprivation values that create cardinal comparability across dimensions. Hence the issues raised in this section, in a preliminary and intuitive way at this stage, have far-reaching implications for multidimensional analyses.
The properties we will present in section 2.5 define certain characteristics that a poverty measure may fulfill. Underlying many properties is an assumption that the poverty measure itself is cardinally meaningful. Based on this assumption, a change in the underlying 1 achievement matrix will change the poverty measure in desired and predictable ways according to the properties. In order to generate a cardinally meaningful multidimensional poverty index, it is necessary to treat indicators correctly according to their scale of measurement, as we have seen already. But it is also necessary to compare and aggregate a set of indicators (i) across people and (ii) across dimensions. Neither of these steps is trivial.[67] It must be recalled that variables used to construct unidimensional poverty measures, as defined in section 2.1, already entail comparisons across themselves as component dimensions and across people and households. In terms of interpersonal comparisons, the same household income level is normally assumed to be associated with the same level of individual welfare or poverty.[68] Components of such measures (sources of income, or consumption / expenditure on different goods) are usually assumed to be additive as cardinal variables having a common unit (usually, a currency like pounds or rupees), hence prices (adjusted where necessary) are used as weights. Equivalence scales may be used to augment comparability across households.
Multidimensional poverty measures, like income poverty measures, entail a basic assumption that the indicators are interpersonally comparable. Additionally, counting-based measures further assume that the same deprivation score is associated with the same level of poverty for different people. This assumption implies that deprivations have been made comparable. Comparability across dimensions must be obtained in order to generate cardinally meaningful deprivation scores and associated multidimensional poverty measures. But how? Multidimensional poverty measures may contain fundamentally distinct components that are not measured in the same units and may have no natural means of conversion into a common variable.
Empirically, the mechanics by which apparent comparability has been created in counting-based measures are clear (Chapter 4). When data are dichotomous and interpersonally comparable, the application of deprivation values is understood to create cardinal comparability across dimensions. When data are ordinal or ordered categorical, deprivation cutoffs are used to dichotomize the data; and those cutoffs, together with deprivation values, establish cardinal comparability across deprivations. In the case of appropriately scaled cardinal data, comparability across deprivations is created by the weights and the deprivation cutoffs.
Let us start with the most straightforward case: the deprivation matrix. This presents dichotomous values either because the original indicator is dichotomous (access to electricity), or because a cardinal, ordinal or ordered categorical variable has been dichotomized by the application of a deprivation cutoff into two categories: deprived and non-deprived.
As mentioned above, we can consider dichotomous deprivations to be (trivially) cardinal.[69] More precisely, the deprivation cutoff establishes a ‘natural zero’ in the sense that any person whose achievement meets or exceeds the natural zero is non-deprived, and anyone who does not, is deprived. We require the natural zeros to be comparable states across dimensions for the purposes of poverty measurement. But how are the ‘one’ values, reflecting deprived states, comparable? Deprivation values (the vector of relative weights), create an explicit value for each of the deprived states, across the set of possible finite dimensional comparisons. After the deprivation values have been applied, deprivations may be cardinally compared across dimensions. The reason we draw the reader’s attention at this stage to the assumptions underlying cardinal comparisons across people and across dimensions, is in order that they might observe how differently multidimensional poverty measurement techniques, such as those surveyed in Chapter 3, undertake and justify such comparisons, and also so that some readers might be encouraged to explore these important issues further—both theoretically and empirically.

2.5 Properties for Multidimensional Poverty Measures

In selecting one poverty measurement methodology from a set of options, a policy maker thinks through how a poverty measure should behave in different situations in order to be a ‘good’ measure of poverty and support policy goals. Then she asks which measure meets these requirements. For example, should the poverty measure increase or decrease if the achievement of a poor person rises while the achievements of other people remain unchanged? Should poverty comparisons change when achievements are expressed in different units of measurement? Should the measure of poverty in a more populous country with a larger number of poor people be higher than the poverty measure in a small country with a smaller number of, but proportionally more, poor people?
A policy maker seeking to ameliorate poverty should have a good understanding of the various normative principles that her chosen poverty measure embodies, just as a pilot of a plane must have a sound understanding of how a particular plane responds to different operations. If the policy maker has a good understanding of the principles embodied by alternative measures then she will be able to choose the measure most closely reflecting publicly desirable ethical principles and most appropriate for the application—just as a pilot will choose the best way to fly the plane for a particular journey.
The normative judgments embodied by a poverty measure are reflected in its mathematical properties, including its structure and its response to changes in its argument. The axiomatic method, formally introduced to this field by Sen (1976),[70] refers to measures that have been designed based on principles that are taken by the researcher as axioms, i.e. as ‘statements that are accepted true without proof’.[71] In this section, we define and discuss the various properties proposed in the literature on multidimensional poverty measures. A consensus may emerge around some properties, thus they may be considered as axioms, while others may remain optional. In what follows, we use the words ‘principles’ and ‘properties’ interchangeably. The set of properties for multidimensional measurement has been built upon its unidimensional counterpart. However, as noted by Alkire and Foster (2011a), there is one vitally important difference: in the multidimensional context, the identification step is no longer elementary and properties must be viewed as restrictions on the overall poverty methodology, 1, that is, as joint restrictions on the identification and the aggregation method. For simplicity, in what follows, we state the properties in terms of the poverty index 1, which entails two assumptions. First, that any multidimensional poverty index is associated with an identification function. Second, that the identification function relies on a functional form of the type 1.[72] We classify the properties into four categories.[73]
The first set of properties requires that a poverty measure should not change under certain transformations of the achievement matrix. We refer to these as invariance properties, which are symmetry, replication invariance, scale invariance and two alternative focus properties, poverty focus and deprivation focus. The second set requires poverty to either increase or decrease with certain changes in the achievement matrix. We refer to these as dominance properties, which are monotonicity, transfer, rearrangement, and dimensional transfer properties. The third set of principles relates overall poverty to either groups of people or groups of dimensions and, thus is called subgroup properties. Other properties, that guarantee that the measure behaves within certain usual, convenient parameters, we refer to as technical properties. Each of the four following sections provides a formal outline and intuitive interpretations of each set of properties.

2.5.1  Invariance Properties

The first invariance principle is symmetry. Symmetry requires that each person in a society is treated anonymously so that only deprivations matter and not the identity of the person who is deprived. Hence this property is also often referred to as anonymity. As long as the deprivation profile of the entire society remains unchanged, swapping achievement vectors across people should not change overall poverty.[74] This type of rearrangement can be obtained by pre-multiplying the achievement matrix by a permutation matrix of appropriate order.
Symmetry: If an achievement matrix is obtained from achievement matrix as , where is a permutation matrix of appropriate order, then .
Example: Suppose the initial achievement matrix is and the permutation matrix is . Then: . Note that the first and the second person (rows) in matrix have swapped their positions. However, as long as the deprivation cutoffs remain unchanged, there is no reason for the level of poverty to be different for these two societies. Hence, .
The second invariance principle, replication invariance, requires that if the population of a society is replicated or cloned with the same achievement vectors a finite number of times, then poverty should not change.[75] In other words, the replication invariance property requires the level of poverty in a society to be standardized by its population size so that societies with different population sizes are comparable to each other, as are societies whose populations change over time. Thus, this property is also known as the principle of population.[76]
Replication Invariance: If an achievement matrix is obtained from another achievement matrix by replicating a finite number of times, then .
Example: Suppose we are required to compare the level of poverty in two societies with the initial achievement matrices and . Note that the population sizes in these societies are different. In order to make them comparable, we replicate twice to obtain and thrice to obtain . By replication invariance, we know that and . Thus, it is equivalent comparing to and to .
The third invariance principle, scale invariance[77] requires that the evaluation of poverty should not be affected by merely changing the scale of the indicators. For example, if the duration of completed schooling is an indicator, then deprivation in education, thus overall poverty, should be the same regardless of whether duration is measured in years or in months, provided the deprivation cutoff is correspondingly adjusted. The scale of any indicator in an achievement matrix can be altered by post-multiplying the achievement matrix by a diagonal matrix 1 of appropriate order (1, the number of dimensions). If a diagonal element is equal to one, then the scale of the respective indicator does not change. The diagonal elements of 1 need not be the same because different indicators may have different scales and units of measurement. A weaker version of the scale invariance principle, referred to as ‘unit consistency’, has been proposed by Zheng (2007) in the context of unidimensional poverty measurement and extended to the multidimensional context by Chakravarty and D’Ambrosio (2013). This principle requires that poverty comparisons, but not necessarily poverty values, should not change if the scales of the dimensions are altered.[78] The scale invariance property implies the unit consistency property, but the converse does not hold.
Scale Invariance: If an achievement matrix is obtained by post-multiplying the achievement matrix by a diagonal matrix such that , and the deprivation cutoff vector is obtained from such that , then .Unit
Consistency: For two achievement matrices and and two deprivation cutoff vectors