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

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

Table 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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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.

9.1.3.1  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.

9.1.3.2  A Decomposable Measure

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

(9.1)

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

(9.2)

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

(9.3)

where  is the mean of all elements in .

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

(9.4)

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

(9.5)

9.1.3.3  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

(9.6)

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

(9.7)

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.

9.1.3.4  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

(9.8)

Similarly, for  and :

(9.9)

 

(9.10)

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

(9.11)

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

(9.12)

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

(9.13)

 

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

In 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

As 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

To 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:

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.

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

This 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

Note 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

A 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

		(9.19)
		(9.20)

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.

9.4.1.1  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.

9.4.1.2  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 .

9.4.1.3  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.

9.4.1.4  Aggregation

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

(9.22)

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.

(9.23)

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,

(9.24)

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

(9.25)

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.

(9.26)

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.

9.4.3.1  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).

9.4.3.2 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 :

(9.27)

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:

(9.28)

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

(9.29)

9.4.3.3 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

(9.30)

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)

9.4.3.4 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 9.4.1.3, 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

(9.32)

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

(9.33)

and

(9.34)

9.4.3.5 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:

(9.35)

Similarly,

(9.36)

and

(9.37)

9.4.3.6 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

We 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.

9.4.4.1  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.

9.4.4.2 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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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

(8.1)

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

(8.2)

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,

(8.3)

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:

(8.4)

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

		(8.5)
		(8.6)
		(8.7)
			(8.8)

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

(8.9)

The unbiased sample estimate of  is

(8.10)

and the standard error of the Adjusted Headcount Ratio is

(8.11)

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

(8.12)

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

(8.13)

and so the unbiased standard error is

(8.14)

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

(8.15)

where, .

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

(8.16)

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

(8.17)

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

(8.18)

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

(8.19)

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).

		(8.20)
		(8.21)
		(8.22)
		(8.23)

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,

		(8.24)
		(8.25)

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

		(8.26)
		(8.27)
		(8.28)
		(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

(8.30)

where ,  and .

The standard errors of  and  are

	(8.31)

 (8.32)

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

(8.33)

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

(8.34)

where

(8.35)

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

(8.36)

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

(8.37)

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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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.

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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 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: 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:   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]

 

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] 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

We 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 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.

 

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. 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. 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.

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 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 (with ), 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 15^oC and 20^oC is the same as the difference between 20^oC and 25^oC, one cannot say that 20^oC is twice as hot as 10^oC because 0^oC does not mean ‘no temperature’. That is, the Celsius scale (and Fahrenheit scale) lack a natural zero. Also, the difference between 15^oC and 20^oC and between 20^oC and 25^oC is also precisely the same if measured in Fahrenheit (59^oF and 68^oF vs. 68^oF and 77^oF) 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 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 (with ), as this preserves the ratio differences. Examples of ratio-scale variables are age, height, weight, and temperature in Kelvin, as 0^o 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.

 

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. . 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 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, , 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 , 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 .[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. 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] 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 of appropriate order (, 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 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.

 

The fourth invariance principle is focus. The primary difference between the measurement of welfare and inequality and the measurement of poverty is that the latter is concerned with the base or bottom of the distribution whereas welfare and inequality are concerned with the entire distribution. The focus principle is crucial because it requires the poverty measure to respond only to the achievements of the poor. The poverty focus principle requires that poverty should not change if there is an improvement in any achievement of a non-poor person. This principle is a natural extension of the focus principle in the single dimensional context, which requires that overall poverty remains unchanged with an increase in the income of the non-poor.

However, in the multidimensional framework, the terms ‘deprived’ and ‘poor’ are not synonymous. Someone can be poor yet not deprived in every single indicator. The poverty focus principle does not cover the situation where a poor person’s achievement increases in a dimension in which that person is not deprived. If poverty were to change in this case, the non-deprived attainments would compensate deprived attainments, creating a form of substitutability across dimensions. Practically, this could encourage a policy maker who is enthusiastically interested in reducing the poverty figures to assist the poor in their non-deprived dimensions, instead of addressing the dimensions in which they are deprived. Thus, a second focus principle might be relevant. The deprivation focus principle requires that overall poverty not change if there is an increase in achievement in a non-deprived dimension, regardless of whether it belongs to a poor or a non-poor person. This property prevents poverty from falling when a poor person’s achievements increase in non-deprived dimensions. Note that focus properties motivate the construction of a censored achievement matrix , as described in section 2.1.1.[79]

The focus principle is one example in which it can be verified that the properties of multidimensional poverty measures are, as stated at the beginning of this section, joint restrictions on the identification and the aggregation methods. For example, for the deprivation focus principle to be satisfied, the identification method cannot follow the aggregate achievement approach. Also, as Alkire and Foster note (2011a, 481), the relevance of the two focus principles is connected to the criterion used to identify the poor (within a counting approach to identification). When a union criterion is used to identify the poor, the deprivation focus principle implies the poverty focus principle, whereas when an intersection criterion is used to identify the poor, the poverty focus principle implies the deprivation focus principle. When intermediate criterions are used, neither of the two principles implies the other.

 

The last invariance property ordinality relates to the type of scale of the particular indicator used for measuring each dimension. As we explained in section 2.3, the scale of an indicator affects the type of operations and statistics that can be meaningfully applied, in the sense that statements remain unchanged when all scales in the statement are transformed by admissible transformations. Building on this notion of meaningfulness, ordinality requires the poverty estimate not to change under admissible transformations of the scales of the indicators that compose the poverty measure.[80]

 

More formally, we say that is obtained from as an equivalent representation if there exist appropriate admissible transformations for such that and for all . In other words, an equivalent representation can comprise a set of admissible transformations, each of which is appropriate for each scale type, and which assigns a different set of numbers to the same underlying basic data. For example, an equivalent representation can include monotonic increasing transformations for ordinal variables, linear transformations for interval-scale variables, and proportional changes for ratio-scale variables. We now state the ordinality axiom as follows:

2.5.2 Dominance Properties

This section covers six principles, each of which has a stronger version and a weaker version. The stronger version requires that a poverty measure strictly moves in a particular direction, given certain transformations in the achievements of the poor. The weaker version, does not require a poverty measure to move in a particular direction but ensures that the poverty measure does not move in the opposite (wrong) direction under certain transformations of the achievements.[81] The first dominance principle, monotonicity, requires that if the achievement of a poor person in a deprived dimension increases while other achievements remain unchanged, then overall poverty should decrease. Normatively, this principle considers that improvements in deprived achievements of the poor are good and should be reflected by producing a reduction in poverty. Its weaker version, referred to as weak monotonicity, ensures that poverty should not increase if there is an increase in any person’s achievement in the society.[82]