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 deprivation status from several dimensions. If the values associated with deprivation statuses differ, results based on techniques that treat each 0-1 variable as cardinally equivalent may be affected, because they implicitly impose equal weights. Such exercises should, strictly, employ the 0—Wj variables because it is their relative weights or deprivation values (wj for all j = 1,..., d) 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 n ? d 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.[59]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.[60] 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.[61] 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? The vector of relative weights create an explicit deprivation value for each of the deprived states across the set of possible finite dimensional comparisons. After the deprivation values have been applied, deprivations maybe 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.
1.4