The AF Class of Poverty Measures: Overview and Practicality

The AF methodology of multidimensional poverty measurement creates a class of measures that both draws on the counting approach and extends the FGT class of measures in natural ways.

Constructing these M_α measures entails the following components:

Identification

1. Defining the set of indicators which will be considered in the multidimensional measure. Data for all indicators need to be available for the same person.

2. Setting the deprivation cutoffs for each indicator, namely the level of achievement considered sufficient (normatively) in order to be non-deprived in each indicator.

3. Applying the cutoffs to ascertain whether each person is deprived or not in each indicator.

4. Selecting the relative weight or value that each indicator has, such that these sum to one.^[161]

5. Creating the weighted sum of deprivations for each person, which can be called his or her ‘deprivation score’.

6. Determining (normatively) the poverty cutoff, namely, the proportion of weighted deprivations a person needs to experience in order to be considered multidimen- sionally poor, and identifying each person as multidimensionally poor or not according to the selected poverty cutoff.

Aggregation

7. Censoring deprivations of the non-poor and computing the proportion of people who have been identified as multidimensional^ poor in the population.

8. Computing the average share of weighted indicators in which poor people are de­prived. This entails adding up the deprivation scores of the poor and dividing them by the total number of poor people. This is the average intensity of multidimensional poverty (A), also sometimes called the breadth of poverty.

9. Computing the M₀ measure as the product of the two previous partial indices: M₀ = H ? A. Analogously, M₀ can be obtained as the mean of the vector of censored deprivation scores, which is also the sum of the weighted deprivations that poor people experience, divided by the total population.

When all indicators are ratio scale, computing M₁ and M₂ entails the following components:

10. Computing the average poverty gap across all instances in which poor persons are deprived, or G. This entails computing the normalized (deprivation) gap as defined in equation (2.2): g_ij = ^z-^xj, where x_ij is censored at z_j for each person and indicator. In words, for a person who is deprived in a given indicator, the normalized gap is the difference between the deprivation cutoff and the person’s achievement for the indicator, divided by is deprivation cutoff; if the person’s achievement does not fall short of the deprivation cutoff, the normalized gap is zero. The average poverty gap is the mean of poor people’s weighted normalized deprivation gaps in those dimensions in which poor people are deprived and is one of the partial indices. This depth of multidimensional poverty is denoted by G.

11. Computing the M₁ measure as the product of three partial indices: M₁ = H ? A ? G. Analogously, M₁ can be obtained as the sum of the weighted deprivation gaps that poor people experience, divided by the total population.

12.

13. Computing the M₂ measure as the product of the following partial indices: M₂ = H ? A ? S. Analogously, M₂ can be obtained as the sum of the weighted squared deprivation gaps that poor people experience, divided by the total population.

Note that in all three cases (M₀, M₁, and M₂) the deprivations experienced by people who have not been identified as poor (i.e. those whose deprivation score is below the poverty cutoff) are censored, hence not included; this censoring of the deprivations of the non-poor is consistent with the property of‘poverty focus' which—analogous to the unidimensional case—requires a poverty measure to be independent of the achievements of the non-poor. For further discussion see Alkire and Foster (2011a).

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

All AF measures can also be unpacked to reveal the dimensional depriva­tions contributing the most to poverty for any given group.

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

A few comments on the AF class before we turn to the final key property for policy. All AF measures also have intuitive interpretations. The Adjusted Headcount Ratio (M₀) reflects the proportion of weighted deprivations the poor experience in a society out of the total number of deprivations this society could experience if all people were poor and were deprived in all dimensions. The Adjusted Poverty Gap (M₁) reflects the average weighted deprivation gap experienced by the poor out of the total number of deprivations this society could experience (which is the maximum possible value of the average weighted deprivation gap when H and A and G are all 100%). The Adjusted Squared Poverty Gap Measure (M₂) reflects the average weighted squared gap or poverty severity experienced by the poor out of the total deprivations this society could experience. In all cases, the term ‘adjusted' refers to the fact that all measures incorporate the intensity of multidimensional poverty—which is key to their properties.

Additionally, while each of the AF measures offers a summary statistic of multidi­mensional poverty, they are related to a set of consistent and intuitive partial indices, namely, poverty incidence (H), intensity (A), and a set of subgroup poverty estimates and dimensional deprivation indices (which in the case of the M₀ measure are called censored headcount ratios) and their corresponding percent contributions. Each M_α measure can be unfolded into an array of informative indices.

Among the AF class of measures, the M0 measure is particularly important because it can be implemented with ordinal data. This is critical for real-world applications. It is relevant when poverty is viewed from the capability perspective, for example, since many key functionings are commonly measured using ordinal variables. The M₀ measure satisfies the ordinality property introduced in section 2.5.1. This means that for any monotonic transformation of the ordinal variable and associated cutoff, overall poverty as estimated by M0 will not change. Moreover, M0 has a natural interpretation as a measure of‘unfreedom' and generates a partial ordering that lies between first- and second-order dominance (Chapter 6). Because of its properties and practicality, this book mainly focuses on M0.

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

5.2