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Measuring the depth of multidimensional poverty with ordinal data

Fernando Flores Tavares

Abstract

This paper proposes a positional poverty gap measure of multidimensional poverty within the Alkire-Foster counting framework. The measure captures the depth of deprivations even when indicators are ordinal, unlike the standard poverty gap, which requires cardinal variables. The proposed method draws on the fuzzy set literature and introduces a distribution-based measure of deprivation depth using the empirical cumulative distribution of each indicator, with the most deprived group as the benchmark. For each deprived individual, the method assigns a score based on the individual's relative position in the distribution. Depth is thus expressed as a difference in distributional positions, motivating the label positional poverty gap. The paper demonstrates that this measure preserves the identification and aggregation structure of the counting approach and satisfies its axiomatic properties when the reference distribution remains fixed over time. The framework remains flexible because it accommodates different identification rules, deprivation cutoffs, and variable types. Overall, it offers a simple, meaningful, and theoretically grounded way to incorporate depth into multidimensional poverty measurement with ordinal data.

Measuring the depth of multidimensional poverty with ordinal data

Abstract

This paper proposes a positional poverty gap measure of multidimensional poverty within the Alkire-Foster counting framework. The measure captures the depth of deprivations even when indicators are ordinal, unlike the standard poverty gap, which requires cardinal variables. The proposed method draws on the fuzzy set literature and introduces a distribution-based measure of deprivation depth using the empirical cumulative distribution of each indicator, with the most deprived group as the benchmark. For each deprived individual, the method assigns a score based on the individual's relative position in the distribution. Depth is thus expressed as a difference in distributional positions, motivating the label positional poverty gap. The paper demonstrates that this measure preserves the identification and aggregation structure of the counting approach and satisfies its axiomatic properties when the reference distribution remains fixed over time. The framework remains flexible because it accommodates different identification rules, deprivation cutoffs, and variable types. Overall, it offers a simple, meaningful, and theoretically grounded way to incorporate depth into multidimensional poverty measurement with ordinal data.
Paper Structure (20 sections, 1 theorem, 24 equations, 3 figures, 5 tables)

This paper contains 20 sections, 1 theorem, 24 equations, 3 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Notation
  4. Indicator development process
  5. Indicators construction
  6. Admissible transformations
  7. Comparability across indicators and individuals
  8. Poverty identification
  9. Intensity
  10. Positional Poverty Gap
  11. Overall multidimensional poverty index
  12. Individual poverty degrees
  13. Aggregated multidimensional poverty
  14. Properties
  15. Illustrative application
  16. ...and 5 more sections

Key Result

Theorem 1

Given weights $w$, cutoffs $(z,k)$, and $P_\alpha$ with $\alpha \geq 1$, the methodology with positional depth scores satisfies: symmetry, replication invariance, bounds/normalisation, ordinal invariance, and deprivation focus (anchored or in-sample); poverty focus under anchored CDFs for any $k$, a

Figures (3)

  • Figure 1: P by poverty line k
  • Figure 2: Individual positional depth scores by deprivation intensity
  • Figure 3: Rank concordance between positional poverty gap and AF poverty gap among the poor ($k = 1/3$, union cutoff), Bangladesh MICS 2019.

Theorems & Definitions (1)

  • Theorem 1