Inequality Comparisons with Ordinal Data

• Published date: 01 September 2021
• Author: Stephen P. Jenkins
• Date: 01 September 2021
• DOI: http://doi.org/10.1111/roiw.12489

© 2020 The Authors. Review of Income and Wealth published by John Wiley & Sons Ltd on behalf of

International Association for Research in Income and Wealth

547

INEQUALITY COMPARISONS WITH ORDINAL DATA

by Stephen p. JenkinS*

LSE, ISER (University of Essex), and IZA

Non-intersection of appropriately defined Generalized Lorenz (GL) curves is equivalent to a unani-

mous ranking of distributions of ordinal data by all Cowell and Flachaire (Economica, 2017) indices of

inequality and by a new index based on GL curve areas. Comparisons of life satisfaction distributions

for six countries reveal a substantial number of unanimous rankings. The GL dominance criteria are

compared with other criteria including the dual-H dominance criteria of Gravel, Magdalou, and Moyes

(Economic Theory, 2020).

JEL Codes: D31, D63, I31

Keywords: inequality, ordinal data, Generalized Lorenz dominance, H dominance, Hammond transfers,

life satisfaction, World Values Survey

1. introduction

Cowell and Flachaire (2017) provide an approach to measuring inequality of

ordinal data such as life satisfaction, happiness, and self-assessed health status that

differs significantly from the approach taken in most recent research. This paper

builds on Cowell and Flachaire’s work by adding dominance results and a new

inequality index, illustrates them using cross-national data about life satisfaction

distributions, and compares the approach with others.

Since the critique by Allison and Foster (2004), most economists have accepted

that it is inappropriate to assess ordinal data inequality using the tools developed

to assess the inequality of cardinal data on income and wealth. The latter methods

associate greater inequality with greater dispersion about the mean, but the mean

is an improper benchmark for an ordinal variable. For ordinal variables, Allison

and Foster (2004) propose instead that greater inequality means greater spread

about the median and they demonstrate that, for distributions with the same

median, a unanimous ordering by all indices incorporating this concept is equiva-

lent to “S-dominance”—a particular configuration of cumulative distribution

functions.1 Allison and Foster (2004) and other researchers, including Abul Naga

1See also Kobus (2015) for characterization results.

Note: The research was part-supported by core funding of the Research Centre on Micro-Social

Change at the Institute for Social and Economic Research by the University of Essex and the UK

Economic and Social Research Council (award ES/L009153/1). I gratefully acknowledge the hospitality

of the School of Economics, University of Queensland, where I was based when writing the first ver-

sion of this paper, and thank the anonymous referees for their helpful comments, especially “referee 2,”

and Yongsheng Xu for clarificatory discussion.

*Correspondence to: Stephen P. Jenkins, Department of Social Policy, London School of

Economics and Political Science, Houghton Street, London, WC2A 2AE, UK (s.jenkins@lse.ac.uk).

Review of Income and Wealth

Series 67, Number 3, September 2021

DOI: 10.1111/roiw.12489

This is an open access article under the terms of the Creative Commons Attribution License, which

permits use, distribution and reproduction in any medium, provided the original work is properly

cited.

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Review of Income and Wealth, Series 67, Number 3, September 2021

548

© 2020 The Authors. Review of Income and Wealth published by John Wiley & Sons Ltd on behalf of

International Association for Research in Income and Wealth

and Yalcin (2008) and Apouey (2007), have developed inequality indices consistent

with S-dominance. A distinguishing feature of the Allison-Foster approach is that

it measures inequality in terms of polarization: “inequality” is maximized when

half the population has the lowest value on the ordinal scale and half the popula-

tion has the largest value. Cowell and Flachaire’s (2017) inequality indices are dif-

ferent because greater inequality reflects greater spread in a sense other than greater

polarization. However, no dominance results currently exist for Cowell-Flachaire

indices.

I show that non-intersection of appropriately defined Generalized Lorenz

(GL) curves is equivalent to a unanimous ranking of distributions by all Cowell

and Flachaire (2017) indices of inequality and by a new index based on areas below

GL curves. The results are not restricted to distributions with the same median.

Comparisons of life satisfaction distributions for six countries derived from World

Values Survey data show that the new dominance results reveal a substantial num-

ber of unanimous inequality rankings.

I use Cowell and Flachaire’s “peer-inclusive downward-looking” definition of

individual status (explained below) because this definition is consistent with the

focus in the median-related inequality measurement literature. (Cumulative distri-

bution functions are the building blocks in common.) There are analogous results

for Cowell and Flachaire’s “peer-inclusive upward-looking” status definition but,

for brevity, I summarize these in the Appendix.2

I also demonstrate that rankings according to the GL criteria differ from

rankings according to the dual H-dominance results of Gravel et al. (2020) based

on the concept of Hammond transfers. I compare the elementary transformations

that underlie each approach and provide empirical illustrations of how they order

World Values Survey life satisfaction distributions differently.

Various supplementary materials cited in the main text are reported in the

Appendix.

2. cowell-Flachaire inequality indiceS For ordinal data

The well-being of each of N individuals is measured on an ordinal scale char-

acterized by a set of numerical labels (l1, l2, …, lK), with –∞ < l1< l2 < … < lK <

∞, and K≥ 3. Thus, the distribution of well-being is summarized by an ordered

categorical variable. The proportion of individuals in the kth category is denoted

fk with 0≤fk≤1 and

∑K

k=1

f

k

=

1

. The proportion of individuals in the kth category

or lower is Fk, with

F

k=

∑k

j=1

fj and FK=1.

Cowell and Flachaire (2017) propose a two-step approach to inequality mea-

surement for ordinal data. First, decide how to summarize “status,” si, for each

individual i= 1, 2, 3, …, N. In particular, Cowell and Flachaire’s “peer-inclusive

downward-looking” status of an individual with scale level k is Fk, and hence does

not depend on the specific values attached to (l1, l2, …, lK). That is, the measure is

scale independent.

2No researchers have used Cowell and Flachaire’s (2017) “peer-exclusive” definitions in applied

work, not even the authors themselves.