Inequality Comparisons with Ordinal Data

Non-intersection of appropriately-defined Generalized Lorenz (GL) curves is equivalent to a unanimous 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 inequality rankings.


Introduction
provide a new 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.
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 equivalent to 'S-dominance' -a particular configuration of cumulative distribution functions. 1 Allison and Foster (2004) and other researchers, including Abul Naga 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 population has the largest value. Cowell and Flachaire's (2017) inequality indices are different -and hence complementary to the median-based onesbecause 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.  Cowell-Flachaire (2017) inequality indices are Hammond-transfer-consistent.

Cowell-Flachaire inequality indices for ordinal data
The well-being of each of N individuals is measured on an ordinal scale characterized 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 k th category is denoted fk with 0 ≤ fk ≤ 1 and ∑ = 1

=1
. (1) The smaller that α is, the greater the weight that is put on small status values relative to high status values. Cowell and Flachaire also cite a closely-related class of 'Atkinson-like' indices, A(α): (2) The next section presents a tractable method based on Generalized Lorenz curve comparisons for assessing whether one distribution of status is unambiguously more (un)equal than another regardless of the differences in social judgements encapsulated in different inequality indices.
The GL curve is drawn using straight lines to connect adjacent points of the form {m/n, GL(s, m/n)}. The vertices of the curve are at {p0 = 0, 0} and {pk, ∑ =1 } for each k = 1, …, K with pk = Fk. The GL ordinate at p = 1 is the arithmetic mean of the status distribution. Figure 1 provides an illustrative example for the case K = 4.
<Figure 1 near here> The 45° ray from (0,0) to (1,1) is the GL curve representing complete equality -when all individuals have the same scale value and hence the same status. With inequality, the GL curve lies below the 45° ray and, intuitively, the further below the ray the curve is, the greater is inequality.
Indeed, one can formally demonstrate that a unanimous ranking of a pair of distributions in terms of their equality (or inequality) is equivalent to the non-crossing of their GL curves: Result 1: For two status distributions s and s′, The proof of Result 1 follows directly from Shorrocks (1983, Theorem 2). Result 2 follows from Result 1 and the relationship between D(.) and W(.).
The connection between GL curve location and inequality suggests a new index of inequality for ordinal data, J. With reference to Figure 1, J is the ratio of area A to area A + B; equivalently, J equals 1 minus twice area B. It is Generalized Lorenz-consistent because a ranking of a pair of distributions by J is the same as the ranking by all D(.) ∈ D when the two GL curves do not cross. Using the expression for the vertices of the GL curve, and applying the Trapezium Formula, one can show that: The minimum value of J is 0, achieved when there is perfect inequality.
The inequality dominance results relate to GL curves, not to Lorenz curves as some readers might expect. The reason is that the mean is an inappropriate reference point with ordinal data (Allison and Foster 2004), and hence also shares of the total are not a suitable building-block for inequality measurement in this context. Differences between maximum and observed status are what matters for Cowell-Flachaire indices. The situation considered here has analogies with the measurement of poverty. Non-intersection of two Three Is of Poverty (TIP) curves is equivalent to a unanimous ranking according to all 'generalized poverty gap' poverty indices (Jenkins and Lambert 1997). But a TIP curve shows, at each p, the vertical distance between two GL curves, one for the distribution of income censored above at the poverty line and the other for the distribution in which every income equals the poverty line (a distribution with perfect equality). Index J, based on the area between two GL curves, is analogous to the Shorrocks (1995) modified-Sen poverty index (twice the area beneath a TIP curve, i.e. twice the area between two GL curves).
Polarized distributions and uniform distributions provide potential maximuminequality benchmarks for ordinal data. As mentioned earlier, inequality indices in the Allison-Foster (2004)  3 The results cited in this paragraph require that N is sufficiently large so that any difference in the number of individuals in each category is negligible, with attention restricted to the two populated categories in the case of a polarized distribution. 4 Similarly, one can also show that the GL curve for a uniform distribution over K+1 levels lies everywhere on or below the GL curve for a uniform distribution over K levels. Illustrating these results, Appendix Figure A1 shows Generalized Lorenz curves for a polarized distribution and uniform distributions with K = 3, 4, 5, and 10.
Hence, according to GL-consistent indices, inequality in a non-uniform distribution may be larger or smaller than in the uniform distribution.

Empirical illustration: ranking countries by life satisfaction inequality
To illustrate the inequality dominance results, I use data about life satisfaction from the mid-   median life satisfaction is lower in South Africa than in the other countries (7 rather than 8).
S-dominance applies only to distributions with a common median -a restriction that does not apply to GL-dominance.
To derive a complete inequality ordering of the 6 countries an inequality index must be used. However, different indices incorporate different social judgements about how to assess differences in different parts of the life satisfaction distribution. It is therefore important to use a portfolio of indices to check the robustness of rankings. I report estimates for 6 GL-consistent indices in Figure 3: I(α) for α = 0, 0.25, 0.5, 0.75, and 0.9, plus J. For comparison, I also include 3 S-dominance-consistent indices from the Abul Naga and Yalcin (2007) class. ANY(1, 1) weights observations in categories above and below the median equally; ANY(4, 1) is more sensitive to above-median spread than below-median spread; and ANY(1, 4) is the opposite (i.e. relatively bottom-sensitive). Figure 3 shows point estimates and 95% confidence intervals. I estimate standard errors using a repeated half-sample bootstrap approach in order to appropriately account for the sample weights (Saigo et al. 2001;Van Kerm 2013) with 500 bootstrap replications.
<Figure 3 near here> Consider the I(α) estimates. Figure 3 confirms that inequality is distinctly greater in South Africa than in every other country. For example, according to I(0), South Africa's inequality is 4% larger than NZ's (with the null hypothesis of no difference decisively rejected: test statistic = 5.2). 7 According to I(0.9), the difference is 6% (test statistic = 4.8).

Summary and conclusions
Cowell and Flachaire's (2017) innovative approach to inequality measurement with ordinal data complements the predominant approach to date that conceptualizes greater inequality as greater spread around the median. This paper builds on Cowell and Notes Entries above the diagonal summarize checks for F-dominance (first-order dominance): '>', x F-dominates y; '<', y F-dominates x; '-', no dominance. *: NZ Sdominates CA. Entries below the diagonal summarize GL dominance checks: '>', x is more equal than y; '<', y is more equal than x; '-', no dominance.

Inequality comparisons with ordinal data
Stephen P. Jenkins

Analysis in the case of peer-inclusive upward-looking status
The main text focuses on Cowell and Flachaire's (2017) peer-inclusive downward-looking status. This note explains how the analysis needs to be modified if Cowell and Flachaire's peer-inclusive upward-looking status definition is used instead.
According to Cowell and Flachaire (2017), your peer-inclusive downward-looking status is the fraction of individuals with the same status as you or lower, with higher values corresponding to higher status. Your peer-inclusive upward-looking status is measured by the proportion of individuals with the same status as you or higher, with lower values corresponding to higher status. Let u represent this distribution of status.
Distribution u can be represented by the survivor function -the proportions of individuals in each category k or higher, = ∑ = , k = 1, …, K, with S1 = 1 and SK = fK.
The GL curve is defined as in (3)  Results 1 and 2 are as stated earlier except that u replaces s and one uses the revised definition of the GL curve.
The area-based inequality index for the peer-inclusive upward-looking status case is which is calculated using the revised definition of the GL curve.