Comparing latent inequality with ordinal data

TL;DR

This paper introduces new methods for comparing latent distributions using only ordinal data, providing identification results and inference tools without assuming parametric forms.

Contribution

It offers novel identification results and inference methods for comparing latent inequalities from ordinal data, applicable to empirical health studies.

Findings

Identification of between-group inequality evidence

Detection of within-group inequality differences

Development of confidence sets for quantile comparisons

Abstract

We propose new ways to compare two latent distributions when only ordinal data are available and without imposing parametric assumptions on the underlying continuous distributions. First, we contribute identification results. We show how certain ordinal conditions provide evidence of between-group inequality, quantified by particular quantiles being higher in one latent distribution than in the other. We also show how other ordinal conditions provide evidence of higher within-group inequality in one distribution than in the other, quantified by particular interquantile ranges being wider in one latent distribution than in the other. Second, we propose an "inner" confidence set for the quantiles that are higher for the first latent distribution. We also describe frequentist and Bayesian inference on features of the ordinal distributions relevant to our identification results. Our contributions are illustrated by empirical examples with mental health and general health.