Measuring the magnitude of health inequality between 2 population subgroup proportions

Abstract

In this paper, we evaluate 11 measures of inequality, d(p1, p2), between 2 proportions p1 and p2, some of which are new to the health disparities literature. These measures are selected because they are continuous, nonnegative, equal to 0 if and only if |p1 - p2| = 0, and maximal when |p1 - p2| = 1. They are also symmetrical [d(p1, p2) = d(p2, p1)] and complement-invariant [d(p1, p2) = d(1 - p2, 1 - p1)]. To study intermeasure agreement, 5 of the 11 measures, including the absolute difference, are retained, because they remain finite and are maximal if and only if |p1 - p2| = 1. Even when the 2 proportions are assumed to be drawn at random from a shared distribution-interpreted as the absence of an avoidable difference-the expected value of d(p1, p2) depends on the shape of the distribution (and the choice of d) and can be quite large. To allow for direct comparisons among measures, we propose a standard measurement unit akin to a z score. For skewed underlying beta distributions, 4 of the 5 retained measures, once standardized, offer more conservative assessments of the magnitude of inequality than the absolute difference. We conclude that, even for measures that share the highlighted mathematical properties, magnitude comparisons are most usefully assessed relative to an elicited or estimated underlying distribution for the 2 proportions.