Risk ratios and Scanlan’s HRX

Abstract

Risk ratios are distribution function tail ratios and are widely used in health disparities research. Let A and D denote advantaged and disadvantaged populations with cdfs F A (x) and F D (x) respectively, F A (x)≤F D (x). Consider a selection setting where those selected have x>c a critical value. Scanlan observed in empirical data that as c is lowered the failure ratio FR(c)=F D (c)/F A (c) and success ratio SR(c)=[1−F D (c)]/[1−F A (c)] can both be increasing with decreasing c, a surprising result Scanlan calls Heuristic Rule X (HRX). The consequences of HRX for disparities research have not been well understood, and appear to be often ignored. The analytical conditions for HRX to hold have not, heretofore, been proven. In the normal case, HRX holds if the variances are equal. In general, HRX is not a robust condition. Settings and distributional conditions necessary for HRX to hold are discussed, including mixture settings of normal and other variables.