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 F R(c)=F D (c)/F A (c) and success ratio S R(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.

中文翻译：

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风险比率和斯堪伦的HRX

风险比是分布函数的尾部比，广泛用于健康差异研究中。令A和D分别表示cdfs FA（x）和FD（x），FA（x）≤FD（x）的有利和不利人群。考虑选择设置，其中选择的那些具有x> ca临界值。斯堪兰在经验数据中观察到，随着c降低，失败率FR（c）= FD（c）/ FA（c）和成功率SR（c）= [1-FD（c）] / [1-FA（c ）]都可以随着c的减小而增加，这是Scanlan称之为启发式规则X（HRX）的令人惊讶的结果。HRX对差异研究的影响尚未得到很好的理解，并且似乎经常被忽略。迄今为止，尚未证明HRX的分析条件。在正常情况下，如果方差相等，则HRX成立。通常，HRX并非可靠条件。