Rank verification for exponential families

Abstract

Many statistical experiments involve comparing multiple population groups. For example, a public opinion poll may ask which of several political candidates commands the most support; a social scientific survey may report the most common of several responses to a question; or, a clinical trial may compare binary patient outcomes under several treatment conditions to determine the most effective treatment. Having observed the “winner” (largest observed response) in a noisy experiment, it is natural to ask whether that candidate, survey response or treatment is actually the “best” (stochastically largest response). This article concerns the problem of rank verification—post hoc significance tests of whether the orderings discovered in the data reflect the population ranks. For exponential family models, we show under mild conditions that an unadjusted two-tailed pairwise test comparing the first two-order statistics (i.e., comparing the “winner” to the “runner-up”) is a valid test of whether the winner is truly the best. We extend our analysis to provide equally simple procedures to obtain lower confidence bounds on the gap between the winning population and the others, and to verify ranks beyond the first.