The Surprising Power of Algebraic Subgroup Selection in Multiple Testing

In this paper, we analyze the effects of the strategic selection of an algebraic subgroup for use in the permutation- or group invariance-based Westfall \& Young maxT multiple testing method. We report the surprising observation that a tiny subgroup can produce a version of the maxT method that dramatically outperforms the method based on entire group or on a large number of Monte Carlo draws. To explain these findings, we characterize the power of the maxT method based on a strategically selected subgroup and the entire group in a Gaussian location model with $p$ tests and $n$ observations. By studying the relative efficiency, we find that the power difference is largest in high dimensional settings where $n^{-1/2} \log p$ is large.