Multiple Point Hypothesis Test Problems and Effective Numbers of Tests for Control of the Family-Wise Error Rate

Abstract

Abtsrcat We consider a specific class of multiple test problems, consisting of M simultaneous point hypothesis tests in local statistical experiments. Under certain structural assumptions the global hypothesis contains exactly one element ϑ* (say), and ϑ* is least favourable parameter configuration with respect to the family-wise error rate (FWER) of multiple single-step tests, meaning that the FWER of such tests becomes largest under ϑ*. Further-more, it turns out that concepts of positive dependence are applicable to the involved test statistics in many practically relevant cases, in particular, for multivariate normal and chi-square distributions. Altogether, this allows for a relaxation of the adjustment for multiplicity by making use of the intrinsic correlation structure in the data. We represent product-type bounds for the FWER in terms of a relaxed Šidák-type correction of the overall significance level and compute “effective numbers of tests”. Our methodology can be applied to a variety of simultaneous inference problems for multi-dimensional (location) parameters, as frequently occurring in analysis of variance models or in the context of simultaneous categorical data analysis. For example, simultaneous chi-square tests for association of categorical features are ubiquitous in genetic association studies. In this type of model, Moskvina and Schmidt (2008) gave a formula for an effective number of tests utilizing Pearson's haplotypic correlation coeffcient as a linkage disequilibrium measure. Their result follows as a corollary from our general theory and will be gen- eralized.