Multiple point hypothesis test problems and effective numbers of tests

We consider a special class of multiple testing problems, consisting of M simultaneous point hypothesis tests in local statistical experiments. Under certain structural assumptions the global hypothesis contains exactly one element v* (say), and v* 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 v*. Furthermore, 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-squared 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 éSidák-type correction of the overall significance level and compute effective numbers of tests. Our methodology can be applied to a variety of simultaneous location parameter problems, as in analysis of variance models or in the context of simultaneous categorical data analysis. For example, simultaneous chisquare tests for association of categorical features are ubiquitous in genomewide 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 coefficient as a linkage disequilibrium measure. Their result follows as a corollary from our general theory and will be generalized.