Simultaneous test procedures in terms of p-value copulae

Abstract

At least since [1], a broad class of multiple comparison procedures, so-called simultaneous test procedures (STPs), is established in the statistical literature. Elements of an STP are a testing family, consisting of a set of null hypotheses and corresponding test statistics, and a common critical constant. The latter threshold with which each of the test statistics has to be compared is calculated under the (joint) intersection hypothesis of all nulls. Under certain structural assumptions, the so-constructed STP provides strong control of the family-wise error rate. More recently, a general method to construct STPs in the case of asymptotic (joint) normality of the family of test statistics has been developed in [2], and numerical solutions to compute the critical constant in such cases were provided. Here, we propose to look at the problem from a different perspective. We will show that the threshold can equivalently be expressed by a quantile of the copula of the family of pvalues associated with the test statistics, assuming that each of these p-values is marginally uniformly distributed on the unit interval under the corresponding null hypothesis. This offers the opportunity to exploit the rich and growing literature on copula-based modeling of multivariate dependency structures for multiple testing problems and in particular for the construction of STPs in non-Gaussian situations.