Weighted multiple testing procedure for grouped hypotheses with k-FWER control

Abstract

In this paper, k-FWER (generalized familywise error rate) control for grouped hypotheses testing is considered. We offer the weights for the p-values in each group, by maximizing an objective function, which is the expectation of the proportion of rejected hypotheses. This objective function utilizes not only the information of the proportion of true null hypotheses, but also the null and non-null distributions of the p-values in each group. When this information is known prior, our weighted testing procedure controls k-FWER for arbitrarily dependent p-values. When this information is unknown, and is estimated from the data, our procedure asymptotically controls k-FWER under the weak dependence assumption of the p-values in each group. The new procedure is shown to be more powerful than some existing procedures both in theory and simulations. For illustration, the proposed procedure is applied to analyse the adequate yearly progress data.