Powerful Partial Conjunction Hypothesis Testing via Conditioning

Research questions across a diverse array of fields are formulated as a Partial Conjunction Hypothesis (PCH) test, which combines information across $m$ base hypotheses to determine whether some subset is non-null. However, standard methods for testing a PCH can be highly conservative. In this paper, we introduce the conditional PCH (cPCH) test, a new framework for testing a single PCH that directly corrects the conservativeness of standard approaches by conditioning on certain order statistics of the base p-values. Under distributional assumptions commonly encountered in PCH testing, the cPCH test produces a p-value that is nearly uniform. Through simulations, we demonstrate that the cPCH test uniformly outperforms standard single PCH tests and maintains Type I error control even under model misspecification, and can in certain situations also be used to outperform state-of-the-art PCH multiple testing procedures. Finally, we illustrate an application of the cPCH test on a replicability analysis of four microarray studies.