Powerful partial conjunction hypothesis testing via conditioning

Summary A partial conjunction hypothesis test combines information across a set of base hypotheses to determine whether some subset is nonnull. Partial conjunction hypothesis tests arise in a diverse array of fields, but standard partial conjunction hypothesis testing methods can be highly conservative, leading to low power especially in low-signal settings commonly encountered in applications. In this paper, we introduce the conditional partial conjunction hypothesis test, a new method for testing a single partial conjunction hypothesis 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 partial conjunction hypothesis testing, the proposed test is valid and produces nearly uniformly distributed $ p $-values under the null, i.e., the $ p $-values are only very slightly conservative. We demonstrate that our proposed test matches or outperforms existing single partial conjunction hypothesis tests with particular power gains in low-signal settings, maintains Type-I error control even under model misspecification and can be used to outperform state-of-the-art multiple partial conjunction hypothesis testing procedures in certain settings, particularly when side information is present. Finally, we illustrate an application of our proposed test through a replicability analysis across DNA microarray studies.