Querying multiple sets of $p$-values through composed hypothesis testing

Motivation: Combining the results of different experiments to exhibit complex patterns or to improve statistical power is a typical aim of data integration. The starting point of the statistical analysis often comes as sets of $p$-values resulting from previous analyses, that need to be combined in a flexible way to explore complex hypotheses, while guaranteeing a low proportion of false discoveries. Results: We introduce the generic concept of {\sl composed hypothesis}, which corresponds to an arbitrary complex combination of simple hypotheses. We rephrase the problem of testing a composed hypothesis as a classification task, and show that finding items for which the composed null hypothesis is rejected boils down to fitting a mixture model and classify the items according to their posterior probabilities. We show that inference can be efficiently performed and provide a thorough classification rule to control for type I error. The performance and the usefulness of the approach are illustrated on simulations and on two different applications combining data from different types. The method is scalable, does not require any parameter tuning, and provided valuable biological insight on the considered application cases. Availability: We implement the QCH methodology in the \texttt{qch} R package hosted on CRAN.