Multiple testing of partial conjunction null hypotheses, with application to replicability analysis of high dimensional studies

The partial conjunction null hypothesis is tested in order to discover a signal that is present in multiple studies. The standard approach of carrying out a multiple test procedure on the partial conjunction $p$-values can be extremely conservative. We suggest alleviating this conservativeness, by eliminating many of the conservative partial conjunction $p$-values prior to the application of a multiple test procedure. This leads to the following two step procedure: first, select the set with partial conjunction $p$-values below a selection threshold; second, within the selected set only, apply a family-wise error rate or false discovery rate controlling procedure on the conditional partial conjunction $p$-values. Applying our procedure to multiple genome wide association studies of Crohn's disease, we make many more discoveries than other recent approaches. We prove that the conditional PC $p$-values are valid for certain classes of one-parametric statistical models (including one-parameter natural exponential families), and provide the conditions for which the FDR controlling procedures considered will be below the nominal level. We also compare the proposed methodology with the other recent approaches by means of computer simulations.