Multiple testing of partial conjunction null hypotheses with conditional $p$-values based on combination test statistics

The partial conjunction null hypothesis is tested in order to discover a signal that is present in multiple studies. We propose methods for multiple testing of partial conjunction null hypotheses which make use of conditional $p$-values based on combination test statistics. Specific examples comprise the Fisher combination function and the Stouffer combination function. The conditional validity of the corresponding $p$-values is proved for certain classes of one-parametric statistical models, including one-parameter natural exponential families. The standard approach of carrying out a multiple test procedure on the (unconditional) 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. By means of computer simulations and real data analyses, we compare the proposed methodology with other recent approaches.