Testing partial conjunction hypotheses under dependency, with applications to meta-analysis

In many statistical problems the hypotheses are naturally divided into groups, and the investigators are interested to perform group-level inference, possibly along with inference on individual hypotheses. We consider the goal of discovering groups containing $u$ or more signals with group-level false discovery rate (FDR) control. This goal can be addressed by multiple testing of partial conjunction hypotheses with a parameter $u,$ which reduce to global null hypotheses for $u=1.$ We consider the case where the partial conjunction $p$-values are combinations of within-group $p$-values, and obtain sufficient conditions on (1) the dependencies among the $p$-values within and across the groups, (2) the combining method for obtaining partial conjunction $p$-values, and (3) the multiple testing procedure, for obtaining FDR control on partial conjunction discoveries. We consider separately the dependencies encountered in the meta-analysis setting, where multiple features are tested in several independent studies, and the $p$-values within each study may be dependent. Based on the results for this setting, we generalize the procedure of Benjamini, Heller, and Yekutieli (2009) for assessing replicability of signals across studies, and extend their theoretical results regarding FDR control with respect to replicability claims.