  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.
