False discovery rate control with compound p-values

In the setting of multiple testing, compound p-values generalize p-values by asking for superuniformity to hold only on average across all true nulls. We study the properties of the Benjamini–Hochberg procedure applied to compound p-values. Under independence, we show that the false discovery rate (FDR) is at most 1.93α, where α is the nominal level, and exhibit a distribution for which the FDR is 76α. If additionally all nulls are true, then the upper bound can be improved to α+2α2, with a corresponding worst-case lower bound of α+α2/4. Under positive dependence, on the other hand, we demonstrate that FDR can be inflated by a factor of O(logm), where m is the number of hypotheses. We provide numerous examples of settings where compound p-values arise in practice, either because we lack sufficient information to compute non-trivial p-values, or to facilitate a more powerful analysis.