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 \emph{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 $\frac{7}{6}α$. If additionally all nulls are true, then the upper bound can be improved to $α+ 2α^2$, with a corresponding worst-case lower bound of $α+ α^2/4$. Under positive dependence, on the other hand, we demonstrate that FDR can be inflated by a factor of $O(\log m)$, 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.