False discovery rate control with compound p-values
Rina Foygel Barber, Richard J Samworth
Journal of the Royal Statistical Society: Series B (Statistical Methodology) · Fri
Abstract
In the setting of multiple testing, compound p-values generalize p-values by asking for superuniformity to hold onlyon averageacross 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 most1.93α, whereαis the nominal level, and exhibit a distribution for which the FDR is76α. If additionally all nulls are true, then the upper bound can be improved toα+2α2, with a corresponding worst-case lower bound ofα+α2/4. Under positive dependence, on the other hand, we demonstrate that FDR can be inflated by a factor ofO(logm), wheremis 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.
Keywords
Benjamini—Hochberg procedure, compound p-values, false discovery rate
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