E-values as unnormalized weights in multiple testing

Standard weighted multiple testing methods require the weights to deterministically add up to the number of hypotheses being tested. We show that this normalization is not required when the weights are not constants, but are themselves e-values obtained from independent data. This could result in a massive increase in power, especially if the non-null hypotheses have e-values much larger than one. More broadly, we study how to combine an e-value and a p-value, and design multiple testing procedures where both e-values and p-values are available for every hypothesis. For false discovery rate control, analogous to the Benjamini-Hochberg procedure with p-values (p-BH) and the recent e-BH procedure for e-values, we propose two new procedures: ep-BH and pe-BH with finite sample validity under different dependence assumptions. These procedures are designed based on several admissible combining functions for p/e-values, which also yield methods for family-wise error rate control. We demonstrate the practical power benefits with a case study with RNA-Seq and microarray data.