τ-censored weighted Benjamini–Hochberg procedures under independence

Abstract

In the field of multiple hypothesis testing, auxiliary information can be leveraged to enhance the efficiency of test procedures. A common way to make use of auxiliary information is by weighting p -values. However, when the weights are learned from data, controlling the finite-sample false discovery rate becomes challenging, and most existing weighted procedures only guarantee false discovery rate control in an asymptotic limit. In a recent study conducted by Ignatiadis & Huber (2021) , a novel τ -censored weighted Benjamini–Hochberg procedure was proposed to control the finite-sample false discovery rate. The authors employed the cross-weighting approach to learn weights for the p -values. This approach randomly splits the data into several folds and constructs a weight for each p -value P i using the p -values outside the fold containing P i . Cross-weighting does not exploit the p -value information inside the fold and only balances the weights within each fold, which may result in a loss of power. In this article, we introduce two methods for constructing data-driven weights for τ -censored weighted Benjamini–Hochberg procedures under independence. They provide new insight into masking p -values to prevent overfitting in multiple testing. The first method utilizes a leave-one-out technique, where all but one of the p -values are used to learn a weight for each p -value. This technique masks the information of a p -value in its weight by calculating the infimum of the weight with respect to the p -value. The second method uses partial information from each p -value to construct weights and utilizes the conditional distributions of the null p -values to establish false discovery rate control. Additionally, we propose two methods for estimating the null proportion and demonstrate how to integrate null-proportion adaptivity into the proposed weights to improve power.