Improving the adjusted Benjamini--Hochberg method using e-values in knockoff-assisted variable selection

Considering the knockoff-based multiple testing framework of Barber and Candès [2015], we revisit the method of Sarkar and Tang [2022] and identify it as a specific case of an un-normalized e-value weighted Benjamini-Hochberg procedure. Building on this insight, we extend the method to use bounded p-to-e calibrators that enable more refined and flexible weight assignments. Our approach generalizes the method of Sarkar and Tang [2022], which emerges as a special case corresponding to an extreme calibrator. Within this framework, we propose three procedures: an e-value weighted Benjamini-Hochberg method, its adaptive extension using an estimate of the proportion of true null hypotheses, and an adaptive weighted Benjamini-Hochberg method. We establish control of the false discovery rate (FDR) for the proposed methods. While we do not formally prove that the proposed methods outperform those of Barber and Candès [2015] and Sarkar and Tang [2022], simulation studies and real-data analysis demonstrate large and consistent improvement over the latter in all cases, and better performance than the knockoff method in scenarios with low target FDR, a small number of signals, and weak signal strength. Simulation studies and a real-data application in HIV-1 drug resistance analysis demonstrate strong finite sample FDR control and exhibit improved, or at least competitive, power relative to the aforementioned methods.

💡 Research Summary

This paper revisits the knockoff‑based multiple‑testing framework introduced by Barber and Candès (2015) and shows that the two‑stage procedure of Sarkar and Tang (2022) can be interpreted as a special case of an un‑normalized e‑value weighted Benjamini–Hochberg (BH) method. By viewing the first‑stage p‑values through a p‑to‑e calibration function, the authors convert them into e‑values that serve as data‑driven weights for the second‑stage p‑values. They propose three new procedures: (1) a basic e‑value weighted BH (ep‑BH), (2) an adaptive ep‑BH that incorporates Storey’s estimate of the proportion of true nulls (π₀), and (3) a weighted adaptive BH that normalizes the e‑values to sum to the number of hypotheses and then applies a data‑driven scaling factor (δ̂₀).

The key technical insight is that bounded calibrators—functions g: