Choosing the nominal level post-hoc with knockoffs using e-values

The knockoff filter is a powerful tool for controlled variable selection with false discovery rate (FDR) control. In this paper, we leverage e-values to allow the nominal FDR level to be switched post-hoc, after looking at the data and applying the knockoff procedure. This approach addresses a significant limitation of standard knockoffs: while frequently used in high-dimensional regressions, they often lack power in low-dimensional and sparse signal settings. One of the main reasons for this is that the knockoff filter requires a minimum number of selections that depends strictly on the nominal FDR level. By utilizing e-values, we can increase the nominal level in cases where the original procedure makes no discoveries, or decrease it to improve precision when discoveries are abundant. These improvements come without any costs, meaning the results of our post-hoc procedure are always more informative than those of the original knockoff filter. We extend this methodology to recently proposed derandomized knockoff procedures and demonstrate its utility in variable selection problems relevant to drug development using real clinical trial data.

💡 Research Summary

The paper introduces a novel post‑hoc adjustment framework for the knockoff filter that leverages e‑values to modify the nominal false discovery rate (FDR) level after the data have been examined. Traditional knockoff methods require a minimum number of discoveries equal to 1/α, where α is the pre‑specified target FDR. This constraint can be overly conservative in low‑dimensional or sparse‑signal settings, often yielding no selections when the true signal is modest. Moreover, analysts cannot retrospectively change α without violating the theoretical guarantees, limiting flexibility in exploratory studies.

The authors adopt the recent e‑value paradigm—non‑negative quantities that act as evidence measures and enjoy a multiplicative super‑martingale property—to construct variable‑specific e‑values from the knockoff statistics (W_i) and their signs (ε_i). By applying the e‑Closure principle, they define a data‑dependent adjusted level α̂ that satisfies
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