One-at-a-time knockoffs: controlled false discovery rate with higher power

TL;DR

The paper introduces one-at-a-time knockoffs (OATK), a new method for variable selection in linear regression that controls false discovery rate with higher power and computational efficiency compared to existing approaches.

Contribution

OATK simplifies and relaxes the knockoff filter by generating knockoffs one-at-a-time, enabling higher power, better computational efficiency, and additional enhancements for FDR control.

Findings

OATK asymptotically controls FDR under mild conditions.

OATK achieves higher power than existing methods like BC.

OATK offers computational advantages and flexibility for further improvements.

Abstract

We propose one-at-a-time knockoffs (OATK), a new methodology for detecting important explanatory variables in linear regression models while controlling the false discovery rate (FDR). For each explanatory variable, OATK generates a knockoff design matrix that preserves the Gram matrix by replacing one-at-a-time only the single corresponding column of the original design matrix. OATK is a substantial relaxation and simplification of the knockoff filter by Barber and Cand\`es (BC), which simultaneously generates all columns of the knockoff design matrix to satisfy a much larger set of constraints. To test each variable's importance, statistics are then constructed by comparing the original vs. knockoff coefficients. Under a mild correlation assumption on the original design matrix, OATK asymptotically controls the FDR at any desired level. Moreover, OATK consistently achieves (often substantially) higher power than BC and other approaches across a variety of simulation examples and a real genetics dataset. Generating knockoffs one-at-a-time also has substantial computational advantages and facilitates additional enhancements, such as conditional calibration or derandomization, to further improve power and consistency of FDR control. OATK can be viewed as the conditional randomization test (CRT) generalized to fixed-design linear regression problems, and can generate fine-grained p-values for each hypothesis.