One-at-a-time knockoffs.

We propose one-at-a-time knockoffs (OATK), a new methodology for detecting important explanatory variables in linear regression problems (including ridge and lasso) 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, 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 versus knockoff coefficients. Under a mild correlation assumption on the original design matrix, we prove that OATK asymptotically controls the FDR at any desired level. Moreover, numerical results across a variety of simulation examples and a real genetics data set demonstrate that OATK provides good FDR control and regularly achieves (often substantially) higher power than existing approaches. In particular, in a genome-wide association study, OATK yielded more uniquely discovered genetic mutations associated with virological response to HIV therapeutics that were corroborated in clinical studies. 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.