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Model-free knockoffs: high-dimensional variable selection that controls the false discovery rate
Many contemporary large-scale applications involve building interpretable models linking a large set of potential covariates to a response in a nonlinear fashion, such as when the response is binary. Although this modeling problem has been extensively studied, it remains unclear how to effectively control the fraction of false discoveries even in high-dimensional logistic regression, not to mention general high-dimensional nonlinear models. To address such a practical problem, we propose a new framework of model-free knockoffs, which reads from a different perspective the knockoff procedure (Barber and Candès, 2015) originally designed for controlling the false discovery rate in linear models. The key innovation of our method is to construct knockoff variables probabilistically instead of geometrically. This enables model-free knockoffs to deal with arbitrary (and unknown) conditional models and any dimensions, including when the dimensionality p exceeds the sample size n, while the original knockoffs procedure is constrained to homoscedastic linear models with n greater than or equal to p. Our approach requires the design matrix be random (independent and identically distributed rows) with a covariate distribution that is known, although we show our procedure to be robust to unknown/estimated distributions. As we require no knowledge/assumptions about the conditional distribution of the response, we effectively shift the burden of knowledge from the response to the covariates, in contrast to the canonical model-based approach which assumes a parametric model for the response but very little about the covariates. To our knowledge, no other procedure solves the controlled variable selection problem in such generality, but in the restricted settings where competitors exist, we demonstrate the superior power of knockoffs through simulations. Finally, we apply our procedure to data from a case-control study of Crohn’s disease in the United Kingdom, making twice as many discoveries as the original analysis of the same data.