Controlling False Discovery Rate Using Gaussian Mirrors

Abstract

Simultaneously, finding multiple influential variables and controlling the false discovery rate (FDR) for linear regression models is a fundamental problem. We here propose the Gaussian Mirror (GM) method, which creates for each predictor variable a pair of mirror variables by adding and subtracting a randomly generated Gaussian perturbation, and proceeds with a certain regression method, such as the ordinary least-square or the Lasso (the mirror variables can also be created after selection). The mirror variables naturally lead to test statistics effective for controlling the FDR. Under a mild assumption on the dependence among the covariates, we show that the FDR can be controlled at any designated level asymptotically. We also demonstrate through extensive numerical studies that the GM method is more powerful than many existing methods for selecting relevant variables subject to FDR control, especially for cases when the covariates are highly correlated and the influential variables are not overly sparse.

Keywords:

• False discovery rate
• Genome-wide association study
• High-dimensional linear regression
• Post-selection inference

Acknowledgment

Liu’s research was supported in part by the NSF grants DMS-1903139 and DMS2015411. Zhao’s research was supported in part by the NSF Grant IIS-1633283. The authors are grateful for discussions and helpful comments from Lucas Janson, Buyu Lin, Chenguang Dai, and Wenshuo Wang.

Additional information

Funding

Liu’s research was supported in part by the NSF grants DMS-1903139 and DMS2015411. Zhao’s research was supported in part by the NSF Grant IIS-1633283. The authors are grateful for discussions and helpful comments from Lucas Janson, Buyu Lin, Chenguang Dai, and Wenshuo Wang.