A Decorrelating and Debiasing Approach to Simultaneous Inference for High-Dimensional Confounded Models

Motivated by the simultaneous association analysis with the presence of latent confounders, this article studies the large-scale hypothesis testing problem for the high-dimensional confounded linear models with both non-asymptotic and asymptotic false discovery control. Such model covers a wide range of practical settings where both the response and the predictors may be confounded. In the presence of the high-dimensional predictors and the unobservable confounders, the simultaneous inference with provable guarantees becomes highly challenging, and the unknown strong dependence among the confounded covariates makes the challenge even more pronounced. This article first introduces a decorrelating procedure that shrinks the confounding effect and weakens the correlations among the predictors, then performs debiasing under the decorrelated design based on some biased initial estimator. Following that, an asymptotic normality result for the debiased estimator is established and standardized test statistics are then constructed. Furthermore, a simultaneous inference procedure is proposed to identify significant associations, and both the finite-sample and asymptotic false discovery bounds are provided. The non-asymptotic result is general and model-free, and is of independent interest. We also prove that, under minimal signal strength condition, all associations can be successfully detected with probability tending to one. Simulation and real data studies are carried out to evaluate the performance of the proposed approach and compare it with other competing methods. Supplementary materials for this article are available online.