Global and Simultaneous Hypothesis Testing for High-Dimensional Logistic Regression Models

Abstract

High-dimensional logistic regression is widely used in analyzing data with binary outcomes. In this article, global testing and large-scale multiple testing for the regression coefficients are considered in both single- and two-regression settings. A test statistic for testing the global null hypothesis is constructed using a generalized low-dimensional projection for bias correction and its asymptotic null distribution is derived. A lower bound for the global testing is established, which shows that the proposed test is asymptotically minimax optimal over some sparsity range. For testing the individual coefficients simultaneously, multiple testing procedures are proposed and shown to control the false discovery rate and falsely discovered variables asymptotically. Simulation studies are carried out to examine the numerical performance of the proposed tests and their superiority over existing methods. The testing procedures are also illustrated by analyzing a dataset of a metabolomics study that investigates the association between fecal metabolites and pediatric Crohn’s disease and the effects of treatment on such associations. Supplementary materials for this article are available online.

keyword:

• False discovery rate
• Global testing
• Large-scale multiple testing
• Minimax lower bound

Supplementary Materials

In the online supplemental materials, we prove Theorem 3, 5, Proposition 1, and the technical lemmas. The technical results and simulations concerning the two-sample tests discussed in Section 4 are also included.