Hypothesis testing for high-dimensional sparse binary regression

TL;DR

This paper investigates the limits of hypothesis testing in high-dimensional sparse binary regression models, revealing new phenomena related to design matrix sparsity and proposing optimal testing procedures.

Contribution

It introduces the concept of a design matrix sparsity index affecting detection boundaries and develops rate-optimal tests for different sparsity regimes.

Findings

Detection boundary depends on design matrix sparsity and signal strength.

Generalized likelihood ratio test is optimal in dense regimes.

Extended Higher Criticism Test is optimal in sparse regimes.

Abstract

In this paper, we study the detection boundary for minimax hypothesis testing in the context of high-dimensional, sparse binary regression models. Motivated by genetic sequencing association studies for rare variant effects, we investigate the complexity of the hypothesis testing problem when the design matrix is sparse. We observe a new phenomenon in the behavior of detection boundary which does not occur in the case of Gaussian linear regression. We derive the detection boundary as a function of two components: a design matrix sparsity index and signal strength, each of which is a function of the sparsity of the alternative. For any alternative, if the design matrix sparsity index is too high, any test is asymptotically powerless irrespective of the magnitude of signal strength. For binary design matrices with the sparsity index that is not too high, our results are parallel to those in the Gaussian case. In this context, we derive detection boundaries for both dense and sparse regimes. For the dense regime, we show that the generalized likelihood ratio is rate optimal; for the sparse regime, we propose an extended Higher Criticism Test and show it is rate optimal and sharp. We illustrate the finite sample properties of the theoretical results using simulation studies.