Statistical Inference for High-Dimensional Generalized Linear Models With Binary Outcomes

Abstract

This article develops a unified statistical inference framework for high-dimensional binary generalized linear models (GLMs) with general link functions. Both unknown and known design distribution settings are considered. A two-step weighted bias-correction method is proposed for constructing confidence intervals (CIs) and simultaneous hypothesis tests for individual components of the regression vector. Minimax lower bound for the expected length is established and the proposed CIs are shown to be rate-optimal up to a logarithmic factor. The numerical performance of the proposed procedure is demonstrated through simulation studies and an analysis of a single cell RNA-seq dataset, which yields interesting biological insights that integrate well into the current literature on the cellular immune response mechanisms as characterized by single-cell transcriptomics. The theoretical analysis provides important insights on the adaptivity of optimal CIs with respect to the sparsity of the regression vector. New lower bound techniques are introduced and they can be of independent interest to solve other inference problems in high-dimensional binary GLMs.

KEYWORDS:

• Adaptivity
• Confidence interval
• Hypothesis testing
• Link functions
• Optimality
• Weighting

Acknowledgements

We would like to thank the editor, associate editor, and two anonymous referees for helpful suggestions that significantly improved the presentation of the results. This work was completed while Rong Ma was a PhD student in the biostatistics program at the University of Pennsylvania.

Supplementary Materials

In the supplement, we prove all the main theorems and the technical lemmas. Some additional discussions about assumptions and numerical studies are also included.

Additional information

Funding

Tony Cai’s research was supported in part by NSF grant DMS-2015259 and NIH grants R01-GM129781 and R01-GM123056. Zijian Guo’s research was supported in part by NSF grants DMS-1811857 and DMS-2015373 and NIH grants R01-GM140463 and R01-LM013614.