Test of Significance for High-Dimensional Thresholds with Application to Individualized Minimal Clinically Important Difference

Abstract

This work is motivated by learning the individualized minimal clinically important difference, a vital concept to assess clinical importance in various biomedical studies. We formulate the scientific question into a high-dimensional statistical problem where the parameter of interest lies in an individualized linear threshold. The goal is to develop a hypothesis testing procedure for the significance of a single element in this parameter as well as of a linear combination of this parameter. The difficulty dues to the high-dimensional nuisance in developing such a testing procedure, and also stems from the fact that this high-dimensional threshold model is nonregular and the limiting distribution of the corresponding estimator is nonstandard. To deal with these challenges, we construct a test statistic via a new bias-corrected smoothed decorrelated score approach, and establish its asymptotic distributions under both null and local alternative hypotheses. We propose a double-smoothing approach to select the optimal bandwidth in our test statistic and provide theoretical guarantees for the selected bandwidth. We conduct simulation studies to demonstrate how our proposed procedure can be applied in empirical studies. We apply the proposed method to a clinical trial where the scientific goal is to assess the clinical importance of a surgery procedure. Supplementary materials for this article are available online.

KEYWORDS:

• Bandwidth selection
• High-dimensional statistical inference
• Kernel method
• Nonstandard asymptotics

Supplemental Materials

The supplementary materials include the technical proofs, some more detailed theoretical results and discussions, and additional numerical results.

Acknowledgments

The authors would like to thank the Editor, an Associate Editor, and two reviewers for their insightful comments which have helped improve the manuscript substantially.

Additional information

Funding

Ning is supported in part by National Science Foundation (NSF) CAREER award DMS-1941945 and NSF award DMS-1854637. Zhao is supported in part by NSF award DMS-2122074 and a startup grant from the University of Wisconsin-Madison.