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

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 of this paper is to develop a hypothesis testing procedure for the significance of a single element in this high-dimensional parameter as well as for the significance of a linear combination of this parameter. The difficulty dues to the high-dimensionality of the nuisance component 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 the null and local alternative hypotheses. In addition, we propose a double-smoothing approach to select the optimal bandwidth parameter in our test statistic and provide theoretical guarantees for the selected bandwidth. We conduct comprehensive simulation studies to demonstrate how our proposed procedure can be applied in empirical studies. Finally, we apply the proposed method to a clinical trial where the scientific goal is to assess the clinical importance of a surgery procedure.