A Sparse Random Projection-Based Test for Overall Qualitative Treatment Effects

Abstract

In contrast to the classical “one-size-fits-all” approach, precision medicine proposes the customization of individualized treatment regimes to account for patients’ heterogeneity in response to treatments. Most of existing works in the literature focused on estimating optimal individualized treatment regimes. However, there has been less attention devoted to hypothesis testing regarding the existence of overall qualitative treatment effects, especially when there are a large number of prognostic covariates. When covariates do not have qualitative treatment effects, the optimal treatment regime will assign the same treatment to all patients regardless of their covariate values. In this article, we consider testing the overall qualitative treatment effects of patients’ prognostic covariates in a high-dimensional setting. We propose a sample splitting method to construct the test statistic, based on a nonparametric estimator of the contrast function. When the dimension of covariates is large, we construct the test based on sparse random projections of covariates into a low-dimensional space. We prove the consistency of our test statistic. In the regular cases, we show the asymptotic power function of our test statistic is asymptotically the same as the “oracle” test statistic which is constructed based on the “optimal” projection matrix. Simulation studies and real data applications validate our theoretical findings. Supplementary materials for this article are available online.

Keywords:

• High-dimensional testing
• Optimal treatment regime
• Precision medicine
• Qualitative treatment effects
• Sparse random projection

Acknowledgments

The authors thank the editor, the AE, and two referees for their helpful suggestions that significantly improved the quality of the article.

Additional information

Funding

The research of Chengchun Shi and Rui Song is partially supported by grant NSF-DMS-1555244 and grant NCI P01 CA142538. The research of Wenbin Lu is partially supported by grant NCI P01 CA142538.