Proper Inference for Value Function in High-Dimensional Q-Learning for Dynamic Treatment Regimes

ABSTRACT

Dynamic treatment regimes are a set of decision rules and each treatment decision is tailored over time according to patients’ responses to previous treatments as well as covariate history. There is a growing interest in development of correct statistical inference for optimal dynamic treatment regimes to handle the challenges of nonregularity problems in the presence of nonrespondents who have zero-treatment effects, especially when the dimension of the tailoring variables is high. In this article, we propose a high-dimensional Q-learning (HQ-learning) to facilitate the inference of optimal values and parameters. The proposed method allows us to simultaneously estimate the optimal dynamic treatment regimes and select the important variables that truly contribute to the individual reward. At the same time, hard thresholding is introduced in the method to eliminate the effects of the nonrespondents. The asymptotic properties for the parameter estimators as well as the estimated optimal value function are then established by adjusting the bias due to thresholding. Both simulation studies and real data analysis demonstrate satisfactory performance for obtaining the proper inference for the value function for the optimal dynamic treatment regimes. Supplementary materials for this article are available online.

KEYWORDS:

• Hard threshold
• Q-learning
• Value function inference
• Variable selection

Supplementary Material

The online supplement contains additional simulation results, and the proofs for the theorems discussed in the article.

Acknowledgment

The authors thank the associate editor and two referees for their constructive comments that led to a significantly improved article.

Additional information

Funding

This work is supported in part by the National Natural Science Foundation of China (11771072 and 11371083), Foundation from China Scholarship Council (No. 201406625026), National Institute of Health and National Science Foundation (2R01NS073671-05A1, 1R01GM124104-01A1, NCI-P01-CA142538, NSF-DMS-1555244).