Smoothed Estimation on Optimal Treatment Regime Under Semisupervised Setting in Randomized Trials

A treatment regime refers to the process of assigning the most suitable treatment to a patient based on their observed information. However, prevailing research on treatment regimes predominantly relies on labeled data, which may lead to the omission of valuable information contained within unlabeled data, such as historical records and healthcare databases. Current semisupervised works for deriving optimal treatment regimes either rely on model assumptions or struggle with high computational burdens for even moderate-dimensional covariates. To address this concern, we propose a semisupervised framework that operates within a model-free context to estimate the optimal treatment regime by leveraging the abundant unlabeled data. Our proposed approach encompasses three key steps. First, we employ a single-index model to achieve dimension reduction, followed by kernel regression to impute the missing outcomes in the unlabeled data. Second, we propose various forms of semisupervised value functions based on the imputed values, incorporating both labeled and unlabeled data components. Lastly, the optimal treatment regimes are derived by maximizing the semisupervised value functions. We establish the consistency and asymptotic normality of the estimators proposed in our framework. Furthermore, we introduce a perturbation resampling procedure to estimate the asymptotic variance. Simulations confirm the advantageous properties of incorporating unlabeled data in the estimation for optimal treatment regimes. A practical data example is also provided to illustrate the application of our methodology. This work is rooted in the framework of randomized trials, with additional discussions extending to observational studies.