Efficient semi-supervised estimation of optimal individualized treatment regimes with survival outcome

Individualized treatment regimes (ITRs) represent decision-making frameworks that tailor treatment assignments to individual patient characteristics. The value function of an ITR quantifies the expected outcome under a counterfactual scenario in which such a treatment rule is applied. However, estimating optimal ITRs for survival data remains a significant challenge when outcomes are right-censored and only a subset of patients has complete outcome information due to time and cost constraints. To overcome this challenge, we formulate the problem within a semi-supervised learning framework and adopt an induced missingness perspective to model partially observed survival outcomes. We propose an imputation-based semi-supervised approach that is robust and adaptable to various imputation models. Specifically, we employ a flexible single-index kernel smoothing imputation technique to effectively utilize unlabeled data in multidimensional covariate settings. The proposed estimators for the parameters indexing the optimal ITRs are shown to be consistent and asymptotically normal. Moreover, semi-supervised estimation enhances efficiency by reducing asymptotic variance relative to supervised estimation. Numerical experiments on both simulated and real datasets demonstrate the superior performance of our proposed semi-supervised approach.