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Abstract:
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Recent exploration of optimal individualized decision rules (IDRs) for patients in precision medicine has attracted a lot of attention due to heterogeneous responses of patients to different treatments. In the existing literature of precision medicine, an optimal IDR is defined as a decision function mapping from the patients' covariate space into the treatment space that maximizes the expected outcome of each individual. Motivated by the concept of Optimized Certainty Equivalent (OCE) introduced originally in \cite{ben1986expected}, we propose a decision-rule based optimized covariates dependent equivalent (CDE) for individualized decision making problems. Our proposed IDR-CDE broadens the existing expected mean outcome framework in precision medicine and enriches the previous concept of the OCE. Under a functional margin description of the decision rule modeled by an indicator function as in the literature of large-margin classifiers, the empirical minimization problem for estimating the optimal IDRs involves a discontinuous objective function. We show that, under a mild condition at the population level, the epigraphical formulation of this empirical optimization problem is a difference-of-convex (dc) constrained dc program. A dc algorithm is adopted to solve the resulting dc program. Numerical experiments demonstrate that our overall approach outperforms existing methods in estimating optimal IDRs under heavy tail distributions of the data. In addition to providing a risk based approach for individualized medical treatments, which is new in the area of precision medicine, the main contributions of this work in general are: the broadening of the concept of the OCE, the epigraphical description of the empirical IDR-CDE minimization problem, its equivalent dc formulation, and the sequential convergence proof of the dc algorithm for a (special) dc constrained dc program.
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