Estimating the Optimal Linear Combination of Biomarkers using Spherically Constrained Optimization
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In the context of a binary classification problem, the optimal linear combination of biomarkers is estimated by maximizing an empirical estimate of the area under the receiver operating characteristic curve. For multi-category outcomes, the optimal biomarker combination can similarly be obtained by maximization of the empirical hypervolume under the manifold (HUM). Since the empirical HUM is discontinuous, non-differentiable, and possibly multi-modal, solving this maximization problem requires a global optimization technique. The recently proposed smoothed approximation of the empirical HUM partially addresses this issue, as it is differentiable over the domain, but the objective function still remains non-concave and possibly multi-modal. Estimation of the optimal coefficient vector using existing global optimization techniques is computationally expensive, becoming prohibitive as the number of biomarkers and the number of outcome categories increases. We propose an efficient derivative-free black-box optimization technique based on pattern search to solve this problem. In both simulation studies and a benchmark real data application, the pro-posed method achieves better performance and greatly reduced computational time as compared to existing methods
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