Estimating the Optimal Linear Combination of Biomarkers using Spherically Constrained Optimization

TL;DR

This paper introduces a novel, efficient derivative-free optimization method for estimating the best linear combination of biomarkers to maximize classification performance, applicable to multi-category medical diagnosis.

Contribution

It proposes a pattern search-based global optimization technique that improves computational efficiency over existing methods for maximizing the hypervolume under the manifold in multi-category classification.

Findings

The proposed method outperforms existing algorithms in simulation studies.

It effectively predicts swallowing difficulty after radiation therapy.

The method is demonstrated on real medical data for disease diagnosis.

Abstract

In the context of a binary classification problem, the optimal linear combination of continuous predictors can be estimated by maximizing an empirical estimate of the area under the receiver operating characteristic (ROC) curve (AUC). For multi-category responses, the optimal predictor combination can similarly be obtained by maximization of the empirical hypervolume under the manifold (HUM). This problem is particularly relevant to medical research, where it may be of interest to diagnose a disease with various subtypes or predict a multi-category outcome. Since the empirical HUM is discontinuous, non-differentiable, and possibly multi-modal, solving this maximization problem requires a global optimization technique. Estimation of the optimal coefficient vector using existing global optimization techniques is computationally expensive, becoming prohibitive as the number of predictors 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. Through extensive simulation studies, we demonstrate that the proposed method achieves better performance compared to existing methods including the step-down algorithm. Finally, we illustrate the proposed method to predict swallowing difficulty after radiation therapy for oropharyngeal cancer based on radiation dose to various structures in the head and neck.