Optimal linear combination of biomarkers for multi-category diagnosis

The receiver operating characteristic (ROC) curve and the area under the ROC curve have been popularly employed in evaluating the diagnosis accuracy for diseases with binary outcome categories and have been naturally used as the utility measures for finding the 'optimal' linear combination of multiple biomarkers, in the hope to improve the diagnostic accuracy based on each single biomarker. For diseases with more than two outcome categories, the ROC manifold and the hypervolume under the ROC manifold (HUM) have been analogously proposed as diagnostic accuracy measures. However, finding optimal combinations of biomarkers based on the HUM criterion is less easily feasible in computation, especially when the number of disease categories is more than three and the number of biomarkers is large. In this study, we propose two new indices for evaluating the diagnostic accuracy for multi-category diagnosis, which are related to the lower and upper bounds of HUM, and involve only diagnostic accuracies for comparing adjacent pairs of outcome categories. We then propose finding the optimal linear combinations of biomarkers for multi-category diagnosis using the new indices as the criterion functions. Simulations and real data examples show that the optimal linear combinations identified by the new proposal perform quite well in diagnostic accuracy and can be much more efficient in computation than the HUM-based method.