Evaluating joint confidence region of hypervolume under ROC manifold and generalized Youden index

In biomarker evaluation/diagnostic studies, the hypervolume under the receiver operating characteristic manifold ((Formula presented.)) and the generalized Youden index ((Formula presented.)) are the most popular measures for assessing classification accuracy under multiple classes. While (Formula presented.) is frequently used to evaluate the overall accuracy, (Formula presented.) provides direct measure of accuracy at the optimal cut-points. Simultaneous evaluation of (Formula presented.) and (Formula presented.) provides a comprehensive picture about the classification accuracy of the biomarker/diagnostic test under consideration. This article studies both parametric and non-parametric approaches for estimating the confidence region of (Formula presented.) and (Formula presented.) for a single biomarker. The performances of the proposed methods are investigated by an extensive simulation study and are applied to a real data set from the Alzheimer's Disease Neuroimaging Initiative.