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 ( ${\text{HUM}}_K$ ) and the generalized Youden index ( $J_K$ ) are the most popular measures for assessing classification accuracy under multiple classes. While ${\text{HUM}}_K$ is frequently used to evaluate the overall accuracy, $J_K$ provides direct measure of accuracy at the optimal cut-points. Simultaneous evaluation of ${\text{HUM}}_K$ and $J_K$ 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 ${\text{HUM}}_K$ and $J_K$ 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.