• bootstrap;
  • hypervolume under the ROC manifold;
  • maximum rank correlation estimator;
  • multi-category classification;
  • volume under the ROC surface


The Receiver Operating Characteristic (ROC) curve and the Area Under the ROC Curve (AUC) are effective statistical tools for evaluating the accuracy of diagnostic tests for binary-class medical data. However, many real-world biomedical problems involve more than two categories. The Volume Under the ROC Surface (VUS) and Hypervolume Under the ROC Manifold (HUM) measures are extensions for the AUC under three-class and multiple-class models. Inference methods for such measures have been proposed recently. We develop a method of constructing a linear combination of markers for which the VUS or HUM of the combined markers is maximized. Asymptotic validity of the estimator is justified by extending the results for maximum rank correlation estimation that are well known in econometrics. A bootstrap resampling method is then applied to estimate the sampling variability. Simulations and examples are provided to demonstrate our methods.