Hypervolume under ROC manifold for discrete biomarkers with ties

Medical multi-category diagnostic problems may involve discrete biomarkers. Many traditional accuracy measures are based on the assumption that all biomarkers follow continuous distributions and consequently may underestimate the true discrimination ability of the discrete markers. In particular, we focus on Hypervolume Under ROC Manifold (HUM) in this paper and propose an extension of the familiar continuous version of HUM to incorporate discrete biomarkers with ties. Statistical estimation and inference procedures are proposed along with asymptotic properties. We carry out simulation studies to examine the performance of our proposed estimators for the new HUM measure. A real medical example is analysed to illustrate our methodology.