The Meaning and Use of the Volume Under a Three-Class ROC Surface (VUS)

Previously, we have proposed a method for three-class receiver operating characteristic (ROC) analysis based on decision theory. In this method, the volume under a three-class ROC surface (VUS) serves as a figure-of-merit (FOM) and measures three-class task performance. The proposed three-class ROC analysis method was demonstrated to be optimal under decision theory according to several decision criteria. Further, an optimal three-class linear observer was proposed to simultaneously maximize the signal-to-noise ratio (SNR) between the test statistics of each pair of the classes provided certain data linearity condition. Applicability of this three-class ROC analysis method would be further enhanced by the development of an intuitive meaning of the VUS and a more general method to calculate the VUS that provides an estimate of its standard error. In this paper, we investigated the general meaning and usage of VUS as a FOM for three-class classification task performance. We showed that the VUS value, which is obtained from a rating procedure, equals the percent correct in a corresponding categorization procedure for continuous rating data. The significance of this relationship goes beyond providing another theoretical basis for three-class ROC analysis—it enables statistical analysis of the VUS value. Based on this relationship, we developed and tested algorithms for calculating the VUS and its variance. Finally, we reviewed the current status of the proposed three-class ROC analysis methodology, and concluded that it extends and unifies decision theoretic, linear discriminant analysis, and psychophysical foundations of binary ROC analysis in a three-class paradigm.