Comparison of two correlated ROC surfaces at a given pair of true classification rates

The receiver operating characteristic (ROC) curve is typically employed when one wants to evaluate the discriminatory capability of a continuous or ordinal biomarker in the case where two groups are to be distinguished, commonly the "healthy" and the "diseased." There are cases for which the disease status has three categories. Such cases employ the ROC surface, which is a natural generalization of the ROC curve for three classes. In this paper, we explore new methodologies for comparing two continuous biomarkers that refer to a trichotomous disease status, when both markers are applied to the same patients. Comparisons based on the volume under the surface have been proposed before, but that measure is often not clinically relevant. Here, we focus on comparing two correlated ROC surfaces at given pairs of true classification rates, which are more relevant to patients and physicians. We propose delta-based parametric techniques, power transformations to normality, and bootstrap-based smooth nonparametric techniques to investigate the performance of an appropriate test. We evaluate our approaches through an extensive simulation study and apply them to a real dataset from prostate cancer screening.