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A Method of Estimating and Comparing Volumes Under Receiver Operating Characteristic (ROC) Surfaces

Receiver operating characteristic (ROC) analysis has been widely used to evaluate the performance of diagnostic tests with dichotomous outcomes. Recently the ROC methodology has been extended to three- and multiple-group classification problems. The methods typically involve construction of an ROC surface and calculation of the volume under the surface (VUS). The volume, which measures the discriminatory accuracy of a test, can be used for statistical inference. While theory and computational methods of the ROC surface analysis have been well studied and developed for some multi-class problems, very little research has been done on a classification problem commonly seen in DNA amplification assays that use an internal control. In this article, we propose using the maximum likelihood estimator (MLE) to estimate the VUS for the aforementioned diagnostic problem. We show that the MLE is a consistent estimator, having an asymptotic normal distribution. A numerical method based on Monte Carlo simulation is proposed for computing the MLE. The performance of the MLE is also evaluated using simulations. The theoretical findings aided with the numerical tool allows us to effectively evaluate diagnostic tests. The use of the method is illustrated through an example.