ROC curves and the generalization to multiple classes

The present work focuses on the study and extension of ROC analysis methodology for multiple-class classification problems. In clinical medical research, the need for developing an approach to measure the diagnostic accuracy of biomedical tests in classifying the true status of a patient is a critical point when doing both diagnosis and prognosis. In a two-category classification setting, the ROC analysis is the natural approach and the Area Under the Curve (AUC) is a summary measure of the diagnostic accuracy. However, many real classification problems rely to more than two classes; thus, the ROC manifold generalization of curve and the hypervolume (HUM) generalization of area recently appeared in the literature to address classification problems with more than two classes. Motivated by a real research question arose during a four-class classification study for early detection of colorectal cancer, we review the literature on ROC analysis and on its extension to multiple classes. Then, we develop a new estimator of the accuracy measure of a diagnostic marker. We derive the analytical form of the HUM estimator and the analytical representation of its variance. To assess the performance of the proposed estimator and compare it with the two alternatives existing in the literature, we perform simulation exercises and empirical applications. The first application deals with the topic that initially moved our interest, the early detection of colorectal cancer patients; the second concerns the classification of synovial tissue inflammatory cells, a typical case study in the biostatistics literature. Finally, in the last part of our work, we suggest a statistical method to combine multiple tests for multicategory classification. The novelty of our approach is the use of the classification accuracy (HUM) of the combined marker as the objective function to be maximized. The methodology is evaluated trough a simulation study and two empirical applications.