Receiver operator characteristic (ROC) analysis has become a standard tool in the design and evaluation of two-class classification problems. It allows for an analysis that incorporates all possible priors, costs, and operating points, which is important in many real problems, where conditions are often nonideal. Extending this to the multiclass case is attractive, conferring the benefits of ROC analysis to a multitude of new problems. Even though the ROC analysis extends theoretically to the multiclass case, the exponential computational complexity as a function of the number of classes is restrictive. In this paper, we show that the multiclass ROC can often be simplified considerably because some ROC dimensions are independent of each other. We present an algorithm that analyzes interactions between various ROC dimensions, identifying independent classes, and groups of interacting classes, allowing the ROC to be decomposed. The resulting decomposed ROC hypersurface can be interrogated in a similar fashion to the ideal case, allowing for approaches such as cost-sensitive and Neyman-Pearson optimization, as well as the volume under the ROC. An extensive bouquet of examples and experiments demonstrates the potential of this methodology.