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Optimality of a utility-based performance metric for ROC analysis

We previously introduced a utility-based ROC performance metric, the "surface-averaged expected cost" (SAEC), to address difficulties which arise in generalizing the well-known area under the ROC curve (AUC) to classification tasks with more than two classes. In a two-class classification task, the SAEC can be shown explicitly to be twice the area above the conventional ROC curve (1-AUC) divided by the arclength along the ROC curve. In the present work, we show that for a variety of two-class tasks under the binormal model, the SAEC obtained for the proper decision variable (the likelihood ratio of the latent decision variable) is less than that obtained for the conventional decision variable (i.e., using the latent decision variable directly). We also justify this result using a readily derived property of the arclength along the ROC curve under a given data model. Numerical studies as well as theoretical analysis suggest that the behavior of the SAEC is consistent with that of the AUC performance metric, in the sense that the optimal value of this quantity is achieved by the ideal observer.