An Analytic Expression for the Binormal Partial Area under the ROC Curve

Abstract

The partial area under the receiver operating characteristic (ROC) curve (pAUC) is a useful summary measure for diagnostic studies. Unlike most summary measures that are functions of the ROC curve, researchers have not been aware of an analytic expression available for computing the pAUC for an ROC curve based on a latent binormal model. Instead, the pAUC has been computed using numerical integration or a rational polynomial approximation. Our purpose is to provide analytic expressions for the two forms of pAUC. We discuss the two fundamentally different types of pAUC. We present analytic expressions for both types, provide corresponding proofs, and illustrate their application with an example comparing the performances of spin echo and cine magnetic resonance imaging for detecting thoracic aortic dissection. We provide an example of using the pAUC as the outcome in a multireader multicase analysis. We find that using the pAUC results in a more significant finding. We have provided analytic expressions for both types of pAUC, making it easier to compute the pAUCs corresponding to binormal ROC curves.