Regression Models for Convex ROC Curves

Summary. The performance of a diagnostic test is summarized by its receiver operating characteristic (ROC) curve. Under quite natural assumptions about the latent variable underlying the test, the ROC curve is convex. Empirical data on a test's performance often comes in the form of observed true positive and false positive relative frequencies under varying conditions. This paper describes a family of regression models for analyzing such data. The underlying ROC curves are specified by a quality parameter μ and a shape parameter Δ and are guaranteed to be convex provided Δ > 1. Both the position along the ROC curve and the quality parameter Δ are modeled linearly with covariates at the level of the individual. The shape parameter μ enters the model through the link functions log(pμ) ‐ log(1 ‐ pμ) of a binomial regression and is estimated either by search or from an appropriate constructed variate. One simple application is to the meta‐analysis of independent studies of the same diagnostic test, illustrated on some data of Moses, Shapiro, and Littenberg (1993). A second application, to so‐called vigilance data, is given, where ROC curves differ across subjects and modeling of the position along the ROC curve is of primary interest.