Regression models for convex ROC curves

Abstract

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 Δ and a shape parameter μ 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(p(μ)) - log(1 - p(μ)) 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.