Lehmann Family of ROC Curves

Mithat Gonen, Memorial Sloan-Kettering Cancer Center
Glenn Heller, Memorial Sloan-Kettering Cancer Center

Abstract

Receiver operating characteristic (ROC) curves evaluate the discriminatory power of a continuous marker to predict a binary outcome. The most popular parametric model for an ROC curve is the binormal model, which assumes that the marker, after an unspecified monotone transformation, is normally distributed conditional on the outcome. Here we present an alternative to the binormal model based on the Lehmann family, also known as the proportional hazards specification. The resulting ROC curve and its functionals (such as the area under the curve) have simple analytic forms. Closed-form expressions for the functional estimates and their corresponding asymptotic variances are derived. This family accommodates the comparison of multiple markers, covariate adjustments and clustered data through a regression formulation. Evaluation of the underlying assumptions, model fitting and model selection can be performed using any off the shelf proportional hazards statistical software package.