Multivariate mixtures of Polya trees for modeling ROC data

Receiver operating characteristic (ROC) curves provide a graphical measure of diagnostic test accuracy. Because ROC curves are determined using the distributions of diagnostic test outcomes for noninfected and infected populations, there is an increasing trend to develop flexible models for these component distributions. We present methodology for joint nonparametric estimation of several ROC curves from multivariate serologic data. We develop an empirical Bayes approach that allows for arbitrary noninfected and infected component distributions that are modelled using Bayesian multivariate mixtures of finite Polya trees priors. Robust, data-driven inferences for ROC curves and the area under the curve are obtained, and a straightforward method for testing a Dirichlet process versus a more general Polya tree model is presented. Computational challenges can arise when using Polya trees to model large multivariate data sets that exhibit clustering. We discuss and implement practical procedures for addressing these obstacles, which are applied to bivariate data used to evaluate the performances of two ELISA tests for detection of Johne's disease.