Estimation of multiple ordered ROC curves using placement values

In many diagnostic accuracy studies, a priori orders may be available on multiple receiver operating characteristic curves. For example, being closer to delivery, fetal ultrasound measures in the third trimester should be no less accurate than those in the second trimester in predicting small-for-gestational-age births. Such an a priori order should be incorporated in estimating receiver operating characteristic curves and associated summary accuracy statistics, as it can potentially improve statistical efficiency of these estimates. Early work in the literature has mainly taken an indirect approach to this task and has induced the desired a priori order through modeling test score distributions. We instead propose a new strategy that incorporates the order directly through the modeling of receiver operating characteristic curves. We achieve this by exploiting the link between placement value (the relative position of a diseased test score in the healthy score distribution), the cumulative distribution function of placement value, and receiver operating characteristic curve, and by building stochastically ordered random variables through mixture distributions. We take a Bayesian semiparametric approach in using Dirichlet process mixture models so that the placement values can be flexibly modeled. We conduct extensive simulation studies to examine the performance of the proposed methodology and apply the new framework to data from obstetrics and women's health studies.