Joint modeling of sensitivity and specificity.

Sensitivity and specificity are two customary performance measures associated with medical diagnostic tests. Typically, they are modeled independently as a function of risk factors using logistic regression, which provides estimated functions for these probabilities. Change in these probabilities across levels of risk factors is of primary interest and the indirect relationship is often displayed using a receiver operating characteristic curve. We refer to this as analysis of 'first-order' behavior. Here, we consider what we refer to as 'second-order' behavior where we examine the stochastic dependence between the (random) estimates of sensitivity and specificity. To do so, we argue that a model for the four cell probabilities that determine the joint distribution of screening test result and outcome result is needed. Such a modeling induces sensitivity and specificity as functions of these cell probabilities. In turn, this raises the issue of a coherent specification for these cell probabilities, given risk factors, i.e. a specification that ensures that all probabilities calculated under it fall between 0 and 1. This leads to the question of how to provide models that are coherent and mechanistically appropriate as well as computationally feasible to fit, particularly with large data sets. The goal of this article is to illuminate these issues both algebraically and through analysis of a real data set.