Bayesian analysis of a ROC curve for categorical data using a skew-binormal model

In a taste-testing experiment, foods are withdrawn from storage at various times and a panel of tasters is asked to rate the foods on a nine-point hedonic scale. We provide a statistical procedure that can assess the difference between fresh foods and foods withdrawn a few months later. Thus, we have two sets of ordinal data, one for the fresh foods and the other for the stored foods that are withdrawn later. A natural and popular way to compare two withdrawals is to use the receiver operating characteristic (ROC) curve and the area under the curve (AUC). It is a standard practice to use a binormal model to obtain the ROC curve and the AUC, and Bayesian methods have been used. One drawback of the binormal model is that it has non-identifiable parameters. First, we look more carefully at non-identifiability, and we use robust measures to obtain a more non-parametric analysis of the Bayesian binormal model. Second, in a more innovative approach we extend the robust binormal model to a skew-binormal model. Like recent approaches to ROC curve analysis, we also incorporate a stochastic ordering. We use the Gibbs sampler to fit both models in order to estimate the ROC curves and the AUCs. Using both models, these AUCs demonstrate that there is not much practical difference between fresh foods and those withdrawn later, but there are some differences in inference between the two models. We have also shown, using marginal likelihoods, that the skew-binormal model may be better. A small simulation study shows that the skew-binormal model provides improved precision over the binormal model with similar AUCs but somewhat different ROC curves.