  The Receiver Operating Characteristic (ROC) surface is a generalization of
ROC curve and is widely used for assessment of the accuracy of diagnostic tests
on three categories. A complication called the verification bias, meaning that
not all subjects have their true disease status verified often occur in real
application of ROC analysis. This is a common problem since the gold standard
test, which is used to generate true disease status, can be invasive and
expensive. In this paper, we will propose a Bayesian approach for estimating
the ROC surface based on continuous data under a semi-parametric trinormality
assumption. Our proposed method often adopted in ROC analysis can also be
extended to situation in the presence of verification bias. We compute the
posterior distribution of the parameters under trinormality assumption by using
a rank-based likelihood. Consistency of the posterior under mild conditions is
also established. We compare our method with the existing methods for
estimating ROC surface and conclude that our method performs well in terms of
accuracy.
