Zhu, R, Ghosal, S. Bayesian nonparametric estimation of ROC surface under verification bias. Statistics in Medicine. 2019; 38: 3361– 3377. doi: 10.1002/sim.8181

The Receiver Operating Characteristic (ROC) surface, as a generalization of the ROC curve, has been widely used to assess the accuracy of a diagnostic test for three categories. A common problem is verification bias, referring to the situation where not all subjects have their true classes verified. This paper considers the problem of estimating the ROC surface under verification bias.
A Bayesian nonparametric approach is adopted by directly modeling the underlying distributions of the three categories by Dirichlet Process mixture priors. The proposed algorithm is robust since it only imposes a missing at random assumption for the verification process but no assumption on the distributions. The method can accommodate covariates information as well in estimating the ROC surface, which can lead to a more comprehensive understanding of the diagnostic accuracy. It can be adapted and hugely simplified to the case where there is no verification bias, in which case fast computation is possible through the Bayesian bootstrap process.
The proposed method is compared with other commonly used methods by extensive simulations. The proposed method generally outperforms other approaches as suggested by simulations. Applying the method to two real datasets, the key findings are as follows: (i) HE4 has a slightly better diagnosis ability comparing to CA125 in discriminating healthy, early stage and late stage patients of epithelial ovarian cancer. (ii) Serum albumin has a prognostic ability in distinguishing different stages of hepatocellular carcinoma.