Estimation of the volume under the ROC surface in presence of nonignorable verification bias

The volume under the receiver operating characteristic surface (VUS) is useful for measuring the overall accuracy of a diagnostic test when the possible disease status belongs to one of three ordered categories. In medical studies, the VUS of a new test is typically estimated through a sample of measurements obtained by some suitable sample of patients. However, in many cases, only a subset of such patients has the true disease status assessed by a gold standard test. In this paper, for a continuous-scale diagnostic test, we propose four estimators of the VUS which accommodate for nonignorable missingness of the disease status. The estimators are based on a parametric model which jointly describes both the disease and the verification process. Identifiability of the model is discussed. Consistency and asymptotic normality of the proposed estimators are shown, and variance estimation is discussed. The finite-sample behavior is investigated by means of simulation experiments. An illustration is provided.