Direct estimation of volume under the ROC surface with verification bias.

In practice, the receiver operating characteristic (ROC) curve of a diagnostic test is widely used to show the performance of the test for discriminating two-class events. The area under the ROC curve (AUC) is proposed as an index for the assessment of the diagnostic accuracy of the test under consideration. Due to ethical and cost considerations associated with application of gold standard (GS) tests, only a subset of the patients initially tested have verified disease status. Statistical evaluation of the test performance based only on test results from subjects with verified disease status are typically biased. Various AUC estimation methods for tests with verification biased data have been developed over the last few decades. In this article, we develop new direct estimation methods for the volume under the ROC surface (VUS) by extending the AUC estimation methods for two-class diagnostic tests to three-class diagnostic tests in the presence of verification bias. The proposed methods will provide a comprehensive guide to deal with the verification bias in three-class diagnostic test accuracy studies and lead to a better choice of diagnostic tests.