Construction of joint confidence spaces for the optimal true class fraction triplet in the ROC space using alternative biomarker cutoffs.

Hepatocellular carcinoma (HCC) is the most common primary cancer of the liver. Finding new biomarkers for its early detection is of high clinical importance. As with many other diseases, cancer has a progressive nature. In cancer biomarker studies, it is often the case that the true disease status of the recruited individuals exhibits more than two classes. The receiver operating characteristic (ROC) surface is a well-known statistical tool for assessing the biomarkers' discriminatory ability in trichotomous settings. The volume under the ROC surface (VUS) is an overall measure of the discriminatory ability of a marker. In practice, clinicians are often in need of cutoffs for decision-making purposes. A popular approach for computing such cutoffs is the Youden index and its recent three-class generalization. A drawback of such a method is that it treats the data in a pairwise fashion rather than consider all the data simultaneously. The use of the minimized Euclidean distance from the perfection corner to the ROC surface (also known as closest to perfection method) is an alternative to the Youden index that may be preferable in some settings. When such a method is employed, there is a need for inferences around the resulting true class rates/fractions that correspond to the optimal operating point. In this paper, we provide an inferential framework for the derivation of marginal confidence intervals (CIs) and joint confidence spaces (CSs) around the corresponding true class fractions, when dealing with trichotomous settings. We explore parametric and nonparametric approaches for the construction of such CIs and CSs. We evaluate our approaches through extensive simulations and apply them to a real data set that refers to HCC patients.