Optimal threshold selection methods under tree or umbrella ordering

Receiver operating characteristic (ROC) curve is a popular tool for evaluating diagnostic accuracy of biomarkers. In ROC framework, there exist several optimal threshold selection methods for binary classification. For diseases with multi-classes, an important category of scenarios is tree or umbrella ordering in which the marker measurement for one particular class is lower or higher than those for the rest classes. Tree or umbrella ordering has important clinical applications, especially in the molecular diagnostics of cancer subtypes. The ROC curve has been extended to a typical ROC framework for tree or umbrella ordering (denoted as TROC). In this paper, we investigate several methods for optimal threshold selection under tree or umbrella ordering. Simulation studies are carried out to explore the performance of these threshold selection methods. A real microarray data set on lung cancer is analyzed using the proposed methods.