Cutoff estimation and construction of their confidence intervals for continuous biomarkers under ternary umbrella and tree stochastic ordering settings.

Tuberculosis (TB) studies often involve four different states under consideration, namely, "healthy," "latent infection," "pulmonary active disease," and "extra-pulmonary active disease." While highly accurate clinical diagnosis tests do exist, they are expensive and generally not accessible in regions where they are most needed; thus, there is an interest in assessing the accuracy of new and easily obtainable biomarkers. For some such biomarkers, the typical stochastic ordering assumption might not be justified for all disease classes under study, and usual ROC methodologies that involve ROC surfaces and hypersurfaces are inadequate. Different types of orderings may be appropriate depending on the setting, and these may involve a number of ambiguously ordered groups that stochastically exhibit larger (or lower) marker scores than the remaining groups. Recently, there has been scientific interest on ROC methods that can accommodate these so-called "tree" or "umbrella" orderings. However, there is limited work discussing the estimation of cutoffs in such settings. In this article, we discuss the estimation and inference around optimized cutoffs when accounting for such configurations. We explore different cutoff alternatives and provide parametric, flexible parametric, and non-parametric kernel-based approaches for estimation and inference. We evaluate our approaches using simulations and illustrate them through a real data set that involves TB patients.