A New Evaluation Strategy for Diagnostic Tests Under Umbrella or Tree Ordering

In medical research, diagnostic tests (or biomarkers) with continuous values are widely employed to attempt to distinguish between diseased and non-diseased subjects. The diagnostic accuracy of a test can be assessed by its receiver operating characteristic (ROC) curve. For diseases with multiclasses, an important category of scenarios assumes tree or umbrella ordering, where the test measurement for one particular class is lower or higher than those for the other classes. In this paper, we propose a new ROC framework for tree or umbrella ordering, together with a related evaluation strategy. Such a strategy is based on new ROC representations on the plane, denoted as LTROC and UTROC, and new summary indexes. Related statistical inference is also discussed. In particular, we propose simple estimation and interval estimation procedures, in a nonparametric setting. For these procedures, we provide theoretical justification and assess the behaviour in finite samples through simulation experiments. Finally, we illustrate the proposed approach with two real data examples.