Different view of the diagnostics test accuracy measures and optimal cut-off point selection procedure under tree or umbrella ordering

In the realm of medical diagnostic testing, diagnostic tests can assume either binary forms, distinguishing between diseased and non-diseased states, or ordinal forms, categorizing states from non-diseased to various stages (1 to K). Another significant classification scheme for multi-class scenarios is tree or umbrella ordering, which entails several unordered sub-classes (subtypes) within a biomarker. Within tree or umbrella ordering, the classifier assesses whether the marker measurement for one class surpasses or falls below those for the other classes. Although Receiver Operating Characteristic (ROC) curves and summary measures have been adapted to accommodate tree and umbrella ordering, these approaches often yield cut-off points that generate highly sensitive tests for certain disease subtypes while compromising specificity for others. This may not be ideal for all diseases. Hence, in this investigation, we explore diverse measures of diagnostic test accuracy and optimal cut-off point selection procedures under tree or umbrella ordering to foster more specific tests. We present numerical examples and simulation studies and demonstrate the approach using real data on lung cancer.