Two-sided tree ordering: providing diagnostic accuracy measures and optimal cut-off selection

In medical diagnostic testing, tests are typically categorized into two main types: binary, which differentiate between diseased and non-diseased cases, and ordinal, which classify non-diseased cases into stages from stage 1 to stage k. Beyond these, a multi-class classification scheme known as tree or umbrella ordering involves a classifier in which the biomarker measurement for one class is higher or lower than that for the other classes. We introduce a novel concept termed ‘two-sided tree ordering’ for disease states. This innovation extends the Receiver Operating Characteristic (ROC) curves and their summary measures to accommodate this classification type. We propose two approaches for this extended ordering: the direct and weighted approaches. Our method aims to enhance the current diagnostic categorization by considering the complexity of certain diseases that do not adhere to conventional classifications. To validate the effectiveness of our approach, we conducted simulation studies and demonstrated its application using real data on lung cancer. These findings underscore the potential of our methods to improve both the precision and understanding of medical diagnostic testing, especially for complex diseases.