On Kullback-Leibler divergence for medical diagnostics accuracy and cut-point selection criterion under tree or umbrella ordering

Diagnostic testing typically involves two types of classification. Binary tests separate individuals into diseased or non-diseased groups, while multi-class methods, like tree or umbrella ordering, compare one class's biomarker levels to those of other classes. Kullback-Leibler divergence (KL), which measures the difference between two distributions, has been considered a valuable index for assessing the diagnostic performance of biomarkers. In this work, we derive and propose the total rule-in and rule-out Kullback-Leibler divergence (TTKL(c)) as a measure of accuracy, obtained by dichotomizing a continuous biomarker, and as an optimization criterion for cut-off point selection under tree or umbrella ordering. We have established a connection between the proposed TTKL(c) measure and the extended Youden index, which is the most used criterion for cut-off point selection. Additionally, we present both theoretical and numerical derivations for scenarios involving a single cut-off point under extended tree ordering. Graphically, KL divergence is represented through the information graph. Using simulation methods, we conducted a power study to compare the performance of our proposed methods with the extended Youden index under tree ordering, as well as the accuracy of optimal cut-off selection. This analysis provides insights into the effectiveness of TTKL(c) as a robust criterion for selecting cut-off points in multi-class diagnostic settings. A comprehensive data analysis of lung cancer data illustrates the proposed applications.