The underlap coefficient as a measure of a biomarker's discriminatory ability

The first step in evaluating a potential diagnostic biomarker is to examine the variation in its values across different disease groups. In a three-class disease setting, the volume under the receiver operating characteristic surface and the three-class Youden index are commonly used summary measures of a biomarker's discriminatory ability. However, these measures rely on a stochastic ordering assumption for the distributions of biomarker outcomes across the three groups. This assumption can be restrictive, particularly when covariates are involved, and its violation may lead to incorrect conclusions about a biomarker's ability to distinguish between the three disease classes. Even when a stochastic ordering exists, the order may vary across different biomarkers in discovery studies involving dozens or even thousands of candidate biomarkers, complicating automated ranking. To address these challenges and complement existing measures, we propose the underlap coefficient, a novel summary index of a biomarker's ability to distinguish between three (or more) disease groups, and study its properties. Additionally, we introduce Bayesian nonparametric estimators for both the unconditional underlap coefficient and its covariate-specific counterpart. These estimators are broadly applicable to a wide range of biomarkers and populations. A simulation study reveals a good performance of the proposed estimators across a range of conceivable scenarios. We illustrate the proposed approach through an application to an Alzheimer's disease (AD) dataset aimed to assess how four potential AD biomarkers distinguish between individuals with normal cognition, mild impairment, and dementia, and how and if age and gender impact this discriminatory ability.