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

The first step in evaluating a potential diagnostic biomarker is to examine how its values vary across disease stages. In a three-class disease setting, the volume under the receiver operating characteristic surface (VUS) and the three-class Youden index are commonly used summary measures of a biomarker's discriminatory ability. However, the VUS in its classical form and the three-class Youden index are typically defined under a stochastic ordering of biomarker distributions across groups, which can be restrictive and, when violated, may lead to misleading conclusions about a biomarker's discriminatory ability. Moreover, even when a stochastic ordering exists, it may differ across biomarkers, complicating ranking in studies involving multiple biomarkers. To address these challenges, we propose the underlap coefficient (UNL), a summary index free from stochastic ordering assumptions and classification rules, to quantify a biomarker's ability to distinguish between three (or more) disease groups. To account for patient heterogeneity, we develop the covariate-specific UNL. We further introduce Bayesian non-parametric estimators for both the unconditional and covariate-specific UNLs. An extensive simulation study demonstrates good performance of the proposed estimators. We illustrate the methodology by assessing how four Alzheimer's disease biomarkers distinguish between individuals with normal cognition, mild impairment, and dementia, examining the impact of age and gender on discriminatory ability. Data used in the preparation of this paper were obtained from the Alzheimer's Disease Neuroimaging Initiative database.