{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,30]],"date-time":"2026-05-30T14:02:34Z","timestamp":1780149754567,"version":"3.54.0"},"reference-count":21,"publisher":"Oxford University Press (OUP)","issue":"2","license":[{"start":{"date-parts":[[2026,4,9]],"date-time":"2026-04-09T00:00:00Z","timestamp":1775692800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"funder":[{"DOI":"10.13039\/501100000275","name":"Leverhulme Trust","doi-asserted-by":"publisher","award":["RF-2025-200\\9"],"award-info":[{"award-number":["RF-2025-200\\9"]}],"id":[{"id":"10.13039\/501100000275","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006129","name":"FCT","doi-asserted-by":"publisher","award":["UID\/4106\/2025"],"award-info":[{"award-number":["UID\/4106\/2025"]}],"id":[{"id":"10.13039\/100006129","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006129","name":"FCT","doi-asserted-by":"publisher","award":["UID\/PRR\/4106\/2025"],"award-info":[{"award-number":["UID\/PRR\/4106\/2025"]}],"id":[{"id":"10.13039\/100006129","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2026,4,9]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>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\u2019s 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\u2019s 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\u2019s 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\u2019s 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\u2019s Disease Neuroimaging Initiative database.<\/jats:p>","DOI":"10.1093\/biomtc\/ujag085","type":"journal-article","created":{"date-parts":[[2026,5,1]],"date-time":"2026-05-01T12:14:12Z","timestamp":1777637652000},"source":"Crossref","is-referenced-by-count":0,"title":["The underlap coefficient as a measure of a biomarker\u2019s discriminatory ability"],"prefix":"10.1093","volume":"82","author":[{"ORCID":"https:\/\/orcid.org\/0009-0005-4719-7116","authenticated-orcid":false,"given":"Zhaoxi","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Mathematics, University of Edinburgh , Edinburgh, EH9 3FD ,","place":["United Kingdom"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8084-1616","authenticated-orcid":false,"given":"Vanda","family":"In\u00e1cio","sequence":"additional","affiliation":[{"name":"School of Mathematics, University of Edinburgh , Edinburgh, EH9 3FD ,","place":["United Kingdom"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3248-6984","authenticated-orcid":false,"given":"Miguel","family":"de\u00a0Carvalho","sequence":"additional","affiliation":[{"name":"School of Mathematics, University of Edinburgh , Edinburgh, EH9 3FD ,","place":["United Kingdom"]},{"name":"CIDMA\u2014Department of Mathematics, Universidade de Aveiro , Aveiro, 3810-193 ,","place":["Portugal"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2026,5,30]]},"reference":[{"key":"2026053009442815000_bib1","doi-asserted-by":"crossref","first-page":"1767","DOI":"10.1002\/sim.8869","article-title":"The length of the receiver operating characteristic curve and the two cutoff Youden index within a robust framework for discovery, evaluation, and cutoff estimation in biomarker studies involving improper receiver operating characteristic curves","volume":"40","author":"Bantis","year":"2021","journal-title":"Statistics in Medicine"},{"key":"2026053009442815000_bib2","doi-asserted-by":"crossref","first-page":"837","DOI":"10.1177\/0962280219846157","article-title":"Affinity-based measures of biomarker performance evaluation","volume":"29","author":"de\u00a0Carvalho","year":"2020","journal-title":"Statistical Methods in Medical Research"},{"key":"2026053009442815000_bib3","doi-asserted-by":"crossref","first-page":"577","DOI":"10.1080\/01621459.1995.10476550","article-title":"Bayesian density estimation and inference using mixtures","volume":"90","author":"Escobar","year":"1995","journal-title":"Journal of the American Statistical Association"},{"key":"2026053009442815000_bib4","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1214\/aos\/1176342360","article-title":"A Bayesian analysis of some nonparametric problems","volume":"1","author":"Ferguson","year":"1973","journal-title":"The Annals of Statistics"},{"key":"2026053009442815000_bib5","doi-asserted-by":"crossref","first-page":"997","DOI":"10.1007\/s11222-013-9416-2","article-title":"Understanding predictive information criteria for Bayesian models","volume":"24","author":"Gelman","year":"2014","journal-title":"Statistics and Computing"},{"key":"2026053009442815000_bib6","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1198\/016214501750332758","article-title":"Gibbs sampling methods for stick-breaking priors","volume":"96","author":"Ishwaran","year":"2001","journal-title":"Journal of the American statistical Association"},{"key":"2026053009442815000_bib7","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1093\/aje\/kwn099","article-title":"Adjusting for covariates in studies of diagnostic, screening, or prognostic markers: an old concept in a new setting","volume":"168","author":"Janes","year":"2008","journal-title":"American Journal of Epidemiology"},{"key":"2026053009442815000_bib8","doi-asserted-by":"crossref","first-page":"566","DOI":"10.1093\/biostatistics\/kxm050","article-title":"ROC analysis with multiple classes and multiple tests: methodology and its application in microarray studies","volume":"9","author":"Li","year":"2008","journal-title":"Biostatistics (Oxford, England)"},{"key":"2026053009442815000_bib9","doi-asserted-by":"crossref","first-page":"5313","DOI":"10.1002\/sim.9912","article-title":"The diagnostic likelihood ratio function and modified test for trend: Identifying, evaluating, and validating nontraditional biomarkers in case-control studies","volume":"42","author":"Lindner","year":"2023","journal-title":"Statistics in Medicine"},{"key":"2026053009442815000_bib10","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1214\/aos\/1176346412","article-title":"On a class of Bayesian nonparametric estimates: I. 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