{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,11]],"date-time":"2026-10-11T04:52:16Z","timestamp":1791694336788,"version":"4.3.4"},"reference-count":32,"publisher":"Oxford University Press (OUP)","issue":"2","license":[{"start":{"date-parts":[[2023,3,22]],"date-time":"2023-03-22T00:00:00Z","timestamp":1679443200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/pages\/standard-publication-reuse-rights"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2023,5,4]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We formulate the estimation of monotone response surface of multiple factors as the inverse of an iteration of partially ordered classifier ensembles. Each ensemble (called product-of-independent-probability-escalation (PIPE)-classifiers) is a projection of Bayes classifiers on the constrained space. We prove that the inverse of PIPE-classifiers (iPIPE) exists, and propose algorithms to efficiently compute iPIPE by reducing the space over which optimisation is conducted. The methods are applied in analysis and simulation settings where the surface dimension is higher than what the isotonic regression literature typically considers. Simulation shows that iPIPE-based credible intervals achieve nominal coverage probability and are more precise compared to unconstrained estimation.<\/jats:p>","DOI":"10.1093\/jrsssb\/qkad014","type":"journal-article","created":{"date-parts":[[2023,3,22]],"date-time":"2023-03-22T11:50:19Z","timestamp":1679485819000},"page":"497-522","source":"Crossref","is-referenced-by-count":9,"title":["Monotone response surface of multi-factor condition: estimation and Bayes classifiers"],"prefix":"10.1093","volume":"85","author":[{"given":"Ying Kuen","family":"Cheung","sequence":"first","affiliation":[{"name":"Department of Biostatistics, Columbia University , New York, NY , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Keith M","family":"Diaz","sequence":"additional","affiliation":[{"name":"Department of Medicine, Columbia University , New York, NY , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2023,3,22]]},"reference":[{"issue":"4","key":"2023050420355383200_qkad014-B1","doi-asserted-by":"crossref","first-page":"641","DOI":"10.1214\/aoms\/1177728423","article-title":"An empirical distribution function for sampling with incomplete information","volume":"26","author":"Ayer","year":"1955","journal-title":"The Annals of Mathematical Statistics"},{"issue":"405","key":"2023050420355383200_qkad014-B2","first-page":"289","article-title":"Additive isotonic regression","volume":"84","author":"Bacchetti","year":"1989","journal-title":"Journal of the American Statistical Association"},{"issue":"4","key":"2023050420355383200_qkad014-B3","doi-asserted-by":"crossref","first-page":"825","DOI":"10.1111\/j.1467-9868.2008.00663.x","article-title":"Gaussian predictive process models for large spatial data sets","volume":"70","author":"Banerjee","year":"2008","journal-title":"Journal of the Royal Statistical Society: Series B (Statistical Methodology)"},{"key":"2023050420355383200_qkad014-B4","volume-title":"Statistical inference under order restrictions","author":"Barlow","year":"1972"},{"issue":"3","key":"2023050420355383200_qkad014-B5","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1093\/biostatistics\/kxq001","article-title":"Stochastically ordered multiple regression","volume":"11","author":"Bornkamp","year":"2010","journal-title":"Biostatistics"},{"issue":"4","key":"2023050420355383200_qkad014-B6","doi-asserted-by":"crossref","first-page":"607","DOI":"10.1214\/aoms\/1177728420","article-title":"Maximum likelihood estimates of monotone parameters","volume":"26","author":"Brunk","year":"1955","journal-title":"The Annals of Mathematical Statistics"},{"issue":"1","key":"2023050420355383200_qkad014-B7","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1111\/j.2517-6161.1981.tb01150.x","article-title":"Empirical Bayes analysis of survival time data","volume":"43","author":"Burridge","year":"1981","journal-title":"Journal of the Royal Statistical Society. 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