{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,17]],"date-time":"2026-10-17T20:26:21Z","timestamp":1792268781901,"version":"4.4.0"},"reference-count":28,"publisher":"Oxford University Press (OUP)","issue":"1","license":[{"start":{"date-parts":[[2020,8,4]],"date-time":"2020-08-04T00:00:00Z","timestamp":1596499200000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2021,3,2]]},"abstract":"<jats:title>Summary<\/jats:title><jats:p>Penalization of the likelihood by Jeffreys\u2019 invariant prior, or a positive power thereof, is shown to produce finite-valued maximum penalized likelihood estimates in a broad class of binomial generalized linear models. The class of models includes logistic regression, where the Jeffreys-prior penalty is known additionally to reduce the asymptotic bias of the maximum likelihood estimator, and models with other commonly used link functions, such as probit and log-log. Shrinkage towards equiprobability across observations, relative to the maximum likelihood estimator, is established theoretically and studied through illustrative examples. Some implications of finiteness and shrinkage for inference are discussed, particularly when inference is based on Wald-type procedures. A widely applicable procedure is developed for computation of maximum penalized likelihood estimates, by using repeated maximum likelihood fits with iteratively adjusted binomial responses and totals. These theoretical results and methods underpin the increasingly widespread use of reduced-bias and similarly penalized binomial regression models in many applied fields.<\/jats:p>","DOI":"10.1093\/biomet\/asaa052","type":"journal-article","created":{"date-parts":[[2020,6,19]],"date-time":"2020-06-19T11:11:24Z","timestamp":1592565084000},"page":"71-82","source":"Crossref","is-referenced-by-count":201,"title":["Jeffreys-prior penalty, finiteness and shrinkage in binomial-response generalized linear models"],"prefix":"10.1093","volume":"108","author":[{"given":"Ioannis","family":"Kosmidis","sequence":"first","affiliation":[{"name":"Department of Statistics, University of Warwick, Coventry CV4 7AL, U.K"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"David","family":"Firth","sequence":"additional","affiliation":[{"name":"Department of Statistics, University of Warwick, Coventry CV4 7AL, U.K"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2020,8,4]]},"reference":[{"key":"2021030708334478200_B1","volume-title":"Categorical Data Analysis","author":"Agresti,","year":"2013"},{"key":"2021030708334478200_B2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1093\/biomet\/71.1.1","article-title":"On the existence of maximum likelihood estimates in logistic regression models","volume":"71","author":"Albert,","year":"1984","journal-title":"Biometrika"},{"key":"2021030708334478200_B3","first-page":"324","article-title":"Rank analysis of incomplete block designs: I. The method of paired comparisons","volume":"39","author":"Bradley,","year":"1952","journal-title":"Biometrika"},{"key":"2021030708334478200_B4","doi-asserted-by":"crossref","first-page":"903","DOI":"10.1002\/sim.2518","article-title":"Confidence intervals for multinomial logistic regression in sparse data","volume":"26","author":"Bull,","year":"2007","journal-title":"Statist. Med."},{"key":"2021030708334478200_B5","doi-asserted-by":"crossref","first-page":"1659","DOI":"10.1198\/016214508000000779","article-title":"Properties and implementation of Jeffreys\u2019s prior in binomial regression models","volume":"103","author":"Chen,","year":"2008","journal-title":"J. Am. Statist. Assoc."},{"key":"2021030708334478200_B6","doi-asserted-by":"crossref","first-page":"629","DOI":"10.1111\/j.2517-6161.1991.tb01852.x","article-title":"Bias correction in generalized linear models","volume":"B 53","author":"Cordeiro,","year":"1991","journal-title":"J. R. Statist. Soc."},{"key":"2021030708334478200_B7","doi-asserted-by":"crossref","first-page":"542","DOI":"10.1007\/s10985-015-9321-4","article-title":"Penalised logistic regression and dynamic prediction for discrete-time recurrent event data","volume":"21","author":"Elgmati,","year":"2015","journal-title":"Lifetime Data Anal."},{"key":"2021030708334478200_B8","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1007\/978-1-4612-2952-0_15","volume-title":"Advances in GLIM and Statistical Modelling: Proceedings of the GLIM 92 Conference, Munich","author":"Firth,","year":"1992"},{"key":"2021030708334478200_B9","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1093\/biomet\/80.1.27","article-title":"Bias reduction of maximum likelihood estimates","volume":"80","author":"Firth,","year":"1993","journal-title":"Biometrika"},{"key":"2021030708334478200_B10","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1111\/j.2517-6161.1984.tb01288.x","article-title":"Iteratively reweighted least squares for maximum likelihood estimation, and some robust and resistant alternatives","volume":"B 46","author":"Green,","year":"1984","journal-title":"J. R. Statist. Soc."},{"key":"2021030708334478200_B11","author":"Heinze,","year":"2018","journal-title":"logistf: Firth\u2019s Bias-Reduced Logistic Regression"},{"key":"2021030708334478200_B12","doi-asserted-by":"crossref","first-page":"2409","DOI":"10.1002\/sim.1047","article-title":"A solution to the problem of separation in logistic regression","volume":"21","author":"Heinze,","year":"2002","journal-title":"Statist. Med."},{"key":"2021030708334478200_B13","volume-title":"Applied Logistic Regression","author":"Hosmer,"},{"key":"2021030708334478200_B14","doi-asserted-by":"crossref","first-page":"981","DOI":"10.1080\/01621459.1991.10475141","article-title":"On Bayesian analysis of generalized linear models using Jeffreys\u2019s prior","volume":"86","author":"Ibrahim,","year":"1991","journal-title":"J. Am. Statist. Assoc."},{"key":"2021030708334478200_B15","doi-asserted-by":"crossref","first-page":"923","DOI":"10.1093\/biomet\/asx046","article-title":"Median bias reduction of maximum likelihood estimates","volume":"104","author":"Kenne Pagui,","year":"2017","journal-title":"Biometrika"},{"key":"2021030708334478200_B16","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1111\/rssb.12025","article-title":"Improved estimation in cumulative link models","volume":"B 76","author":"Kosmidis,","year":"2014","journal-title":"J. R. Statist. Soc."},{"key":"2021030708334478200_B17","author":"Kosmidis,","year":"2020","journal-title":"brglm2: Bias Reduction in Generalized Linear Models"},{"key":"2021030708334478200_B18","doi-asserted-by":"crossref","first-page":"1097","DOI":"10.1214\/10-EJS579","article-title":"A generic algorithm for reducing bias in parametric estimation","volume":"4","author":"Kosmidis,","year":"2010","journal-title":"Electron. J. Statist."},{"key":"2021030708334478200_B19","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1093\/biomet\/asx079","article-title":"On bias reduction and incidental parameters","volume":"105","author":"Lunardon,","year":"2018","journal-title":"Biometrika"},{"key":"2021030708334478200_B20","volume-title":"Matrix Differential Calculus with Applications in Statistics and Econometrics","author":"Magnus,","year":"1999"},{"key":"2021030708334478200_B21","doi-asserted-by":"crossref","first-page":"864","DOI":"10.1093\/aje\/kwx299","article-title":"Separation in logistic regression: Causes, consequences, and control","volume":"187","author":"Mansournia,","year":"2018","journal-title":"Am. J. Epidemiol."},{"key":"2021030708334478200_B22","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1080\/01621459.1981.10477613","article-title":"Concavity of the log likelihood","volume":"76","author":"Pratt,","year":"1981","journal-title":"J. Am. Statist. Assoc."},{"key":"2021030708334478200_B23","doi-asserted-by":"crossref","first-page":"2302","DOI":"10.1002\/sim.7273","article-title":"Firth\u2019s logistic regression with rare events: Accurate effect estimates and predictions?","volume":"36","author":"Puhr,","year":"2017","journal-title":"Statist. Med."},{"key":"2021030708334478200_B24","volume-title":"R: A Language and Environment for Statistical Computing","year":"2021"},{"key":"2021030708334478200_B25","doi-asserted-by":"crossref","first-page":"533","DOI":"10.1093\/biomet\/90.3.533","article-title":"Modified profile likelihoods in models with stratum nuisance parameters","volume":"90","author":"Sartori,","year":"2003","journal-title":"Biometrika"},{"key":"2021030708334478200_B26","doi-asserted-by":"crossref","first-page":"310","DOI":"10.1111\/j.2517-6161.1981.tb01676.x","article-title":"On the existence of maximum likelihood estimators for the binomial response models","volume":"B 43","author":"Silvapulle,","year":"1981","journal-title":"J. R. Statist. Soc."},{"key":"2021030708334478200_B27","doi-asserted-by":"crossref","first-page":"14516","DOI":"10.1073\/pnas.1810420116","article-title":"A modern maximum-likelihood theory for high-dimensional logistic regression","volume":"116","author":"Sur,","year":"2019","journal-title":"Proc. Nat. Acad. Sci."},{"key":"2021030708334478200_B28","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1093\/pan\/mpi009","article-title":"A solution to separation in binary response models","volume":"13","author":"Zorn,","year":"2005","journal-title":"Polit. Anal."}],"container-title":["Biometrika"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/academic.oup.com\/biomet\/advance-article-pdf\/doi\/10.1093\/biomet\/asaa052\/33860509\/asaa052.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"http:\/\/academic.oup.com\/biomet\/article-pdf\/108\/1\/71\/36441114\/asaa052.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"http:\/\/academic.oup.com\/biomet\/article-pdf\/108\/1\/71\/36441114\/asaa052.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,8]],"date-time":"2024-08-08T00:34:28Z","timestamp":1723077268000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/biomet\/article\/108\/1\/71\/5880219"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,8,4]]},"references-count":28,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,8,4]]},"published-print":{"date-parts":[[2021,3,2]]}},"URL":"https:\/\/doi.org\/10.1093\/biomet\/asaa052","relation":{},"ISSN":["0006-3444","1464-3510"],"issn-type":[{"value":"0006-3444","type":"print"},{"value":"1464-3510","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2021,3,1]]},"published":{"date-parts":[[2020,8,4]]}}}