{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,7]],"date-time":"2026-02-07T23:41:57Z","timestamp":1770507717161,"version":"3.49.0"},"reference-count":42,"publisher":"Oxford University Press (OUP)","issue":"1","license":[{"start":{"date-parts":[[2016,3,20]],"date-time":"2016-03-20T00:00:00Z","timestamp":1458432000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2017,1,1]]},"abstract":"<jats:title>Summary<\/jats:title><jats:p>Hierarchical models allow for heterogeneous behaviours in a population while simultaneously borrowing estimation strength across all subpopulations. Unfortunately, existing likelihood-based methods for fitting hierarchical models have high computational demands, and these demands have limited their adoption in large-scale prediction and inference problems. The paper proposes a moment-based procedure for estimating the parameters of a hierarchical model which has its roots in a method originally introduced by Cochran in 1937. The method trades statistical efficiency for computational efficiency. It gives consistent parameter estimates, competitive prediction error performance and substantial computational improvements. When applied to a large-scale recommender system application and compared with a standard maximum likelihood procedure, the method delivers competitive prediction performance while reducing the sequential computation time from hours to minutes.<\/jats:p>","DOI":"10.1111\/rssb.12165","type":"journal-article","created":{"date-parts":[[2016,3,21]],"date-time":"2016-03-21T04:44:41Z","timestamp":1458535481000},"page":"267-291","source":"Crossref","is-referenced-by-count":27,"title":["Fast Moment-Based Estimation for Hierarchical Models"],"prefix":"10.1093","volume":"79","author":[{"given":"Patrick O.","family":"Perry","sequence":"first","affiliation":[{"name":"New York University \u00a0 USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"286","published-online":{"date-parts":[[2016,3,20]]},"reference":[{"key":"2023021709334711800_rssb12165-cit-0001","doi-asserted-by":"crossref","first-page":"734","DOI":"10.1109\/TKDE.2005.99","article-title":"Toward the next generation of recommender systems: a survey of the state-of-the-art and possible extensions","volume":"17","author":"Adomavicius","year":"2005","journal-title":"IEEE Trans. Knowl. Data Enng"},{"key":"2023021709334711800_rssb12165-cit-0002","doi-asserted-by":"crossref","DOI":"10.1002\/9780470315262.ch1","article-title":"Statistical challenges in internet advertising","author":"Agarwal","year":"2008","journal-title":"Statistical Methods in e-commerce Research"},{"key":"2023021709334711800_rssb12165-cit-0003","first-page":"19","volume-title":"Proc. Conf. Association for Computing Machinery Special Interest Group on Knowledge Discovery and Data Mining, Paris","author":"Agarwal","year":"2009"},{"key":"2023021709334711800_rssb12165-cit-0004","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1509\/jmkr.37.3.363.18779","article-title":"Internet recommendations systems","volume":"37","author":"Ansari","year":"2000","journal-title":"J. Marktng Res."},{"key":"2023021709334711800_rssb12165-cit-0005","doi-asserted-by":"crossref","first-page":"1056","DOI":"10.1016\/j.spl.2011.02.029","article-title":"Sparse variational analysis of linear mixed models for large data sets","volume":"81","author":"Armagan","year":"2011","journal-title":"Statist. Probab. Lett."},{"key":"2023021709334711800_rssb12165-cit-0006","volume-title":"lme4: linear mixed-effects models using Eigen and S4","author":"Bates","year":"2013"},{"key":"2023021709334711800_rssb12165-cit-0007","doi-asserted-by":"crossref","first-page":"2507","DOI":"10.1080\/03610928608829265","article-title":"Large sample inference in random coefficient regression models","volume":"15","author":"Carter","year":"1986","journal-title":"Communs Statist. Theor. Meth."},{"key":"2023021709334711800_rssb12165-cit-0008","doi-asserted-by":"crossref","first-page":"102","DOI":"10.2307\/2984123","article-title":"Problems arising in the analysis of a series of similar experiments","volume":"4","author":"Cochran","year":"1937","journal-title":"J. R. Statist. Soc., suppl."},{"key":"2023021709334711800_rssb12165-cit-0009","doi-asserted-by":"crossref","first-page":"101","DOI":"10.2307\/3001666","article-title":"The combination of estimates from different experiments","volume":"10","author":"Cochran","year":"1954","journal-title":"Biometrics"},{"key":"2023021709334711800_rssb12165-cit-0010","volume-title":"Proc. Association for Computing Machinery Special Interest Group on Information Retrieval Wrkshp Recommender Systems: Algorithms and Evaluation","author":"Condliff","year":"1999"},{"key":"2023021709334711800_rssb12165-cit-0011","doi-asserted-by":"crossref","DOI":"10.1201\/9781482285949","volume-title":"Components of Variance","author":"Cox","year":"2002"},{"key":"2023021709334711800_rssb12165-cit-0012","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1080\/01621459.1981.10477653","article-title":"Estimation in covariance components models","volume":"76","author":"Dempster","year":"1981","journal-title":"J. Am. Statist. Ass."},{"key":"2023021709334711800_rssb12165-cit-0013","first-page":"165","volume-title":"Proc. 5th Conf. Recommender Systems","author":"Dror","year":"2011"},{"key":"2023021709334711800_rssb12165-cit-0014","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":"2023021709334711800_rssb12165-cit-0015","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/1471-2288-12-163","article-title":"Fitting parametric random effects models in very large data sets with application to VHA national data","volume":"12","author":"Gebregziabher","year":"2012","journal-title":"BMC Med. Res. Methodol."},{"key":"2023021709334711800_rssb12165-cit-0016","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1093\/biomet\/73.1.43","article-title":"Multilevel mixed linear model analysis using iterative generalized least squares","volume":"73","author":"Goldstein","year":"1986","journal-title":"Biometrika"},{"key":"2023021709334711800_rssb12165-cit-0017","volume-title":"MovieLens 10M Dataset","author":"GroupLens","year":"2009"},{"key":"2023021709334711800_rssb12165-cit-0018","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":"2023021709334711800_rssb12165-cit-0019","doi-asserted-by":"crossref","DOI":"10.2139\/ssrn.1010107","volume-title":"Sampling for Bayesian computation with large datasets","author":"Huang","year":"2005"},{"key":"2023021709334711800_rssb12165-cit-0501","volume-title":"Programming languages\u2014C. Standard ISO\/IEC 9899:1999","author":"International Standards Organization","year":"1999"},{"key":"2023021709334711800_rssb12165-cit-0502","volume-title":"Programming languages\u2014C++. Standard ISO\/IEC 14882:2003","author":"International Standards Organization","year":"2003"},{"key":"2023021709334711800_rssb12165-cit-0020","doi-asserted-by":"crossref","first-page":"805","DOI":"10.2307\/2530695","article-title":"Unbalanced repeated-measures models with structured covariance matrices","volume":"42","author":"Jennrich","year":"1986","journal-title":"Biometrics"},{"key":"2023021709334711800_rssb12165-cit-0021","first-page":"430","volume-title":"Proc. Int. Conf. Big Data","author":"Khanna","year":"2013"},{"key":"2023021709334711800_rssb12165-cit-0022","volume-title":"The BellKor solution to the Netflix Grand Prize","author":"Koren","year":"2009"},{"key":"2023021709334711800_rssb12165-cit-0023","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1109\/MC.2009.263","article-title":"Matrix factorization techniques for recommender systems","volume":"42","author":"Koren","year":"2009","journal-title":"Computer"},{"key":"2023021709334711800_rssb12165-cit-0024","doi-asserted-by":"crossref","first-page":"793","DOI":"10.1093\/biomet\/asp055","article-title":"Bias reduction in exponential family nonlinear models","volume":"96","author":"Kosmidis","year":"2009","journal-title":"Biometrika"},{"key":"2023021709334711800_rssb12165-cit-0025","doi-asserted-by":"crossref","first-page":"619","DOI":"10.1111\/j.2517-6161.1996.tb02105.x","article-title":"Hierarchical generalized linear models (with discussion)","volume":"58","author":"Lee","year":"1996","journal-title":"J. R. Statist. Soc."},{"key":"2023021709334711800_rssb12165-cit-0026","first-page":"1014","article-title":"Newton-Raphson and EM algorithms for linear mixed-effects models for repeated-measures data","volume":"83","author":"Lindstrom","year":"1988","journal-title":"J. Am. Statist. Ass."},{"key":"2023021709334711800_rssb12165-cit-0027","doi-asserted-by":"crossref","first-page":"817","DOI":"10.1093\/biomet\/74.4.817","article-title":"A fast scoring algorithm for maximum likelihood estimation in unbalanced mixed models with nested random effects","volume":"74","author":"Longford","year":"1987","journal-title":"Biometrika"},{"key":"2023021709334711800_rssb12165-cit-0028","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1080\/01621459.1983.10477920","article-title":"Parametric empirical Bayes inference: theory and applications","volume":"78","author":"Morris","year":"1983","journal-title":"J. Am. Statist. Ass."},{"key":"2023021709334711800_rssb12165-cit-0029","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1007\/s11002-008-9036-3","article-title":"Challenges and opportunities in high-dimensional choice data analyses","volume":"19","author":"Naik","year":"2008","journal-title":"Markt. Lett."},{"key":"2023021709334711800_rssb12165-cit-0030","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1093\/biomet\/52.3-4.447","article-title":"The theory of least squares when the parameters are stochastic and its application to the analysis of growth curves","volume":"52","author":"Rao","year":"1965","journal-title":"Biometrika"},{"key":"2023021709334711800_rssb12165-cit-0503","volume-title":"R: a language and Environment for Statistical Computing","author":"R Core Team","year":"2015"},{"key":"2023021709334711800_rssb12165-cit-0031","doi-asserted-by":"crossref","first-page":"642","DOI":"10.1080\/01621459.1985.10478164","article-title":"Mean squared error properties of empirical Bayes estimators in a multivariate random effects general linear model","volume":"80","author":"Reinsel","year":"1985","journal-title":"J. Am. Statist. Ass."},{"key":"2023021709334711800_rssb12165-cit-0032","volume-title":"Int. J. Mangmnt Sci. Engng Mangmnt","author":"Scott","year":"2016"},{"key":"2023021709334711800_rssb12165-cit-0033","doi-asserted-by":"crossref","DOI":"10.4135\/9781446247600","volume-title":"The SAGE Handbook of Multilevel Modeling","author":"Scott","year":"2013"},{"key":"2023021709334711800_rssb12165-cit-0034","volume-title":"Multilevel Analysis","author":"Snijders","year":"2012"},{"key":"2023021709334711800_rssb12165-cit-0035","doi-asserted-by":"crossref","first-page":"311","DOI":"10.2307\/1913012","article-title":"Efficient inference in a random coefficient regression model","volume":"38","author":"Swamy","year":"1970","journal-title":"Econometrica"},{"key":"2023021709334711800_rssb12165-cit-0036","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-387-21706-2","volume-title":"Modern Applied Statistics with S","author":"Venables","year":"2002"},{"key":"2023021709334711800_rssb12165-cit-0037","doi-asserted-by":"crossref","first-page":"556","DOI":"10.1017\/S0021859600050978","article-title":"The analysis of groups of experiments","volume":"28","author":"Yates","year":"1938","journal-title":"J. Agric. Sci."},{"key":"2023021709334711800_rssb12165-cit-0038","first-page":"1913","volume-title":"Advances in Neural Information Processing Systems","author":"Zhang","year":"2009"},{"key":"2023021709334711800_rssb12165-cit-0039","first-page":"47","volume-title":"Proc. 30th A. Int. Association for Computing Machinery Special Interest Group on Information Retrieval Conf. Research and Development in Information Retrieval","author":"Zhang","year":"2007"}],"container-title":["Journal of the Royal Statistical Society Series B: Statistical Methodology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1111%2Frssb.12165","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1111%2Frssb.12165","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/79\/1\/267\/49226958\/jrsssb_79_1_267.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/79\/1\/267\/49226958\/jrsssb_79_1_267.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,6,15]],"date-time":"2024-06-15T04:40:03Z","timestamp":1718426403000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/jrsssb\/article\/79\/1\/267\/7041920"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,3,20]]},"references-count":42,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2016,3,20]]},"published-print":{"date-parts":[[2017,1,1]]}},"URL":"https:\/\/doi.org\/10.1111\/rssb.12165","relation":{},"ISSN":["1369-7412","1467-9868"],"issn-type":[{"value":"1369-7412","type":"print"},{"value":"1467-9868","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2017,1]]},"published":{"date-parts":[[2016,3,20]]}}}