{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,1]],"date-time":"2026-07-01T01:40:18Z","timestamp":1782870018453,"version":"3.54.5"},"reference-count":34,"publisher":"Oxford University Press (OUP)","issue":"1","license":[{"start":{"date-parts":[[2016,7,5]],"date-time":"2016-07-05T00:00:00Z","timestamp":1467676800000},"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":[[2017,3,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n               <jats:p>It is traditionally assumed that the random effects in mixed models follow a multivariate normal distribution, making likelihood-based inferences more feasible theoretically and computationally. However, this assumption does not necessarily hold in practice which may lead to biased and unreliable results. We introduce a novel diagnostic test based on the so-called gradient function proposed by Verbeke and Molenberghs (2013) to assess the random-effects distribution. We establish asymptotic properties of our test and show that, under a correctly specified model, the proposed test statistic converges to a weighted sum of independent chi-squared random variables each with one degree of freedom. The weights, which are eigenvalues of a square matrix, can be easily calculated. We also develop a parametric bootstrap algorithm for small samples. Our strategy can be used to check the adequacy of any distribution for random effects in a wide class of mixed models, including linear mixed models, generalized linear mixed models, and non-linear mixed models, with univariate as well as multivariate random effects. Both asymptotic and bootstrap proposals are evaluated via simulations and a real data analysis of a randomized multicenter study on toenail dermatophyte onychomycosis.<\/jats:p>","DOI":"10.1111\/biom.12551","type":"journal-article","created":{"date-parts":[[2016,7,5]],"date-time":"2016-07-05T15:16:39Z","timestamp":1467731799000},"page":"63-71","source":"Crossref","is-referenced-by-count":38,"title":["Diagnosing Misspecification of the Random-Effects Distribution in Mixed Models"],"prefix":"10.1093","volume":"73","author":[{"given":"Reza","family":"Drikvandi","sequence":"first","affiliation":[{"name":"I-BioStat, KU Leuven , Leuven , Belgium"},{"name":"Department of Mathematics, Imperial College London , London , U.K."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Geert","family":"Verbeke","sequence":"additional","affiliation":[{"name":"I-BioStat, KU Leuven , Leuven , Belgium"},{"name":"I-BioStat, Universiteit Hasselt , Hasselt , Belgium"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Geert","family":"Molenberghs","sequence":"additional","affiliation":[{"name":"I-BioStat, KU Leuven , Leuven , Belgium"},{"name":"I-BioStat, Universiteit Hasselt , Hasselt , Belgium"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2016,7,5]]},"reference":[{"key":"2024011603224090800_biom12551-bib-0001","doi-asserted-by":"crossref","first-page":"639","DOI":"10.1016\/j.csda.2003.12.009","article-title":"Examples in which misspecification of a random effects distribution reduces efficiency, and possible remedies","volume":"47","author":"Agresti","year":"2004","journal-title":"Computational Statistics and Data Analysis"},{"key":"2024011603224090800_biom12551-bib-0002","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1111\/j.0006-341X.1999.00117.x","article-title":"A general maximum likelihood analysis of variance components in generalized linear models","volume":"55","author":"Aitkin","year":"1999","journal-title":"Biometrics"},{"key":"2024011603224090800_biom12551-bib-0003","doi-asserted-by":"crossref","first-page":"318","DOI":"10.1198\/tast.2010.09244","article-title":"A note on the indeterminacy of the random-effects distribution in hierarchical models","volume":"64","author":"Alonso","year":"2010","journal-title":"The American Statistician"},{"key":"2024011603224090800_biom12551-bib-0004","doi-asserted-by":"crossref","first-page":"4474","DOI":"10.1016\/j.csda.2008.02.033","article-title":"A family of tests to detect misspecifications in the random-effects structure of generalized linear mixed models","volume":"52","author":"Alonso","year":"2008","journal-title":"Computational Statistics and Data Analysis"},{"key":"2024011603224090800_biom12551-bib-0005","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1093\/biomet\/76.2.375","article-title":"Likelihood inference for mixtures: Geometrical and other constructions of monotone step-length algorithms","volume":"76","author":"B\u00f6hning","year":"1989","journal-title":"Biometrika"},{"key":"2024011603224090800_biom12551-bib-0006","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1093\/biostatistics\/3.3.347","article-title":"A Monte Carlo EM algorithm for generalized linear mixed models with flexible random effects distribution","volume":"3","author":"Chen","year":"2002","journal-title":"Biostatistics"},{"key":"2024011603224090800_biom12551-bib-0007","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1007\/s11749-009-0148-8","article-title":"Goodness-of-fit tests in mixed models","volume":"18","author":"Claeskens","year":"2009","journal-title":"Test"},{"key":"2024011603224090800_biom12551-bib-0008","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1111\/j.1365-2133.1996.tb15653.x","article-title":"A 12-week treatment for dermatophyte toe onychomycosis: Terbinafine 250\u2009mg\/day vs. itraconazole 200\u2009mg\/day\u2014A double-blind comparative trial","volume":"134","author":"De Backer","year":"1996","journal-title":"British Journal of Dermatology"},{"key":"2024011603224090800_biom12551-bib-0009","article-title":"A goodness-of-fit test for the random-effects distribution in mixed models","volume-title":"Statistical Methods in Medical Research","author":"Efendi","year":"2014"},{"key":"2024011603224090800_biom12551-bib-0010","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4899-3095-8","volume-title":"Number-Theoretic Methods in Statistics","author":"Fang","year":"1994"},{"key":"2024011603224090800_biom12551-bib-0011","doi-asserted-by":"crossref","DOI":"10.1201\/9781420011579","volume-title":"Longitudinal Data Analysis: A Handbook of Modern Statistical Methods","author":"Fitzmaurice","year":"2008"},{"key":"2024011603224090800_biom12551-bib-0012","doi-asserted-by":"crossref","first-page":"973","DOI":"10.1093\/biomet\/88.4.973","article-title":"Misspecified maximum likelihood estimates and generalised linear mixed models","volume":"88","author":"Heagerty","year":"2001","journal-title":"Biometrika"},{"key":"2024011603224090800_biom12551-bib-0013","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1111\/j.1541-0420.2008.01103.x","article-title":"Diagnosis of random-effect model misspecification in generalized linear mixed models for binary response","volume":"65","author":"Huang","year":"2009","journal-title":"Biometrics"},{"key":"2024011603224090800_biom12551-bib-0014","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1093\/biomet\/48.3-4.419","article-title":"Computing the distribution of quadratic forms in normal variables","volume":"48","author":"Imhof","year":"1961","journal-title":"Biometrika"},{"key":"2024011603224090800_biom12551-bib-0015","doi-asserted-by":"crossref","first-page":"1137","DOI":"10.1214\/aos\/1013699997","article-title":"Goodness-of-fit tests for mixed model diagnostics","volume":"29","author":"Jiang","year":"2001","journal-title":"The Annals of Statistics"},{"key":"2024011603224090800_biom12551-bib-0016","doi-asserted-by":"crossref","first-page":"624","DOI":"10.1214\/aos\/1176347130","article-title":"Assessing normality in random effects models","volume":"17","author":"Lange","year":"1989","journal-title":"The Annals of Statistics"},{"key":"2024011603224090800_biom12551-bib-0017","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1214\/aos\/1176346059","article-title":"The geometry of mixture likelihoods: a general theory","volume":"11","author":"Lindsay","year":"1983","journal-title":"The Annals of Statistics"},{"key":"2024011603224090800_biom12551-bib-0018","doi-asserted-by":"crossref","first-page":"783","DOI":"10.1214\/aos\/1176346245","article-title":"The geometry of mixture likelihoods, part II: the exponential family","volume":"11","author":"Lindsay","year":"1983","journal-title":"The Annals of Statistics"},{"key":"2024011603224090800_biom12551-bib-0019","doi-asserted-by":"crossref","first-page":"3125","DOI":"10.1002\/sim.3157","article-title":"The impact of a misspecified random-effects distribution on the estimation and the performance of inferential procedures in generalized linear mixed models","volume":"27","author":"Liti\u00e8re","year":"2008","journal-title":"Statistics in Medicine"},{"key":"2024011603224090800_biom12551-bib-0020","doi-asserted-by":"crossref","first-page":"3105","DOI":"10.1002\/sim.3153","article-title":"A likelihood reformulation method in non-normal random effects models","volume":"27","author":"Liu","year":"2008","journal-title":"Statistics in Medicine"},{"key":"2024011603224090800_biom12551-bib-0021","doi-asserted-by":"crossref","first-page":"388","DOI":"10.1214\/11-STS361","article-title":"Misspecifying the shape of a random effects distribution: Why getting it wrong may not matter","volume":"26","author":"McCulloch","year":"2011","journal-title":"Statistical Science"},{"key":"2024011603224090800_biom12551-bib-0022","doi-asserted-by":"crossref","first-page":"270","DOI":"10.1111\/j.1541-0420.2010.01435.x","article-title":"Prediction of random effects in linear and generalized linear models under model misspecification","volume":"67","author":"McCulloch","year":"2011","journal-title":"Biometrics"},{"key":"2024011603224090800_biom12551-bib-0023","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1198\/106186006X96854","article-title":"Use of the probability integral transformation to fit nonlinear mixed-effects models with nonnormal random effects","volume":"15","author":"Nelson","year":"2006","journal-title":"Journal of Computational and Graphical Statistics"},{"key":"2024011603224090800_biom12551-bib-0024","doi-asserted-by":"crossref","first-page":"755","DOI":"10.1093\/biomet\/79.4.755","article-title":"The effects of mixture distribution misspecification when fitting mixed-effects logistic models","volume":"79","author":"Neuhaus","year":"1992","journal-title":"Biometrika"},{"key":"2024011603224090800_biom12551-bib-0025","doi-asserted-by":"crossref","first-page":"5765","DOI":"10.1016\/j.csda.2006.10.003","article-title":"Quasi-Monte Carlo estimation in generalized linear mixed models","volume":"51","author":"Pan","year":"2007","journal-title":"Computational Statistics and Data Analysis"},{"key":"2024011603224090800_biom12551-bib-0026","doi-asserted-by":"crossref","first-page":"1000","DOI":"10.1111\/j.1541-0420.2005.00365.x","article-title":"Goodness-of-fit methods for generalized linear mixed models","volume":"61","author":"Pan","year":"2005","journal-title":"Biometrics"},{"key":"2024011603224090800_biom12551-bib-0027","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1080\/01621459.1981.10477633","article-title":"The analysis of categorical data from complex sample surveys: Chi-squared tests for goodness of fit and independence in two-way tables","volume":"76","author":"Rao","year":"1981","journal-title":"Journal of the American Statistical Association"},{"key":"2024011603224090800_biom12551-bib-0028","doi-asserted-by":"crossref","first-page":"443","DOI":"10.1111\/j.1467-9469.2004.02_101.x","article-title":"Goodness-of-fit tests for mixed models","volume":"31","author":"Ritz","year":"2004","journal-title":"Scandinavian Journal of Statistics"},{"key":"2024011603224090800_biom12551-bib-0029","doi-asserted-by":"crossref","first-page":"485","DOI":"10.1093\/biomet\/77.3.485","article-title":"Hypothesis testing of regression parameters in semiparametric generalized linear models for cluster correlated data","volume":"77","author":"Rotnitzky","year":"1990","journal-title":"Biometrika"},{"key":"2024011603224090800_biom12551-bib-0030","doi-asserted-by":"crossref","first-page":"1003","DOI":"10.1093\/biomet\/93.4.1003","article-title":"A diagnostic test for the mixing distribution in a generalised linear mixed model","volume":"93","author":"Tchetgen","year":"2006","journal-title":"Biometrika"},{"key":"2024011603224090800_biom12551-bib-0031","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1080\/01621459.1996.10476679","article-title":"A linear mixed-effects model with heterogeneity in the random-effects population","volume":"91","author":"Verbeke","year":"1996","journal-title":"Journal of the American Statistical Association"},{"key":"2024011603224090800_biom12551-bib-0032","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1093\/biostatistics\/kxs059","article-title":"The gradient function as an exploratory goodness-of-fit assessment of the random-effects distribution in mixed models","volume":"14","author":"Verbeke","year":"2013","journal-title":"Biostatistics"},{"key":"2024011603224090800_biom12551-bib-0033","doi-asserted-by":"crossref","first-page":"721","DOI":"10.1111\/j.1467-9469.2006.00504.x","article-title":"A simulation-based goodness-of-fit test for random effects in generalized linear mixed models","volume":"33","author":"Waagepetersen","year":"2006","journal-title":"Scandinavian Journal of Statistics"},{"key":"2024011603224090800_biom12551-bib-0034","doi-asserted-by":"crossref","first-page":"795","DOI":"10.1111\/j.0006-341X.2001.00795.x","article-title":"Linear mixed models with flexible distributions of random effects for longitudinal data","volume":"57","author":"Zhang","year":"2001","journal-title":"Biometrics"}],"container-title":["Biometrics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1111%2Fbiom.12551","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1111%2Fbiom.12551","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/academic.oup.com\/biometrics\/article-pdf\/73\/1\/63\/56025181\/biometrics_73_1_63.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/biometrics\/article-pdf\/73\/1\/63\/56025181\/biometrics_73_1_63.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,1,26]],"date-time":"2024-01-26T18:03:59Z","timestamp":1706292239000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/biometrics\/article\/73\/1\/63-71\/7537702"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,7,5]]},"references-count":34,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2016,7,5]]},"published-print":{"date-parts":[[2017,3,1]]}},"URL":"https:\/\/doi.org\/10.1111\/biom.12551","relation":{},"ISSN":["0006-341X","1541-0420"],"issn-type":[{"value":"0006-341X","type":"print"},{"value":"1541-0420","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2017,3]]},"published":{"date-parts":[[2016,7,5]]}}}