{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,21]],"date-time":"2026-03-21T01:20:37Z","timestamp":1774056037019,"version":"3.50.1"},"reference-count":24,"publisher":"Wiley","license":[{"start":{"date-parts":[[2026,3,20]],"date-time":"2026-03-20T00:00:00Z","timestamp":1773964800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"},{"start":{"date-parts":[[2026,3,20]],"date-time":"2026-03-20T00:00:00Z","timestamp":1773964800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/doi.wiley.com\/10.1002\/tdm_license_1.1"}],"funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["UIDB\/00212\/2020"],"award-info":[{"award-number":["UIDB\/00212\/2020"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["UIDB\/04630\/2020"],"award-info":[{"award-number":["UIDB\/04630\/2020"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["UIDB\/00297\/2020"],"award-info":[{"award-number":["UIDB\/00297\/2020"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Math Methods in App Sciences"],"abstract":"<jats:title>ABSTRACT<\/jats:title>\n                  <jats:p>\n                    Variance\u2010component estimation in Gaussian linear mixed models (LMMs) is routinely performed by least squares\/analysis of variance (LS\/ANOVA) and likelihood\u2010based methods (ML\/REML), but these approaches can become numerically fragile and computationally expensive under incomplete and highly unbalanced designs. We introduce\n                    <jats:italic>Bayesian\u2010stochastic approximation expectation\u2010maximization<\/jats:italic>\n                    (BSA\u2010EM), a hybrid procedure tailored to variance\u2010component inference with arbitrary missing patterns. The method targets maximum\u2010a\u2010posteriori (MAP) estimation under an explicit prior on the variance components. Missing responses are handled through a\n                    <jats:italic>deterministic<\/jats:italic>\n                    EM completion: at each iteration, the conditional mean and variance of the missing block given the observed block are computed analytically from the Gaussian covariance partition, avoiding any Monte Carlo E\u2010step or latent\u2010variable simulation. Given these conditional moments, inverse\u2010gamma regularization yields closed\u2010form MAP updates for each variance component, and a Robbins\u2010Monro damping recursion stabilizes the iteration and enforces strictly positive updates in sparse regimes where boundary\/singularity issues are common for likelihood optimization. Extensive Monte Carlo experiments in crossed two\u2010way random\u2010effects designs with progressively removed cells benchmark BSA\u2010EM against LS, ANOVA), ML, REML, SAEM, and online EM. Across designs, BSA\u2010EM remains stable under extreme imbalance and is orders of magnitude faster than iterative likelihood and stochastic\u2010EM baselines. While it is less accurate than REML, it bridges the gap between spectral speed and REML accuracy in sparse layouts, enabling large\u2010scale parametric bootstrap uncertainty quantification and practical prior\u2010sensitivity analyses. Additional  experiments confirm that these computational advantages persist as the number of random\u2010effect levels increases.\n                  <\/jats:p>","DOI":"10.1002\/mma.70695","type":"journal-article","created":{"date-parts":[[2026,3,21]],"date-time":"2026-03-21T00:55:57Z","timestamp":1774054557000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["CMMSE\u00a0\u2013\u00a0A Hybrid EM and Bayesian\u2010Stochastic Approximation Method for Variance Components in Incomplete Mixed Models"],"prefix":"10.1002","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9095-0947","authenticated-orcid":false,"given":"D\u00e1rio","family":"Ferreira","sequence":"first","affiliation":[{"name":"Department of Mathematics and Center of Mathematics and Applications University of Beira Interior  Covilh\u00e3 Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9209-7772","authenticated-orcid":false,"given":"Sandra S.","family":"Ferreira","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Center of Mathematics and Applications University of Beira Interior  Covilh\u00e3 Portugal"}]}],"member":"311","published-online":{"date-parts":[[2026,3,20]]},"reference":[{"key":"e_1_2_12_2_1","first-page":"137","volume-title":"Encyclopedia of Statistical Sciences","author":"LaMotte L. 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D. P.","year":"1967","journal-title":"Statistician"},{"key":"e_1_2_12_13_1","doi-asserted-by":"crossref","first-page":"112","DOI":"10.1016\/j.cam.2018.05.057","article-title":"Exact Critical Values for One\u2010Way Fixed Effects Models With Random Sample Sizes","volume":"354","author":"Nunes C.","year":"2019","journal-title":"Journal of Computational and Applied Mathematics"},{"key":"e_1_2_12_14_1","doi-asserted-by":"crossref","first-page":"477","DOI":"10.2307\/2530872","article-title":"Two\u2010Stage Analysis Based on a Mixed Model: Large\u2010Sample Asymptotic Theory and Small\u2010Sample Simulation Result","volume":"41","author":"Giesbrec F. 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