{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,7]],"date-time":"2025-11-07T09:51:17Z","timestamp":1762509077571,"version":"build-2065373602"},"reference-count":15,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2025,11,7]],"date-time":"2025-11-07T00:00:00Z","timestamp":1762473600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004239","name":"Pozna\u0144 University of Technology","doi-asserted-by":"publisher","award":["0213\/SBAD\/0122"],"award-info":[{"award-number":["0213\/SBAD\/0122"]}],"id":[{"id":"10.13039\/501100004239","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>An extended growth curve model with fixed and random effects is considered. Under the assumption of multivariate normality, the maximum likelihood estimators of the fixed effects and the dispersion matrix are determined in a model with random nuisance parameters, both without any assumption on the covariance structure and under the assumption of compound symmetry. For this purpose, rules for differentiation of symmetric matrices are applied. Furthermore, when the experiments are designed in balanced complete blocks, particular symmetric matrices appear in the likelihood equations, allowing closed-form expressions for the estimators. It is also shown that the vector of sufficient statistics for the fixed effects extended growth curve model is also sufficient for the model with random nuisance parameters. The presented results are illustrated using a real data example.<\/jats:p>","DOI":"10.3390\/sym17111901","type":"journal-article","created":{"date-parts":[[2025,11,7]],"date-time":"2025-11-07T09:14:45Z","timestamp":1762506885000},"page":"1901","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Estimation and Sufficiency Under the Mixed Effects Extended Growth Curve Model with Compound Symmetry Covariance Structure"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6208-8322","authenticated-orcid":false,"given":"Katarzyna","family":"Filipiak","sequence":"first","affiliation":[{"name":"Institute of Mathematics, Pozna\u0144 University of Technology, Piotrowo 3A, 60-965 Pozna\u0144, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5473-3419","authenticated-orcid":false,"given":"Augustyn","family":"Markiewicz","sequence":"additional","affiliation":[{"name":"Department of Mathematical and Statistical Methods, Pozna\u0144 University of Life Sciences, Wojska Polskiego 28, 60-637 Pozna\u0144, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5318-9896","authenticated-orcid":false,"given":"Pawe\u0142","family":"Krajewski","sequence":"additional","affiliation":[{"name":"Institute of Plant Genetics, Polish Academy of Sciences, Strzeszy\u0144ska 34, 60-479 Pozna\u0144, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9113-567X","authenticated-orcid":false,"given":"Hanna","family":"\u0106wiek-Kupczy\u0144ska","sequence":"additional","affiliation":[{"name":"Luxembourg Centre for Systems Biomedicine, University of Luxembourg, Esch-sur-Alzette, Belval Campus 1, Boulevard du Jazz, L-4370 Belvaux, Luxembourg"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,11,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1093\/biomet\/51.3-4.313","article-title":"A generalized multivariate analysis of variance model useful especially for growth curve problems","volume":"51","author":"Potthoff","year":"1964","journal-title":"Biometrika"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Kollo, T., and von Rosen, D. 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Mass Spectrom."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/11\/1901\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,7]],"date-time":"2025-11-07T09:34:30Z","timestamp":1762508070000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/11\/1901"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,7]]},"references-count":15,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2025,11]]}},"alternative-id":["sym17111901"],"URL":"https:\/\/doi.org\/10.3390\/sym17111901","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,11,7]]}}}