{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,12]],"date-time":"2026-10-12T03:45:49Z","timestamp":1791776749879,"version":"4.3.4"},"reference-count":33,"publisher":"Wiley","issue":"5","license":[{"start":{"date-parts":[[2026,10,11]],"date-time":"2026-10-11T00:00:00Z","timestamp":1791676800000},"content-version":"vor","delay-in-days":10,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/doi.wiley.com\/10.1002\/tdm_license_1.1"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Biometrical J"],"published-print":{"date-parts":[[2026,10]]},"abstract":"<jats:title>ABSTRACT<\/jats:title>\n                  <jats:p>In crossover designs, multiple outcomes are often collected to assess treatment effectiveness in a comprehensive manner. Also, interest has grown in comparing intra\u2010subject variability between treatments in such designs. We present three distinct multivariate linear mixed\u2010effects models, with the choice of the model depending on the design and assumptions regarding carryover effects. We discuss maximum likelihood (ML) and restricted ML (REML) approaches for parameter estimation. For a special case that a crossover design is uniform within sequences, we derive the closed\u2010form expressions for the ML and REML estimators of the variance components. We propose general hypothesis testing methods for the treatment effects and the intra\u2010subject variance matrices. The latter situation has not received much attention in the statistical literature. For the special case, we derive the exact test statistics for treatment means comparisons and for testing the null hypothesis of homogeneous intra\u2010subject variance matrices. Simulation studies are conducted to validate the superior performance of the proposed multivariate approach compared with the usual univariate approach in testing treatment effects. Finally, we apply the methodologies to two data\u00a0examples.<\/jats:p>","DOI":"10.1002\/bimj.70193","type":"journal-article","created":{"date-parts":[[2026,10,12]],"date-time":"2026-10-12T03:08:12Z","timestamp":1791774492000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Multivariate Analysis of Data From Crossover Clinical Trials"],"prefix":"10.1002","volume":"68","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6726-392X","authenticated-orcid":false,"given":"Yi","family":"Jiang","sequence":"first","affiliation":[{"name":"Department of Public Health Sciences, College of Medicine The Pennsylvania State University Hershey Pennsylvania USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6488-7809","authenticated-orcid":false,"given":"Vernon M.","family":"Chinchilli","sequence":"additional","affiliation":[{"name":"Department of Public Health Sciences, College of Medicine The Pennsylvania State University Hershey Pennsylvania USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2026,10,11]]},"reference":[{"issue":"5","key":"e_1_2_8_2_1","doi-asserted-by":"crossref","first-page":"507","DOI":"10.1164\/rccm.201103-0405PP","article-title":"Current Controversies and Future Perspectives in Chronic Obstructive Pulmonary Disease","volume":"184","author":"Agust\u00ed A.","year":"2011","journal-title":"American Journal of Respiratory and Critical Care Medicine"},{"issue":"9465","key":"e_1_2_8_3_1","doi-asserted-by":"crossref","first-page":"1159","DOI":"10.1016\/S0140-6736(05)71879-1","article-title":"Epidemiology and Reporting of Randomised Trials Published in PubMed Journals","volume":"365","author":"Chan A.\u2010W.","year":"2005","journal-title":"The Lancet"},{"issue":"15","key":"e_1_2_8_4_1","doi-asserted-by":"crossref","first-page":"1619","DOI":"10.1002\/(SICI)1097-0258(19960815)15:15<1619::AID-SIM326>3.0.CO;2-N","article-title":"Design and Analysis of Intra\u2010Subject Variability in Cross\u2010Over Experiments","volume":"15","author":"Chinchilli V. M.","year":"1996","journal-title":"Statistics in Medicine"},{"key":"e_1_2_8_5_1","unstructured":"Cruz N. A. O. O.Melo andC. A.Martinez.2024. \u201cEstimation of Complex Carryover Effects in Crossover Designs with Repeated Measures.\u201d arXiv:2402.16362v1."},{"issue":"5","key":"e_1_2_8_6_1","doi-asserted-by":"crossref","first-page":"829","DOI":"10.1016\/j.cpr.2011.03.011","article-title":"The End of the Primary Outcome Measure: A Research Agenda for Constructing its Replacement","volume":"31","author":"De Los Reyes A.","year":"2011","journal-title":"Clinical Psychology Review"},{"key":"e_1_2_8_7_1","first-page":"366","article-title":"CONSORT 2010 Statement: Extension to Randomised Crossover Trials","volume":"2019","author":"Dwan K.","year":"2019","journal-title":"BMJ"},{"issue":"1","key":"e_1_2_8_8_1","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1093\/biomet\/80.1.39","article-title":"Bayes Linear Methods for Grouped Multivariate Repeated Measurement Studies with Application to Crossover Trials","volume":"80","author":"Farrow M.","year":"1993","journal-title":"Biometrika"},{"issue":"19","key":"e_1_2_8_9_1","doi-asserted-by":"crossref","first-page":"2554","DOI":"10.1001\/jama.289.19.2554","article-title":"Composite Outcomes in Randomized Trials: Greater Precision but with Greater Uncertainty?","volume":"289","author":"Freemantle N.","year":"2003","journal-title":"JAMA"},{"key":"e_1_2_8_10_1","unstructured":"Grender J. M. andW. D.Johnson.1992. \u201cUsing the GLM Procedure to Analyze Multivariate Response in Crossover Designs.\u201d SAS. pp.1355\u20131360."},{"issue":"1","key":"e_1_2_8_11_1","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1002\/sim.4780120108","article-title":"Analysis of Crossover Designs with Multivariate Response","volume":"12","author":"Grender J. M.","year":"1993","journal-title":"Statistics in Medicine"},{"issue":"423","key":"e_1_2_8_12_1","first-page":"939","article-title":"Exact Inference About the Within\u2010Subject Variability in 2\u00d7$\\times$ 2 Crossover Trials","volume":"88","author":"Guilbaud O.","year":"1993","journal-title":"Journal of the American Statistical Association"},{"key":"e_1_2_8_13_1","doi-asserted-by":"crossref","first-page":"805","DOI":"10.2307\/2530695","article-title":"Unbalanced Repeated\u2010Measures Models with Structured Covariance Matrices","volume":"42","author":"Jennrich R. 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