{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,11]],"date-time":"2026-08-11T20:32:32Z","timestamp":1786480352992,"version":"build-2736575974"},"reference-count":43,"publisher":"Wiley","issue":"10-12","license":[{"start":{"date-parts":[[2026,5,17]],"date-time":"2026-05-17T00:00:00Z","timestamp":1778976000000},"content-version":"vor","delay-in-days":16,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"},{"start":{"date-parts":[[2026,5,1]],"date-time":"2026-05-01T00:00:00Z","timestamp":1777593600000},"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":["Statistics in Medicine"],"published-print":{"date-parts":[[2026,5]]},"abstract":"<jats:title>ABSTRACT<\/jats:title>\n                  <jats:p>The rise of precision medicine has shifted statistical analysis from population\u2010averaged treatment effects to subgroup\u2010specific effects, highlighting the importance of tailoring treatments to distinct patient subgroups. Longitudinal data, in particular, provides an opportunity to study how treatment effects evolve over time. However, methods for analyzing subgroup\u2010specific, time\u2010varying treatment effects in longitudinal settings remain limited. Most current approaches rely on two\u2010step procedures with restrictive assumptions, such as a fixed number of subgroups or prespecified correlation structures. In this paper, we propose a novel semiparametric subgroup\u2010varying\u2010coefficient model that simultaneously estimates all unknowns, including the number of subgroups and subgroup\u2010specific treatment effects, directly from longitudinal data without requiring restrictive assumptions. We develop a penalized Expectation\u2010Maximization (EM) algorithm that simultaneously identifies the subpopulations and estimates the treatment effects in different subgroups. We establish the theoretical properties of the proposed estimators, including consistency and asymptotic normality. The efficacy and robustness of the method are demonstrated through extensive simulations and applications to two real\u2010world datasets: the Standard and New Antiepileptic Drugs study and the Mayo Clinic trial on primary biliary cirrhosis. In both cases, our approach successfully identifies meaningful patient subgroups and reveals subgroup\u2010specific, time\u2010varying treatment sensitivities.<\/jats:p>","DOI":"10.1002\/sim.70600","type":"journal-article","created":{"date-parts":[[2026,5,18]],"date-time":"2026-05-18T05:42:33Z","timestamp":1779082953000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Varying Treatment Effects in Subgroups: A Unified Framework for Longitudinal Data Analysis"],"prefix":"10.1002","volume":"45","author":[{"given":"Pu","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Statistics Beijing Normal University  Beijing China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xinsheng","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Statistics and Data Science School of Management at Fudan University  Shanghai China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jiao","family":"Jin","sequence":"additional","affiliation":[{"name":"School of Statistics Beijing Normal University  Beijing China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2920-6355","authenticated-orcid":false,"given":"Jiakun","family":"Jiang","sequence":"additional","affiliation":[{"name":"Center for Statistics and Data Science Beijing Normal University  Zhuhai China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2026,5,17]]},"reference":[{"key":"e_1_2_13_2_1","doi-asserted-by":"publisher","DOI":"10.1136\/bmj.k4245"},{"key":"e_1_2_13_3_1","doi-asserted-by":"publisher","DOI":"10.1214\/21-AOAS1503"},{"key":"e_1_2_13_4_1","doi-asserted-by":"publisher","DOI":"10.1002\/sim.9386"},{"key":"e_1_2_13_5_1","doi-asserted-by":"publisher","DOI":"10.1002\/sim.9827"},{"key":"e_1_2_13_6_1","doi-asserted-by":"publisher","DOI":"10.1002\/sim.9879"},{"key":"e_1_2_13_7_1","doi-asserted-by":"publisher","DOI":"10.1080\/10543406.2020.1730867"},{"key":"e_1_2_13_8_1","doi-asserted-by":"publisher","DOI":"10.1002\/sim.9225"},{"key":"e_1_2_13_9_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmva.2020.104691"},{"key":"e_1_2_13_10_1","doi-asserted-by":"publisher","DOI":"10.1177\/0962280220904114"},{"key":"e_1_2_13_11_1","doi-asserted-by":"publisher","DOI":"10.1002\/sim.7020"},{"key":"e_1_2_13_12_1","doi-asserted-by":"publisher","DOI":"10.1093\/biostatistics\/kxad014"},{"key":"e_1_2_13_13_1","doi-asserted-by":"publisher","DOI":"10.1177\/0962280220912772"},{"key":"e_1_2_13_14_1","doi-asserted-by":"publisher","DOI":"10.1002\/cjs.11737"},{"key":"e_1_2_13_15_1","doi-asserted-by":"publisher","DOI":"10.1111\/j.1541-0420.2012.01794.x"},{"key":"e_1_2_13_16_1","doi-asserted-by":"publisher","DOI":"10.1111\/biom.13333"},{"key":"e_1_2_13_17_1","doi-asserted-by":"publisher","DOI":"10.1111\/biom.12343"},{"key":"e_1_2_13_18_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-031-50690-1_12"},{"key":"e_1_2_13_19_1","doi-asserted-by":"publisher","DOI":"10.1002\/sim.7064"},{"key":"e_1_2_13_20_1","volume-title":"Topics in Stochastic Processes: Probability and Mathematical Statistics: A Series of Monographs and Textbooks","author":"Ash R. B.","year":"2014"},{"key":"e_1_2_13_21_1","doi-asserted-by":"publisher","DOI":"10.1093\/biomet\/87.3.587"},{"key":"e_1_2_13_22_1","doi-asserted-by":"publisher","DOI":"10.1198\/016214504000001745"},{"key":"e_1_2_13_23_1","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-9868.2008.00656.x"},{"key":"e_1_2_13_24_1","doi-asserted-by":"publisher","DOI":"10.1214\/17-AOS1553"},{"key":"e_1_2_13_25_1","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511618994"},{"key":"e_1_2_13_26_1","first-page":"147","article-title":"Model Selection for Gaussian Mixture Models","volume":"27","author":"Huang T.","year":"2017","journal-title":"Statistica Sinica"},{"key":"e_1_2_13_27_1","doi-asserted-by":"publisher","DOI":"10.18637\/jss.v011.i08"},{"key":"e_1_2_13_28_1","doi-asserted-by":"publisher","DOI":"10.1093\/biomet\/asm053"},{"key":"e_1_2_13_29_1","doi-asserted-by":"publisher","DOI":"10.1198\/jasa.2009.tm08086"},{"key":"e_1_2_13_30_1","doi-asserted-by":"publisher","DOI":"10.1093\/biomet\/asq032"},{"key":"e_1_2_13_31_1","doi-asserted-by":"publisher","DOI":"10.1214\/009053606000000957"},{"key":"e_1_2_13_32_1","doi-asserted-by":"publisher","DOI":"10.1093\/biomet\/asv020"},{"issue":"37","key":"e_1_2_13_33_1","first-page":"1","article-title":"A Randomised Controlled Trial Examining the Longer\u2010Term Outcomes of Standard Versus New Antiepileptic Drugs. The SANAD Trial","volume":"11","author":"Marson A.","year":"2007","journal-title":"Health Technology Assessment\u2010Southampton"},{"key":"e_1_2_13_34_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0140-6736(07)60460-7"},{"key":"e_1_2_13_35_1","doi-asserted-by":"publisher","DOI":"10.1111\/j.1528-1157.2000.tb00317.x"},{"key":"e_1_2_13_36_1","doi-asserted-by":"publisher","DOI":"10.1002\/hep.510300415"},{"key":"e_1_2_13_37_1","doi-asserted-by":"publisher","DOI":"10.1002\/sim.9343"},{"key":"e_1_2_13_38_1","doi-asserted-by":"publisher","DOI":"10.1080\/01621459.2022.2157727"},{"issue":"11","key":"e_1_2_13_39_1","article-title":"Longitudinal Studies","volume":"7","author":"Caruana E. J.","year":"2015","journal-title":"Journal of Thoracic Disease"},{"key":"e_1_2_13_40_1","volume-title":"Empirical Process Theory and Applications","author":"Van de Geer S. A.","year":"2000"},{"key":"e_1_2_13_41_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-2545-2_3"},{"key":"e_1_2_13_42_1","volume-title":"Convergence of Stochastic Processes","author":"Pollard D.","year":"2012"},{"key":"e_1_2_13_43_1","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1176325486"},{"key":"e_1_2_13_44_1","doi-asserted-by":"publisher","DOI":"10.1198\/016214506000000311"}],"container-title":["Statistics in Medicine"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/sim.70600","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/full-xml\/10.1002\/sim.70600","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/sim.70600","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,11]],"date-time":"2026-08-11T20:11:59Z","timestamp":1786479119000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/sim.70600"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,5]]},"references-count":43,"journal-issue":{"issue":"10-12","published-print":{"date-parts":[[2026,5]]}},"alternative-id":["10.1002\/sim.70600"],"URL":"https:\/\/doi.org\/10.1002\/sim.70600","archive":["Portico"],"relation":{},"ISSN":["0277-6715","1097-0258"],"issn-type":[{"value":"0277-6715","type":"print"},{"value":"1097-0258","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,5]]},"assertion":[{"value":"2025-07-05","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2026-05-06","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2026-05-17","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"e70600"}}