{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,25]],"date-time":"2026-09-25T18:14:53Z","timestamp":1790360093295,"version":"4.1.0"},"reference-count":32,"publisher":"Wiley","issue":"18-19","license":[{"start":{"date-parts":[[2025,8,7]],"date-time":"2025-08-07T00:00:00Z","timestamp":1754524800000},"content-version":"vor","delay-in-days":6,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12301368"],"award-info":[{"award-number":["12301368"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Statistics in Medicine"],"published-print":{"date-parts":[[2025,8]]},"abstract":"<jats:title>ABSTRACT<\/jats:title>\n                  <jats:p>Patients affected by neurological disorders usually present substantial heterogeneity in multi\u2010domain biomarkers and clinical measures. This heterogeneity arises from differences in disease stage, unique characteristics, and membership in distinct latent subtypes. Exploring such complex heterogeneity and identifying disease progression\u2010related markers is crucial for early diagnosis and developing timely and targeted interventions. This paper proposes a mixture exponential family trajectory model to integrate markers from multiple modalities to learn the disease progression. We incorporate continuous neuroimaging and microRNA sequencing biomarkers, categorical clinical symptoms, and ordinal cognitive markers using appropriate exponential family distributions with lower\u2010dimensional latent factors. The mixture model assigns subtype\u2010specific parameters to these distributions for each mixture component, enabling the characterization of patients in heterogeneous latent subgroups. The proposed model can also describe the nonlinear trajectory of disease deterioration and provide a temporal sequence of decline for each marker. We develop a Bayesian estimation procedure coupled with efficient Markov chain Monte Carlo (MCMC) sampling schemes to perform statistical inference for the mixture model. The proposed method is assessed through extensive simulation studies and an application to Parkinson's Progression Markers Initiative (PPMI) to learn the temporal ordering and subtypes of neurodegeneration of Parkinson's disease (PD).<\/jats:p>","DOI":"10.1002\/sim.70211","type":"journal-article","created":{"date-parts":[[2025,8,7]],"date-time":"2025-08-07T13:24:36Z","timestamp":1754573076000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A Bayesian Mixture of Exponential Family Factor Models for Uncovering Disease Progression Subtypes"],"prefix":"10.1002","volume":"44","author":[{"ORCID":"https:\/\/orcid.org\/0009-0002-8094-4871","authenticated-orcid":false,"given":"Kai","family":"Kang","sequence":"first","affiliation":[{"name":"Department of Statistics Sun Yat\u2010sen University  Guangzhou China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Qinxia","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Biostatistics, Mailman School of Public Health Columbia University  New York New York USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhanpeng","family":"Xu","sequence":"additional","affiliation":[{"name":"Department of Statistics, School of Computer, Data &amp; Information Sciences University of Wisconsin\u2010Madison  Madison Wisconsin USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yuanjia","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Biostatistics, Mailman School of Public Health Columbia University  New York New York USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2025,8,7]]},"reference":[{"key":"e_1_2_9_2_1","doi-asserted-by":"publisher","DOI":"10.1590\/S1980-57642010DN40400003"},{"key":"e_1_2_9_3_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jalz.2013.10.003"},{"key":"e_1_2_9_4_1","doi-asserted-by":"publisher","DOI":"10.1016\/S1474-4422(09)70299-6"},{"key":"e_1_2_9_5_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.neuroimage.2012.07.059"},{"key":"e_1_2_9_6_1","doi-asserted-by":"publisher","DOI":"10.1038\/s43587-021-00042-6"},{"key":"e_1_2_9_7_1","doi-asserted-by":"publisher","DOI":"10.1093\/biostatistics\/kxac042"},{"key":"e_1_2_9_8_1","doi-asserted-by":"publisher","DOI":"10.1038\/ng1570"},{"issue":"1","key":"e_1_2_9_9_1","first-page":"8244","article-title":"Integrative Analysis Using Coupled Latent Variable Models for Individualizing Prognoses","volume":"17","author":"Schulam P.","year":"2016","journal-title":"Journal of Machine Learning Research"},{"key":"e_1_2_9_10_1","doi-asserted-by":"publisher","DOI":"10.1186\/s40035-021-00249-y"},{"key":"e_1_2_9_11_1","doi-asserted-by":"publisher","DOI":"10.1214\/12-AOAS597"},{"key":"e_1_2_9_12_1","doi-asserted-by":"publisher","DOI":"10.1111\/biom.13108"},{"key":"e_1_2_9_13_1","doi-asserted-by":"crossref","unstructured":"G.LiandI.Gaynanova \u201cA General Framework for Association Analysis of Heterogeneous Data \u201d2018.","DOI":"10.1214\/17-AOAS1127"},{"key":"e_1_2_9_14_1","doi-asserted-by":"publisher","DOI":"10.1111\/biom.12885"},{"key":"e_1_2_9_15_1","doi-asserted-by":"publisher","DOI":"10.1093\/biostatistics\/kxy052"},{"key":"e_1_2_9_16_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.pneurobio.2011.09.005"},{"key":"e_1_2_9_17_1","doi-asserted-by":"publisher","DOI":"10.1177\/0962280217748675"},{"issue":"4","key":"e_1_2_9_18_1","first-page":"1117","article-title":"Mixture Modeling for Longitudinal Data\u201d. 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