{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,3]],"date-time":"2026-04-03T07:23:46Z","timestamp":1775201026689,"version":"3.50.1"},"reference-count":24,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2023,4,27]],"date-time":"2023-04-27T00:00:00Z","timestamp":1682553600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["41271 038"],"award-info":[{"award-number":["41271 038"]}]},{"name":"National Natural Science Foundation of China","award":["R72021106"],"award-info":[{"award-number":["R72021106"]}]},{"name":"National Natural Science Foundation of China","award":["2022B1212010006"],"award-info":[{"award-number":["2022B1212010006"]}]},{"name":"UIC New Faculty Start-up Research Fund","award":["41271 038"],"award-info":[{"award-number":["41271 038"]}]},{"name":"UIC New Faculty Start-up Research Fund","award":["R72021106"],"award-info":[{"award-number":["R72021106"]}]},{"name":"UIC New Faculty Start-up Research Fund","award":["2022B1212010006"],"award-info":[{"award-number":["2022B1212010006"]}]},{"name":"Guangdong Provincial Key Laboratory of Interdisciplinary Research and Application for Data Science, BNU-HKBU United International College (UIC)","award":["41271 038"],"award-info":[{"award-number":["41271 038"]}]},{"name":"Guangdong Provincial Key Laboratory of Interdisciplinary Research and Application for Data Science, BNU-HKBU United International College (UIC)","award":["R72021106"],"award-info":[{"award-number":["R72021106"]}]},{"name":"Guangdong Provincial Key Laboratory of Interdisciplinary Research and Application for Data Science, BNU-HKBU United International College (UIC)","award":["2022B1212010006"],"award-info":[{"award-number":["2022B1212010006"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>A Bayesian semiparametric model framework is proposed to analyze multivariate longitudinal data. The new framework leads to simple explicit posterior distributions of model parameters. It results in easy implementation of the MCMC algorithm for estimation of model parameters and demonstrates fast convergence. The proposed model framework associated with the MCMC algorithm is validated by four covariance structures and a real-life dataset. A simple Monte Carlo study of the model under four covariance structures and an analysis of the real dataset show that the new model framework and its associated Bayesian posterior inferential method through the MCMC algorithm perform fairly well in the sense of easy implementation, fast convergence, and smaller root mean square errors compared with the same model without the specified autoregression structure.<\/jats:p>","DOI":"10.3390\/axioms12050431","type":"journal-article","created":{"date-parts":[[2023,4,27]],"date-time":"2023-04-27T04:30:47Z","timestamp":1682569847000},"page":"431","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Longitudinal Data Analysis Based on Bayesian Semiparametric Method"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3328-3206","authenticated-orcid":false,"given":"Guimei","family":"Jiao","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jiajuan","family":"Liang","sequence":"additional","affiliation":[{"name":"Department of Statistics and Data Science, BNU-HKBU United International College, Zhuhai 519087, China"},{"name":"Guangdong Provincial Key Laboratory of Interdisciplinary Research and Application for Data Science, BNU-HKBU United International College, Zhuhai 519087, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fanjuan","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaoli","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shaokang","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hao","family":"Li","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jing","family":"Jin","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jiali","family":"Cai","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fangjie","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,4,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2992","DOI":"10.1177\/0962280214536537","article-title":"Longitudinal data subject to irregular observation: A review of methods with a focus on visit processes, assumptions, and study design","volume":"25","author":"Pullenayegum","year":"2016","journal-title":"Statist. Methods Med. Res."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Chakraborty, R., Banerjee, M., and Vemuri, B.C. (2017, January 18\u201321). Statistics on the space of trajectories for longitudinal data analysis. Proceedings of the 2017 IEEE 14th International Symposium on Biomedical Imaging (ISBI 2017), Melbourne, VIC, Australia.","DOI":"10.1109\/ISBI.2017.7950684"},{"key":"ref_3","first-page":"769","article-title":"Nonparametric regession analysis of multivariate longitudinal data","volume":"23","author":"Xiang","year":"2013","journal-title":"Stat. Sin."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1168","DOI":"10.1080\/01621459.2015.1076725","article-title":"Bayesian nonparametric longitudinal data analysis","volume":"111","author":"Quintana","year":"2016","journal-title":"J. Am. Statist. Assoc."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1798","DOI":"10.1038\/s41467-019-09785-8","article-title":"An additive Gaussian process regression model for interpretable non-parametric analysis of longitudinal data","volume":"10","author":"Cheng","year":"2019","journal-title":"Nat. Commun."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"921","DOI":"10.2307\/2533846","article-title":"A semiparametric Bayesian approach to the random effects model","volume":"54","author":"Kleinman","year":"1998","journal-title":"Biometrics"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1232","DOI":"10.1080\/01621459.2018.1482756","article-title":"Semiparametric regression analysis of multiple right- and interval-censored events","volume":"114","author":"Gao","year":"2019","journal-title":"J. Am. Statist. Assoc."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1722","DOI":"10.1198\/jasa.2006.s142","article-title":"Semiparametric regression","volume":"101","author":"Lee","year":"2006","journal-title":"J. Am. Statist. Assoc."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1007\/s10985-013-9251-y","article-title":"Profile local linear estimation of generalized semiparametric regression model for longitudinal data","volume":"19","author":"Sun","year":"2013","journal-title":"Lifetime Data Anal."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"689","DOI":"10.2307\/2532783","article-title":"Semiparametric models for longitudinal data with application to CD4 cell numbers in HIV seroconverters","volume":"50","author":"Zeger","year":"1994","journal-title":"Biometrics"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1111\/j.1541-0420.2009.01227.x","article-title":"Bayesian inference in semiparametric mixed models for longitudinal data","volume":"66","author":"Li","year":"2010","journal-title":"Biometrics"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1183","DOI":"10.1080\/00949655.2015.1057732","article-title":"Quantile regression-based Bayesian semiparametric mixed-effects models for longitudinal data with non-normal, missing and mismeasured covariate","volume":"86","author":"Hunag","year":"2016","journal-title":"J. Statist. Comput. Simul."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Li, J., Zhou, J., Zhang, B., and Li, X.R. (2017, January 10\u201313). Estimation of high dimensional covariance matrices by shrinkage algorithms. Proceedings of the 2017 20th International Conference on Information Fusion (Fusion), Xi\u2019an, China.","DOI":"10.23919\/ICIF.2017.8009753"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1630","DOI":"10.1214\/17-AOS1597","article-title":"An MCMC approach to empirical Bayes inference and Bayesian sensitivity analysis via empirical processes","volume":"46","author":"Doss","year":"2018","journal-title":"Ann. Statist."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1080\/10618600.2000.10474879","article-title":"Markov chain sampling methods for Dirichlet process mixture models","volume":"9","author":"Neal","year":"2000","journal-title":"J. Comput. Graph. Statist."},{"key":"ref_16","unstructured":"Kingma, D.P., and Welling, M. (2014, January 14\u201316). Auto-encoding variational Bayes. Proceedings of the International Conference on Learning Representations (ICLR), Banff, AB, Canada."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"146","DOI":"10.1214\/aop\/1176996454","article-title":"I-Divergence geometry of probability distributions and minimization problems","volume":"3","author":"Csiszar","year":"1975","journal-title":"Ann. Probab."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Rasmussen, C.E., and Williams, C.K.I. (2006). Gaussian Processes for Machine Learning, MIT Press.","DOI":"10.7551\/mitpress\/3206.001.0001"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1111\/1467-9868.00282","article-title":"Non-Gaussian Ornstein\u2013Uhlenbeck-based models and some of their uses in financial economics","volume":"63","author":"Shephard","year":"2001","journal-title":"J. Roy. Statist. Soc. (Ser. B)"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1198\/016214505000000727","article-title":"Order-based dependent Dirichlet processes","volume":"101","author":"Griffin","year":"2006","journal-title":"J. Am. Statist. Assoc."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"633","DOI":"10.1080\/01621459.1997.10474014","article-title":"Dirichlet process mixed generalized linear models","volume":"92","author":"Mukhopadhyay","year":"1997","journal-title":"J. Am. Statist. Assoc."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Zimmerman, D.L., and N\u00fa\u00f1ez-Ant\u00f3n, V.A. (2009). Antedependence Models for Longitudinal Data, CRC Press.","DOI":"10.1201\/9781420064278"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1007\/s10463-010-0314-5","article-title":"Bayesian estimation of a covariance matrix with flexible prior specification","volume":"64","author":"Hsu","year":"2012","journal-title":"Ann. Inst. Statist. Math."},{"key":"ref_24","unstructured":"Chen, X. (2019). Longitudinal Data Analysis Based on Bayesian Semiparametric Method. [Master\u2019s Thesis, Lanzhou University]. (In Chinese)."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/5\/431\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:24:22Z","timestamp":1760124262000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/5\/431"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,27]]},"references-count":24,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2023,5]]}},"alternative-id":["axioms12050431"],"URL":"https:\/\/doi.org\/10.3390\/axioms12050431","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,4,27]]}}}