{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T15:49:49Z","timestamp":1753890589460,"version":"3.41.2"},"reference-count":30,"publisher":"Frontiers Media SA","license":[{"start":{"date-parts":[[2023,2,24]],"date-time":"2023-02-24T00:00:00Z","timestamp":1677196800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["frontiersin.org"],"crossmark-restriction":true},"short-container-title":["Front. Appl. Math. Stat."],"abstract":"<jats:sec><jats:title>Introduction<\/jats:title><jats:p>Longitudinal individual response profiles could exhibit a mixture of two or more phases of increase or decrease in trend throughout the follow-up period, with one or more unknown transition points (changepoints). The detection and estimation of these changepoints is crucial. Most of the proposed statistical methods for detecting and estimating changepoints in literature rely on distributional assumptions that may not hold. In this case, a good alternative is to use a robust approach; the quantile regression model. There are methods in the literature to deal with quantile regression models with a changepoint. These methods ignore the within-subject dependence of longitudinal data.<\/jats:p><\/jats:sec><jats:sec><jats:title>Methods<\/jats:title><jats:p>We propose a mixed effects quantile regression model with changepoints to account for dependence structure in the longitudinal data. Fixed effects parameters, in addition to the location of the changepoint, are estimated using the profile estimation method. The stochastic approximation EM algorithm is proposed to estimate the fixed effects parameters exploiting the link between an asymmetric Laplace distribution and the quantile regression. In addition, the location of the changepoint is estimated using the usual optimization methods.<\/jats:p><\/jats:sec><jats:sec><jats:title>Results and discussion<\/jats:title><jats:p>A simulation study shows that the proposed estimation and inferential procedures perform reasonably well in finite samples. The practical use of the proposed model is illustrated using COVID-19 data. The data focus on the effect of global economic and health factors on the monthly death rate due to COVID-19 from 1 April 2020 to 30th April 2021. the results show a positive effect on the monthly number of patients with COVID-19 in intensive care units (ICUs) for both 0.5th and 0.8th quantiles of new monthly deaths per million. The stringency index, hospital beds, and diabetes prevalence have no significant effect on both 0.5th and 0.8th quantiles of new monthly deaths per million.<\/jats:p><\/jats:sec>","DOI":"10.3389\/fams.2023.1106958","type":"journal-article","created":{"date-parts":[[2023,2,24]],"date-time":"2023-02-24T07:50:53Z","timestamp":1677225053000},"update-policy":"https:\/\/doi.org\/10.3389\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["A mixed effects changepoint quantile regression model for longitudinal data with application on COVID-19 data"],"prefix":"10.3389","volume":"9","author":[{"given":"Wafaa I. M.","family":"Ibrahim","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ahmed M.","family":"Gad","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Abd-Elnaser S.","family":"Abd-Rabou","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1965","published-online":{"date-parts":[[2023,2,24]]},"reference":[{"key":"B1","doi-asserted-by":"publisher","first-page":"1181","DOI":"10.1080\/03610918.2018.1491990","article-title":"Ibrahim WIM. An adaptive linear regression approach for modeling heavy-tailed longitudinal data","volume":"49","author":"Gad","year":"2020","journal-title":"Commun. Stat. Simulat. Computat"},{"key":"B2","doi-asserted-by":"publisher","first-page":"1038","DOI":"10.1002\/sim.5557","article-title":"Gao S. Bivariate random change point models for longitudinal outcomes","volume":"32","author":"Yang","year":"2012","journal-title":"Stat. Med"},{"key":"B3","doi-asserted-by":"publisher","first-page":"2074","DOI":"10.1002\/sim.2671","article-title":"Vaida F. Random Changepoint modeling of HIV immunologic responses","volume":"26","author":"Ghosh","year":"2007","journal-title":"Stat. Med"},{"key":"B4","doi-asserted-by":"publisher","first-page":"1052","DOI":"10.1111\/biom.12218","article-title":"Albert P","volume":"70","author":"Mclain","year":"2014"},{"key":"B5","doi-asserted-by":"publisher","first-page":"1625","DOI":"10.1080\/01621459.2012.712425","article-title":"Ying Z. A semiparametric change-point regression model for longitudinal data","volume":"107","author":"Xing","year":"2012","journal-title":"J Am Statistic Assoc"},{"key":"B6","doi-asserted-by":"publisher","first-page":"1015","DOI":"10.1002\/sim.5996","article-title":"Albert P. Identifying multiple change-points in a linear mixed effects model","volume":"33","author":"Lai","year":"2014","journal-title":"Stat Med."},{"key":"B7","doi-asserted-by":"publisher","first-page":"625","DOI":"10.1111\/biom.12313","article-title":"Chappell R. Quantile regression with a change-point model for longitudinal data: an application to the study of cognitive changes in preclinical alzheimer's disease","volume":"71","author":"Li","year":"2015","journal-title":"Biometrics."},{"key":"B8","doi-asserted-by":"publisher","first-page":"251","DOI":"10.1080\/01621459.1996.10476683","volume":"91","author":"Jung","year":"1996","journal-title":"Am Stat Assoc."},{"key":"B9","doi-asserted-by":"publisher","first-page":"756","DOI":"10.1214\/07-AOS564","article-title":"Fygenson M. Inference for censored quantile regression","volume":"37","author":"Wang","year":"2009","journal-title":"Annal Stat"},{"key":"B10","doi-asserted-by":"publisher","first-page":"242","DOI":"10.1111\/j.1541-0420.2010.01436.x","article-title":"Bent line quantile regression with application to an allometric study of land mammls' speed and mass","volume":"67","author":"Li","year":"2011","journal-title":"Biometrics"},{"year":"2011","author":"Sha","journal-title":"On Testing the Change-point in the Longitudinal Bent Line Quantile Regression Model, Graduate school of Arts and sciences, Columbia University, Coulmbia","key":"B11"},{"key":"B12","doi-asserted-by":"publisher","first-page":"437","DOI":"10.1016\/S0167-7152(01)00124-9","article-title":"Moyeed R. Bayesian quantile regression","volume":"54","author":"Yu","year":"2001","journal-title":"Statistics Probabil Lett"},{"key":"B13","doi-asserted-by":"publisher","first-page":"28","DOI":"10.2202\/1557-4679.1186","article-title":"Bottai M. Mixed-eefects models for conditional quantiles with longitudinal data","volume":"5","author":"Liu","year":"2009","journal-title":"Int J Biostatistic"},{"key":"B14","doi-asserted-by":"publisher","first-page":"121","DOI":"10.1007\/s11222-010-9212-1","volume":"22","author":"Meza","year":"2012","journal-title":"Stat Comput"},{"year":"2001","author":"Kotz","journal-title":"The Laplace Distribution and Generalizations. A Revisit with Applications to Communications, Economics, Engineering, and Finance.","key":"B15"},{"key":"B16","doi-asserted-by":"publisher","first-page":"471","DOI":"10.4310\/SII.2017.v10.n3.a10","article-title":"Bandyopadhyay D. Quantile regression in linear mixed models: a stochastic approximation EM approach","volume":"10","author":"Galarza","year":"2017","journal-title":"Stat Interface"},{"key":"B17","doi-asserted-by":"publisher","first-page":"1","DOI":"10.17654\/AS05701","article-title":"Gad AM. Sensitivity analysis index for shared parameter models in longitudinal studies","volume":"17","author":"Abd Elwahab","year":"2019","journal-title":"Adv Applicat Statistics"},{"key":"B18","doi-asserted-by":"publisher","first-page":"4446","DOI":"10.1080\/03610926.2019.1601223","article-title":"Gad AM. A stochastic variant of the EM algorithm to fit mixed (discrete and continuous) longitudinal data with nonignorable missingness","volume":"498","author":"Yaseen","year":"2020","journal-title":"Commun Stat Theory Meth"},{"key":"B19","doi-asserted-by":"publisher","first-page":"303","DOI":"10.1016\/0304-4076(94)01652-G","article-title":"Estimating the asymptotic covariance matrix for quantile regression models: a monte carlo study","volume":"8","author":"Buchinsky","year":"1995","journal-title":"J Econometr."},{"key":"B20","doi-asserted-by":"publisher","first-page":"463","DOI":"10.1111\/1467-9876.00084","article-title":"Zhao LP. Quantile regression methods for longitudinal data with drop-outs: application to CD4 cell counts of patients infected with the human immunodeficiency virus","volume":"46","author":"Lipsitz","year":"1997","journal-title":"Appl Stat"},{"key":"B21","doi-asserted-by":"publisher","first-page":"23","DOI":"10.1111\/1468-0262.00092","article-title":"Buchinsky M. A three-step method for choosing the number of bootstrap repetitions","volume":"68","author":"Andrews","year":"2000","journal-title":"Econometrica"},{"key":"B22","doi-asserted-by":"publisher","first-page":"1127","DOI":"10.1007\/s00148-020-00778-2","article-title":"Impacts of social and economic factors on the transmission of coronavirus disease 2019","volume":"33","author":"Qiu","year":"2020","journal-title":"J Populat Econ"},{"key":"B23","doi-asserted-by":"publisher","first-page":"21","DOI":"10.1098\/rsos.201823","article-title":"Liatsis P. Analyzing the impact of global demographic characteristics over the COVID-19 spread using class rule mining and pattern matching","volume":"8","author":"Khan","year":"2021","journal-title":"Royal Soc Open Sci"},{"key":"B24","doi-asserted-by":"publisher","first-page":"25","DOI":"10.1038\/s41598-021-97114-9","article-title":"Ayele DG. Additive quantile mixed effects modeling with application to longitudinal CD4 count data","volume":"11","author":"Yirga","year":"2021","journal-title":"Scientific Rep"},{"key":"B25","doi-asserted-by":"publisher","first-page":"1837","DOI":"10.1080\/03610918.2019.1610440","article-title":"Tian M. Weighted composite quantile regression for longitudinal mixed effects models with application to AIDS studies","volume":"50","author":"Tian","year":"2021","journal-title":"Commun Stat Simulat Computat"},{"key":"B26","doi-asserted-by":"publisher","first-page":"1281","DOI":"10.1007\/s00362-018-0988-y","article-title":"Lachos VH. Quantile regression for nonlinear mixed effects models: a likelihood based perspective","volume":"1","author":"Galarza","year":"2020","journal-title":"Stat Pap"},{"key":"B27","doi-asserted-by":"publisher","first-page":"p","DOI":"10.1080\/03610918.2022.2112600","article-title":"A CUSUM test for change point in quantile regression for longitudinal data, communications in statistics","volume":"22","author":"Abdelwahab","year":"2022","journal-title":"Simulat Applicat"},{"year":"1999","author":"Politis","article-title":"Wolf Subsampling M, Stanford, USA: Springer","key":"B28"},{"key":"B29","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1111\/j.2517-6161.1977.tb01600.x","article-title":"Rubin DB. Maximum likelihood from incomplete data via the EM algorithm","volume":"39","author":"Dempster","year":"1977","journal-title":"J Royal Stat Soc"},{"key":"B30","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1002\/j.2333-8504.2003.tb01899.x","article-title":"Assessing convergence of the Markov chain Monte Carlo algorithms: A review. ETS Research Report Series, A review","volume":"1","author":"Sinharay","year":"2003","journal-title":"ETS Res Rep Series"}],"container-title":["Frontiers in Applied Mathematics and Statistics"],"original-title":[],"link":[{"URL":"https:\/\/www.frontiersin.org\/articles\/10.3389\/fams.2023.1106958\/full","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,2,24]],"date-time":"2023-02-24T07:50:58Z","timestamp":1677225058000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.frontiersin.org\/articles\/10.3389\/fams.2023.1106958\/full"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,24]]},"references-count":30,"alternative-id":["10.3389\/fams.2023.1106958"],"URL":"https:\/\/doi.org\/10.3389\/fams.2023.1106958","relation":{},"ISSN":["2297-4687"],"issn-type":[{"type":"electronic","value":"2297-4687"}],"subject":[],"published":{"date-parts":[[2023,2,24]]},"article-number":"1106958"}}