{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,20]],"date-time":"2026-01-20T05:59:20Z","timestamp":1768888760759,"version":"3.49.0"},"reference-count":38,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,2,13]],"date-time":"2023-02-13T00:00:00Z","timestamp":1676246400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004826","name":"Beijing Natural Science Foundation","doi-asserted-by":"publisher","award":["Z210003"],"award-info":[{"award-number":["Z210003"]}],"id":[{"id":"10.13039\/501100004826","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004826","name":"Beijing Natural Science Foundation","doi-asserted-by":"publisher","award":["20200201258JC"],"award-info":[{"award-number":["20200201258JC"]}],"id":[{"id":"10.13039\/501100004826","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004826","name":"Beijing Natural Science Foundation","doi-asserted-by":"publisher","award":["12171328"],"award-info":[{"award-number":["12171328"]}],"id":[{"id":"10.13039\/501100004826","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004826","name":"Beijing Natural Science Foundation","doi-asserted-by":"publisher","award":["12071176"],"award-info":[{"award-number":["12071176"]}],"id":[{"id":"10.13039\/501100004826","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004826","name":"Beijing Natural Science Foundation","doi-asserted-by":"publisher","award":["11971064"],"award-info":[{"award-number":["11971064"]}],"id":[{"id":"10.13039\/501100004826","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Technology Developing Plan of Jilin Province","award":["Z210003"],"award-info":[{"award-number":["Z210003"]}]},{"name":"Technology Developing Plan of Jilin Province","award":["20200201258JC"],"award-info":[{"award-number":["20200201258JC"]}]},{"name":"Technology Developing Plan of Jilin Province","award":["12171328"],"award-info":[{"award-number":["12171328"]}]},{"name":"Technology Developing Plan of Jilin Province","award":["12071176"],"award-info":[{"award-number":["12071176"]}]},{"name":"Technology Developing Plan of Jilin Province","award":["11971064"],"award-info":[{"award-number":["11971064"]}]},{"name":"National Nature Science Foundation of China","award":["Z210003"],"award-info":[{"award-number":["Z210003"]}]},{"name":"National Nature Science Foundation of China","award":["20200201258JC"],"award-info":[{"award-number":["20200201258JC"]}]},{"name":"National Nature Science Foundation of China","award":["12171328"],"award-info":[{"award-number":["12171328"]}]},{"name":"National Nature Science Foundation of China","award":["12071176"],"award-info":[{"award-number":["12071176"]}]},{"name":"National Nature Science Foundation of China","award":["11971064"],"award-info":[{"award-number":["11971064"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we discuss regression analysis of bivariate interval-censored failure time data that often occur in biomedical and epidemiological studies. To solve this problem, we propose a kind of general and flexible copula-based semiparametric partly linear additive hazards models that can allow for both time-dependent covariates and possible nonlinear effects. For inference, a sieve maximum likelihood estimation approach based on Bernstein polynomials is proposed to estimate the baseline hazard functions and nonlinear covariate effects. The resulting estimators of regression parameters are shown to be consistent, asymptotically efficient and normal. A simulation study is conducted to assess the finite-sample performance of this method and the results show that it is effective in practice. Moreover, an illustration is provided.<\/jats:p>","DOI":"10.3390\/axioms12020198","type":"journal-article","created":{"date-parts":[[2023,2,14]],"date-time":"2023-02-14T01:41:06Z","timestamp":1676338866000},"page":"198","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Partially Linear Additive Hazards Regression for Bivariate Interval-Censored Data"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0663-3533","authenticated-orcid":false,"given":"Ximeng","family":"Zhang","sequence":"first","affiliation":[{"name":"Center for Applied Statistical Research, College of Mathematics, Jilin University, Changchun 130012, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shishun","family":"Zhao","sequence":"additional","affiliation":[{"name":"Center for Applied Statistical Research, College of Mathematics, Jilin University, Changchun 130012, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4463-004X","authenticated-orcid":false,"given":"Tao","family":"Hu","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Capital Normal University, Beijing 100048, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jianguo","family":"Sun","sequence":"additional","affiliation":[{"name":"Department of Statistics, University of Missouri, Columbia, MO 65211, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,13]]},"reference":[{"key":"ref_1","unstructured":"Sun, J. (2006). The Statistical Analysis of Interval-Censored Failure Time Data, Springer."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1093\/biomet\/85.2.289","article-title":"Additive hazards regression with current status data","volume":"85","author":"Lin","year":"1998","journal-title":"Biometrika"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"649","DOI":"10.1093\/biomet\/89.3.649","article-title":"Efficient estimation in additive hazards regression with current status data","volume":"89","author":"Martinussen","year":"2002","journal-title":"Biometrika"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1080\/03610926.2020.1742922","article-title":"Estimation of the additive hazards model with linear inequality restrictions based on current status data","volume":"51","author":"Feng","year":"2022","journal-title":"Commun. Stat.-Theory Methods"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1131","DOI":"10.1007\/s42952-020-00051-y","article-title":"A new method for regression analysis of interval-censored data with the additive hazards model","volume":"49","author":"Wang","year":"2020","journal-title":"J. Korean Stat. Soc."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"499","DOI":"10.1002\/cjs.11544","article-title":"Estimation of the additive hazards model with interval-censored data and missing covariates","volume":"48","author":"Li","year":"2020","journal-title":"Can. J. Stat."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Wang, T.B., Yopadhyay, D., and Sinha, S. (2022). Efficient estimation of the additive risks model for interval-censored data. arXiv.","DOI":"10.1007\/978-3-031-12366-5_9"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"364","DOI":"10.1002\/bimj.200710418","article-title":"Regression analysis of multivariate interval-censored failure time data with application to tumorigenicity experiments","volume":"50","author":"Tong","year":"2008","journal-title":"Biom. J."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"801","DOI":"10.1093\/biomet\/91.4.801","article-title":"Additive hazards model with multivariate failure time data","volume":"91","author":"Yin","year":"2004","journal-title":"Biometrika"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"549","DOI":"10.1002\/cjs.11647","article-title":"Semiparametric additive frailty hazard model for clustered failure time data","volume":"50","author":"Liu","year":"2022","journal-title":"Can. J. Stat."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1007\/s10985-009-9145-1","article-title":"Additive transformation models for clustered failure time data","volume":"16","author":"Zeng","year":"2010","journal-title":"Lifetime Data Anal."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Yu, M., and Du, M. (2022). Regression analysis of multivariate interval-censored failure time data under transformation model with informative censoring. Mathematics, 10.","DOI":"10.3390\/math10183257"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1093\/biostatistics\/kxz032","article-title":"Copula-based semiparametric regression method for bivariate data under general interval censoring","volume":"22","author":"Sun","year":"2021","journal-title":"Biostatistics"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"886","DOI":"10.1080\/01621459.2019.1593178","article-title":"Copula link-based additive models for right-censored event time data","volume":"115","author":"Marra","year":"2020","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"107550","DOI":"10.1016\/j.csda.2022.107550","article-title":"Copula link-based additive models for bivariate time-to-event outcomes with general censoring scheme","volume":"175","author":"Petti","year":"2022","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"141","DOI":"10.3150\/12-BEJ479","article-title":"Efficient semiparametric estimation in generalized partially linear additive models for longitudinal\/clustered data","volume":"20","author":"Cheng","year":"2014","journal-title":"Bernoulli"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"306","DOI":"10.1111\/sjos.12108","article-title":"Efficient estimation of the partly linear additive hazards model with current status data","volume":"42","author":"Lu","year":"2015","journal-title":"Scand. J. Stat."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"779","DOI":"10.1007\/s42952-019-00036-6","article-title":"An efficient estimation for the parameter in additive partially linear models with missing covariates","volume":"49","author":"Wang","year":"2020","journal-title":"J. Korean Stat. Soc."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1111\/biom.13399","article-title":"Analysis of clustered interval-censored data using a class of semiparametric partly linear frailty transformation models","volume":"78","author":"Lee","year":"2022","journal-title":"Biometrics"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1","DOI":"10.5109\/4844359","article-title":"Partially linear additive hazards regression for clustered and right censored data","volume":"54","author":"Chen","year":"2022","journal-title":"Bulletin of Informatics and Cybernetics"},{"key":"ref_21","first-page":"229","article-title":"Fonctions de repartition an dimensions et leurs marges","volume":"8","author":"Sklar","year":"1959","journal-title":"Publ. Inst. Stat. Univ. Paris"},{"key":"ref_22","unstructured":"Nelson, R.B. (2006). An Introduction to Copulas, Springer Science and Business Media."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Joe, H. (1997). Multivariate Models and Dependence Concepts, CRC Press.","DOI":"10.1201\/b13150"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1093\/biomet\/65.1.141","article-title":"A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence","volume":"65","author":"Clayton","year":"1978","journal-title":"Biometrika"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"698","DOI":"10.1080\/01621459.1960.10483368","article-title":"Bivariate exponential distributions","volume":"55","author":"Gumble","year":"1960","journal-title":"J. Am. Statitical Assoc."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"664","DOI":"10.1080\/01621459.2016.1158113","article-title":"A sieve semiparametric maximum likelihood approach for regression analysis of bivariate interval-censored failure time data","volume":"112","author":"Zhou","year":"2017","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1007\/BF02071384","article-title":"Shape preserving representations and optimality of the bernstein basis","volume":"1","author":"Carnicer","year":"1993","journal-title":"Adv. Comput. Math."},{"key":"ref_28","unstructured":"Burnham, K.P., Anderson, D.R., Burnham, K.P., and Anderson, D.R. (1993). Practical Use of the Information-Theoretic Approach, Springer."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1214\/aos\/1176344552","article-title":"Bootstrap methods: Another look at the jackknife","volume":"7","author":"Efron","year":"1979","journal-title":"Ann. Stat."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Age-Related Eye Disease Study Research Group (1999). The age-related eye disease study (AREDS): Design implications AREDS report no. 1. Control. Clin. Trials, 20, 573.","DOI":"10.1016\/S0197-2456(99)00031-8"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"266","DOI":"10.32614\/RJ-2020-025","article-title":"CopulaCenR: Copula based regression models for bivariate censored data in R","volume":"12","author":"Sun","year":"2020","journal-title":"R J."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"540","DOI":"10.1214\/aos\/1032894452","article-title":"Efficient estimation for the proportional hazards model with interval censoring","volume":"24","author":"Huang","year":"1996","journal-title":"Ann. Stat."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1111\/j.1467-9469.2009.00680.x","article-title":"A spline-based semiparametric maximum likelihood estimation method for the cox model with interval-censored data","volume":"37","author":"Zhang","year":"2010","journal-title":"Scand. J. Stat."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"960","DOI":"10.1080\/01621459.1997.10474050","article-title":"Sieve estimation for the proportional-odds failure-time regression model with interval censoring","volume":"92","author":"Huang","year":"1997","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_35","first-page":"383","article-title":"A frailty model approach for regression analysis of bivariate interval-censored survival data","volume":"23","author":"Wen","year":"2013","journal-title":"Stat. Sin."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Van der Vaart, A., and Wellner, J. (1996). Weak Convergence and Empirical Processes, Springer.","DOI":"10.1007\/978-1-4757-2545-2"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"580","DOI":"10.1214\/aos\/1176325486","article-title":"Convergence rate of sieve estimates","volume":"22","author":"Shen","year":"1994","journal-title":"Ann. Stat."},{"key":"ref_38","unstructured":"Lorentz, G.G. (2013). Bernstein Polynomials, American Mathematical Society."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/2\/198\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:34:02Z","timestamp":1760121242000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/2\/198"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,13]]},"references-count":38,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2023,2]]}},"alternative-id":["axioms12020198"],"URL":"https:\/\/doi.org\/10.3390\/axioms12020198","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,2,13]]}}}