{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:52:47Z","timestamp":1760147567975,"version":"build-2065373602"},"reference-count":47,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,2,12]],"date-time":"2023-02-12T00:00:00Z","timestamp":1676160000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The additive hazard regression model plays an important role when the excess risk is the quantity of interest compared to the relative risks, where the proportional hazard model is better. This paper discusses parametric regression analysis of survival data using the additive hazards model with competing risks in the presence of independent right censoring. In this paper, the baseline hazard function is parameterized using a modified Weibull distribution as a lifetime model. The model parameters are estimated using maximum likelihood and Bayesian estimation methods. We also derive the asymptotic confidence interval and the Bayes credible interval of the unknown parameters. The finite sample behaviour of the proposed estimators is investigated through a Monte Carlo simulation study. The proposed model is applied to liver transplant data.<\/jats:p>","DOI":"10.3390\/sym15020485","type":"journal-article","created":{"date-parts":[[2023,2,13]],"date-time":"2023-02-13T04:07:19Z","timestamp":1676261239000},"page":"485","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Estimation of the Modified Weibull Additive Hazards Regression Model under Competing Risks"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1762-3573","authenticated-orcid":false,"given":"Habbiburr","family":"Rehman","sequence":"first","affiliation":[{"name":"Department of Medicine (Biomedical Genetics), Boston University Chobanian & Avedisian School of Medicine, Boston, MA 02118, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1213-7739","authenticated-orcid":false,"given":"Navin","family":"Chandra","sequence":"additional","affiliation":[{"name":"Department of Statistics, Ramanujan School of Mathematical Sciences, Pondicherry University, Puducherry 605014, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3904-4014","authenticated-orcid":false,"given":"Takeshi","family":"Emura","sequence":"additional","affiliation":[{"name":"Biostatistics Center, Kurume University, Kurume 8300011, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Manju","family":"Pandey","sequence":"additional","affiliation":[{"name":"Department of Zoology and DST Centre for Mathematical Sciences, Institute of Sciences, Banaras Hindu University, Varanasi 221005, India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1111\/j.2517-6161.1972.tb00899.x","article-title":"Regression models and life-tables","volume":"34","author":"Cox","year":"1972","journal-title":"J. R. Stat. Soc. Ser. Methodol."},{"key":"ref_2","unstructured":"Cox, D.R., and Oakes, D. (1984). Analysis of Survival Data, Chapman and Hall\/CRC."},{"key":"ref_3","unstructured":"Breslow, N.E., and Day, N.E. (1987). Statistical Methods in Cancer Research, International Agency for Research on Cancer."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"907","DOI":"10.1002\/sim.4780080803","article-title":"A linear regression model for the analysis of life times","volume":"8","author":"Aalen","year":"1989","journal-title":"Stat. Med."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1093\/biomet\/81.1.61","article-title":"Semiparametric analysis of the additive risk model","volume":"81","author":"Lin","year":"1994","journal-title":"Biometrika"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1712","DOI":"10.1214\/aos\/1176324320","article-title":"Semiparametric analysis of general additive-multiplicative hazard models for counting processes","volume":"23","author":"Lin","year":"1995","journal-title":"Ann. Stat."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Kalbfleisch, J.D., and Prentice, R.L. (2002). The Statistical Analysis of Failure Time Data, John Wiley & Sons.","DOI":"10.1002\/9781118032985"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1007\/s10985-012-9230-8","article-title":"Applying competing risks regression models: An overview","volume":"19","author":"Haller","year":"2013","journal-title":"Lifetime Data Anal."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Lawless, J.F. (2003). Statistical Models and Methods for Lifetime Data, John Wiley & Sons.","DOI":"10.1002\/9781118033005"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Pintilie, M. (2006). Competing Risks: A Practical Perspective, John Wiley & Sons.","DOI":"10.1002\/9780470870709"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2307","DOI":"10.1177\/0962280219892295","article-title":"Comparison of the marginal hazard model and the sub-distribution hazard model for competing risks under an assumed copula","volume":"29","author":"Emura","year":"2020","journal-title":"Stat. Methods Med. Res."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1177\/09622802221136067","article-title":"Simulating time-to-event data subject to competing risks and clustering: A review and synthesis","volume":"32","author":"Meng","year":"2023","journal-title":"Stat. Methods Med. Res."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Zuo, Z., Wang, L., and Lio, Y. (2023). Reliability estimation for dependent left-truncated and right-censored competing risks data with illustrations. Energies, 16.","DOI":"10.3390\/en16010062"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"184","DOI":"10.1093\/biostatistics\/kxj040","article-title":"Parametric regression on cumulative incidence function","volume":"8","author":"Jeong","year":"2007","journal-title":"Biostatistics"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1177\/0008068320150303","article-title":"Parametric analysis of lifetime data with multiple causes of failure using cause specific reversed hazard rates","volume":"67","author":"Anjana","year":"2015","journal-title":"Calcutta Stat. Assoc. Bull."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2128","DOI":"10.1080\/02664763.2019.1577370","article-title":"Parametric inference for quantile event times with adjustment for covariates on competing risks data","volume":"46","author":"Lee","year":"2019","journal-title":"J. Appl. Stat."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"953","DOI":"10.1007\/s42081-021-00110-6","article-title":"Competing risks regression with dependent multiple spells: Monte Carlo evidence and an application to maternity leave","volume":"4","author":"Lipowski","year":"2021","journal-title":"Jpn. J. Stat. Data Sci."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1007\/s42081-021-00142-y","article-title":"Inferences on cumulative incidence function for middle censored survival data with Weibull regression","volume":"5","author":"Rehman","year":"2022","journal-title":"Jpn. J. Stat. Data Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1093","DOI":"10.1111\/j.0006-341X.1999.01093.x","article-title":"Confidence bands for cumulative incidence curves under the additive risk model","volume":"55","author":"Shen","year":"1999","journal-title":"Biometrics"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1081\/STA-120028378","article-title":"Additive hazards model for competing risks analysis of the case-cohort design","volume":"33","author":"Sun","year":"2004","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1007\/s10260-011-0161-4","article-title":"Regression analysis of competing risks data via semi-parametric additive hazard model","volume":"20","author":"Zhang","year":"2011","journal-title":"Stat. Methods Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"11667","DOI":"10.1080\/03610926.2016.1277759","article-title":"An additive subdistribution hazard model for competing risks data","volume":"46","author":"Li","year":"2017","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1963","DOI":"10.1080\/03610918.2017.1332213","article-title":"Additive risks regression model for middle censored exponentiated-exponential lifetime data","volume":"47","author":"Sankaran","year":"2018","journal-title":"Commun. Stat. Simul. Comput."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1109\/TR.2002.805788","article-title":"A modified Weibull distribution","volume":"52","author":"Lai","year":"2003","journal-title":"IEEE Trans. Reliab."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"794","DOI":"10.3390\/math7090794","article-title":"Bayesian inference of \u03b4= P (X< Y) for Burr type XII distribution based on progressively first failure-censored samples","volume":"7","author":"Byrnes","year":"2019","journal-title":"Mathematics"},{"key":"ref_26","unstructured":"Martz, H.F., and Waller, R. (1982). Bayesian Reliability Analysis, John Wiley & Sons."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"107747","DOI":"10.1016\/j.ress.2021.107747","article-title":"Bayes analysis of some important lifetime models using MCMC based approaches when the observations are left truncated and right censored","volume":"214","author":"Ranjan","year":"2021","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"374","DOI":"10.1109\/TR.2005.853036","article-title":"Parameter estimation for a modified Weibull distribution, for progressively type-II censored samples","volume":"54","author":"Ng","year":"2005","journal-title":"IEEE Trans. Reliab."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"647","DOI":"10.1080\/02664760801920846","article-title":"Markov chain Monte Carlo methods for parameter estimation of the modified Weibull distribution","volume":"35","author":"Jiang","year":"2008","journal-title":"J. Appl. Stat."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1080\/00949650802600730","article-title":"A Bayes analysis of modified Weibull distribution via Markov chain Monte Carlo simulation","volume":"80","author":"Upadhyay","year":"2010","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"308","DOI":"10.1093\/comjnl\/7.4.308","article-title":"A simplex method for function minimization","volume":"7","author":"Nelder","year":"1965","journal-title":"Comput. J."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Robert, C.P., Casella, G., and Casella, G. (2010). Introducing Monte Carlo Methods with R, Springer Science & Business Media.","DOI":"10.1007\/978-1-4419-1576-4"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"721","DOI":"10.1109\/TPAMI.1984.4767596","article-title":"Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images","volume":"PAMI-6","author":"Geman","year":"1984","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1093\/biomet\/57.1.97","article-title":"Monte Carlo sampling methods using Markov chains and their applications","volume":"57","author":"Hastings","year":"1970","journal-title":"Biometrika"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Guure, C.B., and Ibrahim, N.A. (2012). Bayesian analysis of the survival function and failure rate of Weibull distribution with censored data. Math. Probl. Eng., 2012.","DOI":"10.1155\/2012\/329489"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Beyersmann, J., Allignol, A., and Schumacher, M. (2012). Competing Risks and Multistate Models with R, Springer Science & Business Media.","DOI":"10.1007\/978-1-4614-2035-4"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Knight, K. (1999). Mathematical Statistics, CRC Press.","DOI":"10.1201\/9781584888567"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1007\/s00180-020-01027-6","article-title":"Likelihood-based analysis of doubly-truncated data under the location-scale and AFT model","volume":"36","author":"Huang","year":"2021","journal-title":"Comput. Stat."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Lunn, D., Jackson, C., Best, N., Spiegelhalter, D., and Thomas, A. (2012). The BUGS Book: A Practical Introduction to Bayesian Analysis, Chapman and Hall\/CRC.","DOI":"10.1201\/b13613"},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Therneau, T.M., and Grambsch, P.M. (2000). Modeling Survival Data: Extending the Cox Model, Springer Science & Business Media.","DOI":"10.1007\/978-1-4757-3294-8"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1016\/j.csda.2017.03.011","article-title":"Competing risk model with bivariate random effects for clustered survival data","volume":"112","author":"Lai","year":"2017","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Klein, J.P., and Moeschberger, M.L. (2003). Survival Analysis: Techniques for Censored and Truncated Data, Springer.","DOI":"10.1007\/b97377"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"462","DOI":"10.1109\/TR.2019.2907518","article-title":"A competing risks model with multiply censored reliability data under multivariate Weibull distributions","volume":"68","author":"Fan","year":"2019","journal-title":"IEEE Trans. Reliab."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Li, Y., and Ye, J. (2022). Analysis for partially accelerated dependent competing risks model with masked data based on copula function. Commun. Stat. Simul. Comput., 1\u201317.","DOI":"10.1080\/03610918.2022.2039199"},{"key":"ref_45","unstructured":"Sinha, S. (1998). Bayesian Estimation, New Age International (P) Limited Publisher."},{"key":"ref_46","unstructured":"Jeffreys, H. (1961). Theory of Probability, Oxford University Press."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"515","DOI":"10.1214\/06-BA117A","article-title":"Prior distributions for variance parameters in hierarchical models (comment on article by Browne and Draper)","volume":"1","author":"Gelman","year":"2006","journal-title":"Bayesian Anal."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/2\/485\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:32:21Z","timestamp":1760121141000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/2\/485"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,12]]},"references-count":47,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2023,2]]}},"alternative-id":["sym15020485"],"URL":"https:\/\/doi.org\/10.3390\/sym15020485","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2023,2,12]]}}}