Joint Frailty Mixing Model for Recurrent Event Data with an Associated Terminal Event: Application to Hospital Readmission Data


  • Goutam Barman Department of Statistics, Krishnagar Government College, West Bengal, India
  • Babulal Seal Department of Mathematics and Statistics, Aliah University, West Bengal, India
  • Shreya Bhunia Department of Mathematics and Statistics, Aliah University, West Bengal, India
  • Proloy Banerjee Department of Mathematics and Statistics, Aliah University, West Bengal, India



Frailty, Proportional hazard model, Proportional intensity model, Mixture distribution, Recurrent events


Recurrent events like repeated hospitalization, cancer tumour recurrences, and many others occur frequently. The follow-up on recurrent events may be stopped by a terminal event like death. It is obvious that if the frequencies of recurrent events are more, then it may lead to a terminal event and in this case terminal event becomes ‘dependent’. In this article, we study a joint modelling and analysis of recurrent events with a dependent terminal event. Here, the proportional intensity model for the recurrent events process and the proportional hazard model for the terminal event time are taken. To account for the association between recurrent events and terminal events, mixing frailty or random effect is studied rather than available pure frailty. In our case, the distribution of frailty is introduced as a mixture of folded normal distribution and gamma distribution rather than using pure gamma distribution. An estimation procedure in the joint frailty model is applied to estimate the parameters of the model. This method is close to the method of minimum chi-square rather than a complicated one. An extensive simulation study has been performed to estimate the model parameters and the performances are evaluated based on bias and MSE criteria. Further from an application point of view, the method is illustrated to a hospital readmission data for colorectal cancer patients.


Prentice RL, Williams BJ, Peterson AV. On the regression analysis of multivariate failure time data. Biometrika 1981; 68: 373-379. DOI:

Pepe MS, Cai J. Some graphical displays and marginal regression analysis for recurrent failure times and time-dependent covariates. J Am Statist Assoc 1993; 88: 811-820. DOI:

Lawless JF, Nadeau C. Some simple robust methods for the analysis of recurrent events. Technometrics 1995; 37: 158-168. DOI:

Sun L, Zhao X, Zhou J. A class of mixed models for recurrent event data. Canad J Statist 2011; 39: 578-590. DOI:

Ghosh D, Lin DY. Nonparametric analysis of recurrent events and death. Biometrics 2000; 56: 554-562. DOI:

Miloslavsky M, Keles S, Van der Laan MJ, et al. Recurrent events analysis in the presence of time-dependent covariates and dependent censoring. J R Statist Soc 2004; 66B: 239-257. DOI:

Zeng D, Lin DY. Semiparametric transformation models with random effects for joint analysis of recurrent and terminal events. Biometrics 2009; 65: 746-752. DOI:

Zeng D, Cai J. A semiparametric additive rate model for recurrent events with informative terminal event. Biometrika 2010; 97: 699-712. DOI:

Wang MC, Qin J, Chiang CT. Analyzing recurrent event data with informative censoring. J Am Statist Assoc 2001; 96: 1057-1065. DOI:

Huang CY, Wang MC. Joint Modeling and estimation of recurrent event process and failure time. J Am Statist Assoc 2004; 99: 1153-1165. DOI:

Ye Y, Kalbfleisch JD, Schaubel DE. Semiparametric analysis of correlated recurrent and terminal events. Biometrics 2007; 63: 78-87. DOI:

Cook RJ, Lawless JF. Marginal analysis of recurrent events and a terminating event. Stat Med 1997; 16: 911-924.<911::AID-SIM544>3.0.CO;2-I DOI:<911::AID-SIM544>3.0.CO;2-I

Ghosh D, Lin DY. Marginal regression models for recurrent and terminal events. Statist Sinica 2002; 12: 663-668.

Cook RJ, Lawless JF, Lakhal CL, et al. Robust estimation of mean functions and treatment effects for recurrent events under event-dependent censoring and termination: Application to skeletal complications in cancer metastatic to bone. J Am Statist Assoc 2009; 104: 60-75. DOI:

Cox DR. Regression models and life-tables, Journal of the Royal Statistical Society. Series B (Methodological) 1972; 34(2): 187-220. DOI:

Hougaard P. Analysis of multivariate survival data, Springer Verlag: New York 2000. DOI:

Duchateau L, Janssen P. The frailty model, Springer: New York, 2008.

Che X, Angus J. A new joint model of recurrent event data with the additive hazards model for the terminal event time. Metrica 2016. DOI:

Liu L, Wolfe RA, Huang X. Shared frailty models for recurrent events and a terminal event. Biometrics 2004; 60: 747-756. DOI:

Huang X, Liu L. A joint frailty model for survival and gap times between recurrent events. Biometrics 2007; 63(2): 747-756. DOI:

Joly P, Commenges D, Letenneur L. A penalized likelihood approach for arbitrarily censored and truncated data: application to age-specific incidence of dementia. Biometrics 1998; 54(1): 185-194. DOI:

Rondeau V, Mathoulin-Pelissier S, Jacqmin-Gadda H, Brouste V, Soubeyran P. Joint frailty models for recurring events and death using maximum penalized likelihood estimation: application on cancer events. Biostatistics 2007; 8(4): 708-721. DOI:

Marquardt D. An algorithm for least-squares estimation of nonlinear parameters. SIAM Journal on Applied Mathematics 1963; 11: 431-441. DOI:

Mazroui Y, Mathoulin-Pelissier S, Soubeyran P and Rondeau V. General joint frailty model for recurrent event data with a dependent terminal event: application to follicular lymphoma data. Statistics in Medicine 2012. DOI:

Toenges G, Jahn-Eimermacher A. Computational issues in fitting joint frailty models for recurrent events with an associated terminal event. Computer Methods and Programs in Biomedicine 2020; 188: 1-13. DOI:

Banerjee P, Goswami A, Bhunia S, Basu S. Determination of Causal Relationship Between Bilirubin and Other Liver Biomarker in Case of Hepatitis C. Biomed Stat Informatics 2021; 6(2): 23-31. DOI:

Peng Y, Jiajia D, Jun Z. Joint analysis of recurrent event data with a dependent terminal event. J Syst Sci Complex 2017; 30: 1443-1458. DOI:

González JR, Fernandez E, Moreno V, Ribes J, Peris M, Navarro M, Cambray M, Borrás JM. Sex differences in hospital readmission among colorectal cancer patients. Journal of Epidemiology and Community Health 2005; 59: 506-511. DOI:

Fathoni M, Gunardi G, Adi-Kusumo F, Hutajulu SH, I Purwanto. Cox Proportional Hazard Regression Interaction Model and Its Application to Determine The Risk of Death in Breast Cancer Patients after Chemotherapy. Int J Stats Med Res 2022; 11: 105-113. DOI:




How to Cite

Barman, G. ., Seal, B. ., Bhunia, S. ., & Banerjee, P. . (2023). Joint Frailty Mixing Model for Recurrent Event Data with an Associated Terminal Event: Application to Hospital Readmission Data. International Journal of Statistics in Medical Research, 12, 213–225.



General Articles