, Volume 81, Issue 5, pp 523–547 | Cite as

Joint analysis of recurrent event data with additive–multiplicative hazards model for the terminal event time

  • Miao Han
  • Liuquan Sun
  • Yutao Liu
  • Jun Zhu


Recurrent event data are often collected in longitudinal follow-up studies. In this article, we propose a semiparametric method to model the recurrent and terminal events jointly. We present an additive–multiplicative hazards model for the terminal event and a proportional intensity model for the recurrent events, and a shared frailty is used to model the dependence between the recurrent and terminal events. We adopt estimating equation approaches for inference, and the asymptotic properties of the resulting estimators are established. The finite sample behavior of the proposed estimators is evaluated through simulation studies. An application to a medical cost study of chronic heart failure patients from the University of Virginia Health System is illustrated.


Additive–multiplicative hazard model Estimating equation Frailty Recurrent events Terminal event 



The authors would like to thank the reviewers for their constructive and insightful comments and suggestions that greatly improved the paper. This research was partly supported by the National Natural Science Foundation of China (Grant Nos. 11601307, 11771431 and 11690015) and Key Laboratory of RCSDS, CAS (No. 2008DP173182).

Compliance with ethical standards

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.


  1. Andersen PK, Gill RD (1982) Cox’s regression model for counting processes: a large sample study. Ann Stat 10:1100–1120MathSciNetCrossRefzbMATHGoogle Scholar
  2. Byar DP (1980) The Veterans administration study of chemoprophylaxis for recurrent stage I bladder tumors: comparisons of placebo, pyridoxine and topical thiotepa. In: Pavone-Macaluso M, Smith PH, Edsmyr F (eds) Bladder tumors and other topics in urological oncology. Plenum, New York, pp 363–370CrossRefGoogle Scholar
  3. Claeskens G, Hjort NL (2008) Model selection and model averaging. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  4. Cook RJ, Lawless JF (1997) Marginal analysis of recurrent events and a terminating event. Stat Med 16:911–924CrossRefGoogle Scholar
  5. Cook RJ, Lawless JF, Lakhal-Chaieb L, Lee KA (2009) 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 Stat Assoc 104:60–75MathSciNetCrossRefzbMATHGoogle Scholar
  6. Ghosh D, Lin DY (2000) Nonparametric analysis of recurrent events and death. Biometrics 56:554–562MathSciNetCrossRefzbMATHGoogle Scholar
  7. Ghosh D, Lin DY (2002) Marginal regression models for recurrent and terminal events. Stat Sin 12:663–688MathSciNetzbMATHGoogle Scholar
  8. Ghosh D, Lin DY (2003) Semiparametric analysis of recurrent events data in the presence of dependent censoring. Biometrics 59:877–885MathSciNetCrossRefzbMATHGoogle Scholar
  9. Han M, Song X, Sun L, Liu L (2016) An additive-multiplicative mean model for marker data contingent on recurrent event with an informative terminal event. Stat Sin 26:1197–1218MathSciNetzbMATHGoogle Scholar
  10. Huang CY, Wang MC (2004) Joint modeling and estimation of recurrent event processes and failure time. J Am Stat Assoc 99:1153–1165MathSciNetCrossRefzbMATHGoogle Scholar
  11. Kalbfleisch JD, Schaubel DE, Ye Y, Gong Q (2013) An estimating function approach to the analysis of recurrent and terminal events. Biometrics 69:366–374MathSciNetCrossRefzbMATHGoogle Scholar
  12. Liang KY, Zeger SL (1986) Longitudinal data analysis using generalized linear models. Biometrika 73:13–22MathSciNetCrossRefzbMATHGoogle Scholar
  13. Lin DY, Wei LJ, Yang I, Ying Z (2000) Semiparametric regression for the mean and rate function of recurrent events. J R Stat Soc B 69:711–730MathSciNetCrossRefzbMATHGoogle Scholar
  14. Lin DY, Ying Z (1995) Semiparametric analysis of general additive–multiplicative hazard models for counting processes. Ann Stat 23:1712–1734MathSciNetCrossRefzbMATHGoogle Scholar
  15. Lin DY, Ying Z (2001) Semiparametric and nonparametric regression analysis of longitudinal data. J Am Stat Assoc 96:103–126MathSciNetCrossRefzbMATHGoogle Scholar
  16. Liu L, Huang X, O’Quigley J (2008) Analysis of longitudinal data in the presence of informative observational times and a dependent terminal event, with application to medical cost data. Biometrics 64:950–958MathSciNetCrossRefzbMATHGoogle Scholar
  17. Liu L, Wolfe RA, Huang X (2004) Shared frailty models for recurrent events and a terminal event. Biometrics 60:747–756MathSciNetCrossRefzbMATHGoogle Scholar
  18. Liu Y, Wu Y, Cai J, Zhou H (2010) Additive-multiplicative rates model for recurrent events. Lifetime Data Anal 16:353–373MathSciNetCrossRefzbMATHGoogle Scholar
  19. Martinussen T, Scheike TH (2002) A flexible additive multiplicative hazard model. Biometrika 89:283–298MathSciNetCrossRefzbMATHGoogle Scholar
  20. Miloslavsky M, Keles S, Van der Laan MJ, Butler S (2004) Recurrent events analysis in the presence of time-dependent covariates and dependent censoring. J R Stat Soc B 66:239–257MathSciNetCrossRefzbMATHGoogle Scholar
  21. Pan Q, Schaubel DE (2009) Flexible estimation of differences in treatment-specific recurrent event means in the presence of a terminating event. Biometrics 65:753–761MathSciNetCrossRefzbMATHGoogle Scholar
  22. Rondeau V, Mathoulin-Pelissier S, Jacqmin-Gadda H, Brouste V, Soubeyran P (2007) Joint frailty models for recurring events and death using maximum penalized likelihood estimation: application on cancer events. Biostatistics 8:708–721CrossRefzbMATHGoogle Scholar
  23. Rondeau V, Mazroui Y, Gonzalez JR (2012) frailtypack: an R package for the analysis of correlated survival data with frailty models using penalized likelihood estimation or parametrical estimation. J Stat Softw 47:1–28CrossRefGoogle Scholar
  24. Schaubel DE, Zeng D, Cai J (2006) A semiparametric additive rates model for recurrent event data. Lifetime Data Anal 12:389–406MathSciNetCrossRefzbMATHGoogle Scholar
  25. Schaubel DE, Zhang M (2010) Estimating treatment effects on the marginal recurrent event mean in the presence of a terminating event. Lifetime Data Anal 16:451–477MathSciNetCrossRefzbMATHGoogle Scholar
  26. Scheike TH, Zhang MJ (2002) An additive-multiplicative Cox–Aalen regression model. Scand J Stat 29:75–88MathSciNetCrossRefzbMATHGoogle Scholar
  27. Sun L, Kang F (2013) An additive-multiplicative rates model for recurrent event data with informative terminal event. Lifetime Data Anal 19:117–137MathSciNetCrossRefzbMATHGoogle Scholar
  28. Sun L, Song X, Zhou J, Liu L (2012) Joint analysis of longitudinal data with informative observation times and a dependent terminal event. J Am Stat Assoc 107:688–700MathSciNetCrossRefzbMATHGoogle Scholar
  29. van der Vaart AW, Wellner JA (1996) Weak convergence and empirical processes. Springer, New YorkCrossRefzbMATHGoogle Scholar
  30. Volinsky CT, Raftery AE (2000) Bayesian information criterion for censored survival models. Biometrics 56:256–262CrossRefzbMATHGoogle Scholar
  31. Wang MC, Qin J, Chiang CT (2001) Analyzing recurrent event data with informative censoring. J Am Stat Assoc 96:1057–1065MathSciNetCrossRefzbMATHGoogle Scholar
  32. Ye Y, Kalbfleisch JD, Schaubel DE (2007) Semiparametric analysis of correlated recurrent and terminal events. Biometrics 63:78–87MathSciNetCrossRefzbMATHGoogle Scholar
  33. Zeng D, Cai J (2010) Semiparametric additive rate model for recurrent events with informative terminal event. Biometrika 97:699–712MathSciNetCrossRefzbMATHGoogle Scholar
  34. Zeng D, Lin DY (2006) Efficient estimation of semiparametric transformation models for counting processes. Biometrika 93:627–640MathSciNetCrossRefzbMATHGoogle Scholar
  35. Zeng D, Lin DY (2009) Semiparametric transformation models with random effects for joint analysis of recurrent and terminal events. Biometrics 65:746–752MathSciNetCrossRefzbMATHGoogle Scholar
  36. Zhao X, Zhou J, Sun L (2011) Semiparametric transformation models with time-varying coefficients for recurrent and terminal events. Biometrics 67:404–414MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Statistics and ManagementShanghai University of Finance and EconomicsShanghaiPeople’s Republic of China
  2. 2.Institute of Applied Mathematics, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingPeople’s Republic of China
  3. 3.University of Chinese Academy of SciencesBeijingPeople’s Republic of China
  4. 4.School of Statistics and MathematicsCentral University of Finance and EconomicsBeijingPeople’s Republic of China

Personalised recommendations