Lifetime Data Analysis

, Volume 23, Issue 2, pp 223–253 | Cite as

Additive mixed effect model for recurrent gap time data



Gap times between recurrent events are often of primary interest in medical and observational studies. The additive hazards model, focusing on risk differences rather than risk ratios, has been widely used in practice. However, the marginal additive hazards model does not take the dependence among gap times into account. In this paper, we propose an additive mixed effect model to analyze gap time data, and the proposed model includes a subject-specific random effect to account for the dependence among the gap times. Estimating equation approaches are developed for parameter estimation, and the asymptotic properties of the resulting estimators are established. In addition, some graphical and numerical procedures are presented for model checking. The finite sample behavior of the proposed methods is evaluated through simulation studies, and an application to a data set from a clinic study on chronic granulomatous disease is provided.


Additive mixed effect model Estimating equation Gap times Random effects Recurrent event data 



The authors thank the Editor, Professor Mei-Ling Ting Lee, an Associate Editor, and two reviewers for their insightful comments and suggestions that greatly improved the article. This research was partly supported by the National Natural Science Foundation of China Grants (No. 11231010, 11171330 and 11101314), Key Laboratory of RCSDS, CAS (No. 2008DP173182) and BCMIIS.


  1. Andersen PK, Gill RD (1982) Cox’s regression model for counting processes: a large sample study. Ann Stat 10:1100–1120MathSciNetCrossRefMATHGoogle Scholar
  2. Cai J, Zeng D (2011) Additive mixed effect model for clustered failure time data. Biometrics 67:1340–1351MathSciNetCrossRefMATHGoogle Scholar
  3. Chang SH, Wang MC (1999) Conditional regression analysis for recurrence time data. J Am Stat Assoc 94:1221–1230MathSciNetCrossRefMATHGoogle Scholar
  4. Chang SH (2004) Estimating marginal effects in accelerated failure time models for serial sojourn among repeated event. Lifetime Data Anal 10:175–190MathSciNetCrossRefMATHGoogle Scholar
  5. Cook RJ, Lawless JF (2007) The statistical analysis of recurrent events. Springer, New YorkMATHGoogle Scholar
  6. Cook RJ, Lawless JF, Lakhal-Chaieb L, Lee K-A (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 104:60–75MathSciNetCrossRefMATHGoogle Scholar
  7. Darlington GA, Dixon SN (2013) Event-weighted proportional hazards modelling for recurrent gap time data. Stat Med 32:124–130MathSciNetCrossRefGoogle Scholar
  8. Du P (2009) Nonparametric modeling of the gap time in recurrent event data. Lifetime Data Anal 15:256–277MathSciNetCrossRefMATHGoogle Scholar
  9. Fleming TR, Harrington DP (1991) Counting processes and survival analysis. Wiley, New YorkMATHGoogle Scholar
  10. Gail MH, Santner TJ, Brown CC (1980) An analysis of comparative carcinogenesis experiments based on multiple times to tumor. Biometrics 36:255–266MathSciNetCrossRefMATHGoogle Scholar
  11. Huang Y, Chen YQ (2003) Marginal regression of gaps between recurrent events. Lifetime Data Anal 9:293–303MathSciNetCrossRefMATHGoogle Scholar
  12. Huang X, Liu L (2007) A joint frailty model for survival and gap times between recurrent events. Biometrics 63:389–397MathSciNetCrossRefMATHGoogle Scholar
  13. Kalbfleisch JD, Prentice RL (2002) The statistical analysis of failure time data, 2nd edn. Wiley, New YorkCrossRefMATHGoogle Scholar
  14. Liang KY, Zeger SL (1986) Longitudinal data analysis using generalized linear models. Biometrika 73:13–22MathSciNetCrossRefMATHGoogle Scholar
  15. Lin DY, Sun W, Ying Z (1999) Nonparametric estimation of the gap time distribution for serial events with censored data. Biometrika 86:59–70MathSciNetCrossRefMATHGoogle Scholar
  16. 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 69:711–730MathSciNetCrossRefMATHGoogle Scholar
  17. Lin DY, Wei LJ, Ying Z (1998) Accelerated failure time models for counting processes. Biometrika 85:605–618MathSciNetCrossRefMATHGoogle Scholar
  18. Lin DY, Wei LJ, Ying Z (2001) Semiparametric transformationmodels for point processes. J Am Stat Assoc 96:620–628CrossRefMATHGoogle Scholar
  19. Lin DY, Ying Z (1994) Semiparametric analysis of the additive risk model. Biometrika 81:61–71MathSciNetCrossRefMATHGoogle Scholar
  20. Lu W (2005) Marginal regression of multivariate event times based on linear transformation models. Lifetime Data Anal 11:389–404MathSciNetCrossRefMATHGoogle Scholar
  21. Luo X, Huang CY (2011) Analysis of recurrent gap time data using the weighted risk-set method and the modified within-cluster resampling method. Stat Med 30:301–311MathSciNetCrossRefGoogle Scholar
  22. Luo X, Huang CY, Wang L (2013) Quantile regression for recurrent gap time data. Biometrics 69:375–385MathSciNetCrossRefMATHGoogle Scholar
  23. McKeague IW, Sasieni PD (1994) A partly parametric additive risk model. Biometrika 81:501–514MathSciNetCrossRefMATHGoogle Scholar
  24. Peña EA, Strawderman R, Hollander M (2001) Nonparametric estimation with recurrent event data. J Am Stat Assoc 96:1299–1315MathSciNetCrossRefMATHGoogle Scholar
  25. Pepe MS, Cai J (1993) Some graphical displays andmarginal regression analyses for recurrent failure times and time-dependent covariates. J Am Stat Assoc 88:811–820CrossRefMATHGoogle Scholar
  26. Pollard D (1990) Empirical processes: theory and applications. Institute of Mathematical Statistics, HaywardMATHGoogle Scholar
  27. Schaubel DE, Cai J (2004) Regression analysis for gap time hazard functions of sequentially ordered multivariate failure time data. Biometrika 91:291–303MathSciNetCrossRefMATHGoogle Scholar
  28. Strawderman RL (2005) The accelerated gap times model. Biometrika 92:647–666MathSciNetCrossRefMATHGoogle Scholar
  29. Sun L, Park D, Sun J (2006) The addicitve hazards model for recurrent gap times. Stat Sinica 16:919–932MATHGoogle Scholar
  30. Sun L, Tong X, Zhou X (2011) A class of Box-Cox transformation models for recurrent event data. Lifetime Data Anal 17:280–301MathSciNetCrossRefMATHGoogle Scholar
  31. Sun L, Zhao X, Zhou J (2011) A class of mixed models for recurrent event data. Can J Stat 39:578–590MathSciNetCrossRefMATHGoogle Scholar
  32. van der Vaart AW, Wellner JA (1996) Weak convergence and empirical processes. Springer, New YorkCrossRefMATHGoogle Scholar
  33. Wang MC, Chang SH (1999) Nonparametric estimation of a recurrent survival function. J Am Stat Assoc 94:146–153MathSciNetCrossRefMATHGoogle Scholar
  34. Wang MC, Chen YQ (2000) Nonparametric and semiparametric trend analysis of stratified recurrence time data. Biometrics 56:789–794CrossRefMATHGoogle Scholar
  35. Yin G (2007) Model checking for additvie hazards model with multivariate survival data. J Multivar Anal 98:1018–1032CrossRefMATHGoogle Scholar
  36. Yin G, Cai J (2004) Additive hazards model with multivariate failure time data. Biometrika 91:801–818MathSciNetCrossRefMATHGoogle Scholar
  37. Zeng D, Cai J (2010) Semiparametric additive rate model for recurrent events with informative terminal event. Biometrika 97:699–712MathSciNetCrossRefMATHGoogle Scholar
  38. Zeng D, Lin DY (2007) Semiparametric transformation models with random effects for recurrent events. J Am Stat Assoc 102:167–180MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsWuhan UniversityWuhanPeople’s Republic of China
  2. 2.Institute of Applied Mathematics, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingPeople’s Republic of China

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