Frailty modelling approaches for semi-competing risks data

  • Il Do HaEmail author
  • Liming Xiang
  • Mengjiao Peng
  • Jong-Hyeon Jeong
  • Youngjo Lee


In the semi-competing risks situation where only a terminal event censors a non-terminal event, observed event times can be correlated. Recently, frailty models with an arbitrary baseline hazard have been studied for the analysis of such semi-competing risks data. However, their maximum likelihood estimator can be substantially biased in the finite samples. In this paper, we propose effective modifications to reduce such bias using the hierarchical likelihood. We also investigate the relationship between marginal and hierarchical likelihood approaches. Simulation results are provided to validate performance of the proposed method. The proposed method is illustrated through analysis of semi-competing risks data from a breast cancer study.


Frailty models Hierarchical likelihood Marginal likelihood Modified likelihood Semi-competing risks 



Dr. Ha’s research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. NRF-2015R1D1A3A01015663). Dr. Xiang’s research was supported in part by the Singapore MOE AcRF (MOE2013-T2-2-118). Dr. Jeong’s research was supported in part by National Institute of Health (NIH) grants 5-U10-CA69651-11. Dr. Lee’s research was supported by an NRF grant funded by Korea government (MEST) (No. 2011-0030810).

Supplementary material

10985_2019_9464_MOESM1_ESM.pdf (64 kb)
Supplementary material 1 (pdf 64 KB)


  1. Aalen O, Borgan O, Gjessing HK (2008) Survival and event history analysis. Springer, New YorkCrossRefzbMATHGoogle Scholar
  2. Andersen PK, Klein JP, Knudsen K, Palacios RT (1997) Estimation of variance in Cox’s regression model with shared gamma frailties. Biometrics 53:1475–1484MathSciNetCrossRefzbMATHGoogle Scholar
  3. Barker P, Henderson R (2005) Small sample bias in the gamma frailty model for univariate survival. Lifetime Data Anal 11:265–284MathSciNetCrossRefzbMATHGoogle Scholar
  4. Breslow NE (1974) Covariance analysis of censored survival data. Biometrics 30:89–99CrossRefGoogle Scholar
  5. Chen YH (2012) Maximum likelihood analysis of semicompeting risks data with semiparametric regression models. Lifetime Data Anal 18:36–57MathSciNetCrossRefzbMATHGoogle Scholar
  6. Engel E, Keen A (1996) Discussion of Lee and Nelder’s paper. J R Stat Soc Ser B 58:656–657Google Scholar
  7. Fan J, Li R (2002) Variable selection for Cox’s proportional hazards model and frailty model. Ann Stat 30:74–99MathSciNetCrossRefzbMATHGoogle Scholar
  8. Fine JP, Jiang H, Chappell R (2001) On semi-competing risks data. Biometrika 88:907–919MathSciNetCrossRefzbMATHGoogle Scholar
  9. Fisher B, Costantino J, Redmond C et al (1989) A randomized clinical trial evaluating tamoxifen in the treatment of patients with node-negative breast cancer who have estrogen receptor-positive tumors. N Engl J Med 320:479–484CrossRefGoogle Scholar
  10. Fisher B, Dignam J, Bryant J et al (1996) Five versus more than five years of tamoxifen therapy for breast cancer patients with negative lymph nodes and estrogen receptor- positive tumors. J Natl Cancer Inst 88:1529–1542CrossRefGoogle Scholar
  11. Gu MG, Sun L, Huang C (2004) A universal procedure for parametric frailty models. J Stat Comput Simul 74:1–13MathSciNetCrossRefzbMATHGoogle Scholar
  12. Ha ID, Lee Y (2003) Estimating frailty models via Poisson hierarchical generalized linear models. J Comput Graph Stat 12:663–681MathSciNetCrossRefGoogle Scholar
  13. Ha ID, Lee Y, Song JK (2001) Hierarchical likelihood approach for frailty models. Biometrika 88:233–243MathSciNetCrossRefzbMATHGoogle Scholar
  14. Ha ID, Lee Y, Pawitan Y (2007) Genetic mixed linear models for twin survival data. Behav Genet 37:621–630CrossRefGoogle Scholar
  15. Ha ID, Noh M, Lee Y (2010) Bias reduction of likelihood estimators in semiparametric frailty models. Scand J of Stat 37:307–320MathSciNetCrossRefzbMATHGoogle Scholar
  16. Ha ID, Sylvester R, Legrand C, MacKenzie G (2011) Frailty modelling for survival data from multi-centre clinical trials. Stat Med 30:2144–2159MathSciNetCrossRefGoogle Scholar
  17. Ha ID, Jeong J-H, Lee Y (2017) Statistical modelling of survival data with random effects: h-likelihood approach. Springer, SingaporeCrossRefzbMATHGoogle Scholar
  18. Heize G, Schemper M (2001) A solution to the problem of monotone likelihood in Cox regression. Biometrics 57:114–119MathSciNetCrossRefzbMATHGoogle Scholar
  19. Lawless JF (2003) Statistical models and methods for lifetime data, 2nd edn. Wiley, New YorkzbMATHGoogle Scholar
  20. Lee Y, Nelder JA (1996) Hierarchical generalized linear models (with discussion). J R Stat Soc Ser B 58:619–678zbMATHGoogle Scholar
  21. Lee Y, Nelder JA (2001) Hierarchical generalised linear models: a synthesis of generalised linear models, random-effect models and structured dispersions. Biometrika 88:987–1006MathSciNetCrossRefzbMATHGoogle Scholar
  22. Lee Y, Nelder JA (2009) Likelihood inference for models with unobservables: another view (with discussion). Stat Sci 24:255–293CrossRefzbMATHGoogle Scholar
  23. Lee KH, Haneuse S, Schrag D, Dominici F (2015) Bayesian semiparametric analysis of semicompeting risks data: investigating hospital readmission after a pancreatic cancer diagnosis. J R Stat Soc Ser C 64:253–273MathSciNetCrossRefGoogle Scholar
  24. Lee Y, Nelder JA, Pawitan Y (2017) Generalised linear models with random effects: unified analysis via h-likelihood, 2nd edn. Chapman and Hall, Baca RatonzbMATHGoogle Scholar
  25. Neyman J, Scott EL (1948) Consistent estimates based on partially consistent observations. Econometrica 16:1–32MathSciNetCrossRefzbMATHGoogle Scholar
  26. Noh M, Lee Y (2007) REML estimation for binary data in GLMMs. J Multivar Anal 98:896–915MathSciNetCrossRefzbMATHGoogle Scholar
  27. Ripatti S, Palmgren J (2000) Estimation of multivariate frailty models using penalized partial likelihood. Biometrics 56:1016–1022MathSciNetCrossRefzbMATHGoogle Scholar
  28. Self SG, Liang KY (1987) Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions. J Am Stat Assoc 82:605–610MathSciNetCrossRefzbMATHGoogle Scholar
  29. Stram DO, Lee JW (1994) Variance components testing in the longitudinal mixed effects model. Biometrics 50:1171–1177CrossRefzbMATHGoogle Scholar
  30. Therneau TM, Grambsch PM (2000) Modelling survival data: extending the Cox model. Springer, New YorkCrossRefzbMATHGoogle Scholar
  31. Therneau TM, Grambsch PM, Pankratz VS (2003) Penalized survival models and frailty. J Comput Graph Stat 12:156–175MathSciNetCrossRefGoogle Scholar
  32. Tierney L, Kadane JB (1986) Accurate approximations for posterior moments and marginal densities. J Am Stat Assoc 81:82–86MathSciNetCrossRefzbMATHGoogle Scholar
  33. Varadhan R, Xue QL, Bandeen-Roche K (2014) Semicompeting risks in aging research: methods, issues and needs. Lifetime Data Anal 20:538–562MathSciNetCrossRefzbMATHGoogle Scholar
  34. Xu J, Kalbfleisch JD, Tai B (2010) Statistical analysis of illness-death processes and semicompeting risks data. Biometrics 66:716–725MathSciNetCrossRefzbMATHGoogle Scholar
  35. Zhang Y, Chen MH, Ibrahim JG, Zeng D, Chen Q, Pan Z, Xue X (2014) Bayesian gamma frailty models for survival data with semi-competing risks and treatment switching. Lifetime Data Anal 20:76–105MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of StatisticsPukyong National UniversityBusanSouth Korea
  2. 2.School of Physical and Mathematical SciencesNanyang Technological UniversitySingaporeSingapore
  3. 3.Department of BiostatisticsUniversity of PittsburghPittsburghUSA
  4. 4.Department of StatisticsSeoul National UniversitySeoulSouth Korea

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