A comparative study with bootstrap resampling technique to uncover behavior of unconditional hazards and survival functions for gamma and inverse Gaussian frailty models


Applications of misspecified models in the field of survival analysis particularly frailty models may result in poor generalization and biases. Since gamma and inverse Gaussian distributions are often used interchangeably as frailty distributions for heterogeneous survival data, clear distinction between them is necessary. Based on closed form expressions of unconditional hazards and survival functions, in this paper we compare the effectiveness of gamma and inverse Gaussian distributions for frailty term in modeling survival data for heterogeneous populations. Different baseline hazards were considered including exponential, Weibull and Gompertz. We derived the closed form expressions for unconditional hazards and survival functions under each baseline distribution for both gamma and inverse Gaussian frailty models. Both graphical and extensive statistical simulation approaches are applied to compare the models. For the inference purpose, real data concerning the East Coast Fever (ECF) transmission dynamics is applied. General overview from the graphical analysis and results from both real and synthetic data indicate that gamma distribution under the Gompertz and Weibull baseline hazards is better compared to inverse Gaussian in modeling survival data for a heterogeneous population. Simulation, graphical and inferential analyses were done using appropriate packages in R language.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8


  1. 1.

    Vaupel, J.W., Manton, K.G., Stallard, E.: Impact of heterogeneity in individual frailty on the dynamics of mortality. Demography 16, 439–454 (1979)

    Article  Google Scholar 

  2. 2.

    Gupta, R.C., Gupta, R.D.: General frailty model and stochastic orderings. J. Stat. Plann. Inference 139, 3277–3287 (2009)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Hanagal, D.D.: Correlated positive stable frailty models. Commun. Stat. Theory Methods. https://doi.org/10.1080/03610926.2020.1736305 (2020) (in press)

  4. 4.

    Govindarajulu, U.S., Lin, H., Lunetta, K.L., D’Agostino, R.: Frailty models: applications to biomedical and genetic studies. Stat. Med. 30(22), 2754–2764 (2011)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Wienke, A.: Frailty Models in Survival Analysis. Chapman & Hall/CRC Biostatistics Series, New York (2011)

    Google Scholar 

  6. 6.

    Hanagal, D.D., Pandey, A., Ganguly, A.: Correlated gamma frailty models for bivariate survival data. Commun. Stat. Simul. Comput. 46(5), 3627–3644 (2017)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Jiang, X., Liu, W., Zhang, B.: A note on the prediction of frailties with misspecified shared frailty models. J. Stat. Comput. Simul. https://doi.org/10.1080/00949655.2020.1811279 (2020) (in press)

  8. 8.

    Weiss, K.M.: Biodemography of variation in human frailty. Demography 27, 185–206 (1990)

    Article  Google Scholar 

  9. 9.

    Heckman, J.J., Singer, B.: Identifiability of the proportional hazard model. Rev. Econ. Stud. 51, 231–241 (1984)

    Article  Google Scholar 

  10. 10.

    Hougaard, P.: Frailty models for survival data. Lifetime Data Anal. 1, 255–273 (1995)

    Article  Google Scholar 

  11. 11.

    Hougaard, P.: Analysis of Multivariate Survival Data. Springer, New York (2000)

    Google Scholar 

  12. 12.

    Agresti, A., Caffo, B., Ohman-Strickland, P.: Examples in which misspecification of a random effects distribution reduces efficiency, and possible remedies. Comput. Stat. Data. Anal. 47(3), 639–653 (2004)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Hanagal, D.D., Bhambure, S.M.: Modeling bivariate survival data using shared inverse Gaussian frailty model. Commun. Stat. Theory Methods 45(17), 4969–4987 (2016)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Hanagal, D.D., Pandey, A.: Correlated gamma frailty models for bivariate survival data based on reversed hazard rate. Int. J. Data Sci. 2(4), 301–324 (2017)

    Article  Google Scholar 

  15. 15.

    East Coast Fever (ECF) transmission dynamics dataset. http://www.vetstat.ugent.be/research/frailty/datasets/

  16. 16.

    Greenwood, M., Yule, G.: An inquiry into the nature of frequency distributions representative of multiple happenings with particular reference to the occurrence of multiple attacks of disease or of repeated accidents. J. R. Stat. Soc. 83(2), 255–279 (1920)

    Article  Google Scholar 

  17. 17.

    Beard, R.E.: Note on some mathematical mortality models. The lifespan of animals. In: Wolstenholme, G.E.W., O’Conner, M. (eds.) Ciba Foundation Colloquium on Ageing, pp. 302–311. Little Brown and Company, Boston (1959)

    Google Scholar 

  18. 18.

    Congdon, P.: Modeling frailty in area mortality. Stat. Med. 14, 1859–1874 (1995)

    Article  Google Scholar 

  19. 19.

    dos Santos, D.M., Davies, R.B., Francis, B.: Nonparametric hazard versus nonparametric frailty distribution in modelling recurrence of breast cancer. J. Stat. Plan. Inference 47, 111–127 (1995)

    Article  Google Scholar 

  20. 20.

    Duchateau, L., Janssen, P.: Frailty Model. Springer, New York (2008)

    Google Scholar 

  21. 21.

    Pandey, A., Lalpawimawha, R.: Comparison of correlated frailty models. Commun. Stat. Simul. Comput. https://doi.org/10.1080/03610918.2020.1770287 (2020) (in press)

  22. 22.

    Hanagal, D.D.: Modeling Survival Data Using Frailty Models, 2nd edn. Springer Nature, Singapore (2019)

    Google Scholar 

  23. 23.

    Hougaard, P.: Life table methods for heterogeneous populations. Biometrika 71, 75–83 (1984)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Manton, K., Stallard, E., Vaupel, J.: Alternative models for heterogeneity of mortality risks among the aged. J. Am. Stat. Assoc. 81, 635–644 (1986)

    Article  Google Scholar 

  25. 25.

    Klein, J.P., Moeschberger, M., Li, Y.H., Wang, S.T., Flournoy, N.: Estimating random effects in the Framingham Heart Study. In: Klein, J.P., Goel, P.K. (eds.) Survival Analysis: State of the Art. Nato Science (Series E: Applied Sciences), vol. 211. Springer, Dordrecht (1992)

    Google Scholar 

  26. 26.

    Keiding, N., Andersen, P., Klein, J.: The role of frailty models and accelerated failure time models in describing heterogeneity due to omitted. Stat. Med. 16(1–3), 215–224 (1997)

    Article  Google Scholar 

  27. 27.

    Price, D.L., Manatunga, A.K.: Modelling survival data with a cured fraction using frailty models. Stat. Med. 20(9–10), 1515–1527 (2001)

    Article  Google Scholar 

  28. 28.

    Economou, P., Caroni, C.: Graphical tests for the assumption of gamma and inverse Gaussian frailty distributions. Lifetime Data Anal. 11, 565–582 (2005)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Kheiri, S., Kimber, A., Meshani, M.R.: Bayesian analysis of an inverse Gaussian correlated frailty model. Comput. Stat. Data Anal. 51(11), 5317–5326 (2007)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Punzo, A.: A new look at the inverse Gaussian distribution with applications to insurance and economic data. J. Appl. Stat. 46(7), 1260–1287 (2019)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Hanagal, D.D.: Modeling Survival Data Using Frailty Models. Chapman & Hall, New York (2011)

    Google Scholar 

  32. 32.

    Rizzle, M.L.: Statistical Computing with R. Chapman & Hall/CRC The R Series, New York (2007)

    Google Scholar 

  33. 33.

    Boos, D.D.: Introduction to the Bootstrap World. Statistical Science, 18(2), Silver Anniversary of the Bootstrap, 168-174 (2003)

  34. 34.

    Chernick, M.R., LaBudde, R.A.: Introduction to Bootstrap Method with Application to R. Wiley, New York (2011)

    Google Scholar 

Download references


The authors would like to thank the referees for their valuable comments and suggestions to improve the quality of the publication.

Author information



Corresponding author

Correspondence to Nihal Ata Tutkun.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ata Tutkun, N., Marthin, P. A comparative study with bootstrap resampling technique to uncover behavior of unconditional hazards and survival functions for gamma and inverse Gaussian frailty models. Math Sci 15, 99–109 (2021). https://doi.org/10.1007/s40096-020-00366-1

Download citation


  • Frailty
  • Conditional hazards
  • Bootstrap resampling technique
  • Laplace transform

Mathematics Subject Classification

  • 62N01