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Shared Gamma Frailty Models

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Modeling Survival Data Using Frailty Models

Part of the book series: Industrial and Applied Mathematics ((INAMA))

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Abstract

The shared gamma frailty model is a more widely used frailty distribution in the literature. In this chapter, we consider frailty distribution as gamma distribution because as the gamma variates are positive, it fits the nonnegative criterion of frailties with no transformation.

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References

  • Arnold, B.C., Beaver, R.J.: The skew Cauchy distribution. Stat. Probab. Lett. 49, 285–290 (2000)

    Article  MathSciNet  Google Scholar 

  • Bennett, S.: Log-logistic regression model for survival data. Appl. Stat. 32(2), 165–171 (1983)

    Article  Google Scholar 

  • Burr, I.W.: Cumulative frequency distribution. Ann. Math. Stat. 13, 215–232 (1942)

    Article  Google Scholar 

  • Clayton, D.G.: A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika 65, 141–151 (1978)

    Article  MathSciNet  Google Scholar 

  • Cox, D.R., Snell, E.J.: Analysis of Binary Data. Chapman and Hall, New York (1989)

    Google Scholar 

  • El-Saidi, M.A., Singh, K.P., Bartolucci, A.A.: A note on a characterization of the generalized log-logistic distribution. Environmetrics 1(4), 337–342 (1990)

    Article  Google Scholar 

  • Gelman, A., Rubin, D.B.: A single series from the Gibbs sampler provides a false sense of security. In: Bernardo, J.M., Berger, J.O., Dawid, A.P., Smith, A.F.M., (eds.), Bayesian Statistics, vol. 4, pp. 625–632. Oxford University Press, Oxford (1992)

    Google Scholar 

  • Geweke, J.: Evaluating the Accuracy of Sampling-Based Approaches to the Calculation of Posterior Moments. In: Bernardo, J.M., Berger, J., Dawid, A.P., Smith, A.F.M. (eds.) Bayesian Statistics, vol. 4, pp. 169–193. Oxford University Press, Oxford (1992)

    Google Scholar 

  • Gupta, R.D., Kundu, D.: A new class of weighted exponential distributions, statistics. J. Theor. Appl. Stat. 43(6), 621–634 (2009)

    MATH  Google Scholar 

  • Hanagal, D.D.: A positive stable frailty regression model in bivariate survival data. J. Indian. Soc. Probab. Stat. 19, 35–44 (2005)

    Google Scholar 

  • Hanagal, D.D.: A gamma frailty regression model in bivariate survival data. IAPQR Trans. 31, 73–83 (2006)

    Google Scholar 

  • Hanagal, D.D.: Positive stable frailty regression models in mixture distributions. In: Proceedings of 3rd International Conference on Reliability and Safety Engineering, 17–19, December, 2007, held at Indian Institute of Technology, Udaipur, India, pp. 350–356 (2007a)

    Google Scholar 

  • Hanagal, D.D.: Gamma frailty regression models in mixture distributions. Econ. Qual. Control. 22(2), 295–302 (2007b)

    Google Scholar 

  • Hanagal,D.D.: Modeling heterogeneity for bivariate survival data by the lognormal distribution. Stat. Probab. Lett. 78(9), 1101–1109 (2008)

    Article  MathSciNet  Google Scholar 

  • Hanagal, D.D.: Correlated compound Poisoon frailty model for the bivariate survival data. Int. J. Stat. Manag. Syst. 5, 127–140 (2010)

    Google Scholar 

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

    Book  Google Scholar 

  • Hanagal, D.D.: Frailty models in public health. In: Rao, A.S.R.S., Pyne S., Rao, C.R. (eds.) Handbook of Statistics. Disease Modelling and Public Health, vol. 37(B), pp. 209–247. Elsevier, Amsterdam (2017)

    Google Scholar 

  • Hanagal, D.D., Bhambure, S.M.: Analysis of kidney infection data using positive stable frailty models. Adv. Reliab. 1, 21–39 (2014)

    Google Scholar 

  • Hanagal, D.D., Bhambure, S.M.: Comparison of shared gamma frailty models using Bayesian approach. Model Assist. Stat. Appl. 10, 25–41 (2015)

    Google Scholar 

  • 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)

    Article  MathSciNet  Google Scholar 

  • Hanagal, D.D., Dabade, A.D.: Modeling hetrogeneity in bivariate survival data by compound Poisson distribution using Bayesian approach. Int. J. Stat. Manag. Syst. 7(1–2), 36–84 (2012)

    Google Scholar 

  • Hanagal, D.D., Dabade, A.D.: Modeling of inverse Gaussian frailty model for bivariate survival data. Commun. Stat. Theory Methods 42(20), 3744–3769 (2013a)

    Article  MathSciNet  Google Scholar 

  • Hanagal, D.D., Dabade, A.D.: Bayesian estimation of parameters and comparison of shared gamma frailty models. Commun. Stat. Simul. Comput. 42(4), 910–931 (2013b)

    Article  MathSciNet  Google Scholar 

  • Hanagal, D.D., Dabade, A.D.: Compound negative binomial shared frailty models for bivariate survival data. Stat. Probab. Lett. 83, 2507–2515 (2013c)

    Article  MathSciNet  Google Scholar 

  • Hanagal, D.D., Dabade, A.D.: A comparative study of shared frailty models for bivariate survival data with generalized exponential baseline distribution. J. Data Sci. 11, 109–142 (2013d)

    Google Scholar 

  • Hanagal, D.D., Dabade, A.D.: Comparisons of frailty models for kidney infection data under Weibull baseline distribution. Int. J. Math. Model. Numer. Optim. 5(4), 342–373 (2014)

    MATH  Google Scholar 

  • Hanagal, D.D., Dabade, A.D.: Comparisons of frailty models for kidney infection data under exponential power baseline distribution. Commun. Stat. Theory Methods 44(23), 5091–5108 (2015)

    Article  MathSciNet  Google Scholar 

  • Hanagal, D.D., Kamble, A.T.: Bayesian estimation in shared gamma frailty models. Int. J. Stat. Reliab. Eng. 1(2), 88–112 (2014a)

    Google Scholar 

  • Hanagal, D.D., Kamble, A.T.: Bayesian estimation in shared inverse Gaussian frailty models. Int. J. Stat. 1, 9–20 (2014b)

    Google Scholar 

  • Hanagal, D.D., Kamble, A.T.: Bayesian estimation in shared positive stable frailty models. J. Data Sci. 13, 615–640 (2014c)

    Google Scholar 

  • Hanagal, D.D., Kamble, A.T.: Bayesian estimation in shared compound Poisson frailty models. J. Reliab. Stat. Stud. 8(1), 159–180 (2015)

    Google Scholar 

  • Hanagal, D.D., Kamble, A.T.: Bayesian estimation in shared compound negative binomial frailty models. Res. Rev. J. Stat. Math. Sci. 2(1), 53–67 (2016)

    Google Scholar 

  • Hanagal, D.D., Pandey, A.: Inverse Gaussian shared frailty for modeling kidney infection data. Adv. Reliab. 1, 1–14 (2014)

    Google Scholar 

  • Hanagal, D.D., Pandey, A.: Gamma frailty models for bivariate survival data. J. Stat. Comput. Simul. 85(15), 3172–3189 (2015a)

    Article  MathSciNet  Google Scholar 

  • Hanagal, D.D., Pandey, A.: Gamma frailty model based on reversed hazard rate. Commun. Stat. Theory Methods 45(7), 2071–2088 (2015b)

    Google Scholar 

  • Hanagal, D.D., Pandey, A.: Gamma shared frailty model based on reversed hazard rate. Commun. Stat., Theory Methods 45(7), 2071–2088 (2016)

    Article  MathSciNet  Google Scholar 

  • Hanagal, D.D., Pandey, A.: Shared frailty models based on reversed hazard rate for modified inverse Weibull distribution as a baseline distribution. Commun. Stat., Theory Methods. 46(1), 234–246 (2017)

    Article  MathSciNet  Google Scholar 

  • Hanagal, D.D., Pandey, A., Sankaran, P.G.: Shared frailty model based on reversed hazard rate for left censoring data. Commun. Stat. Simul. Comput. 46(1), 230–243 (2017a)

    Google Scholar 

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

    Google Scholar 

  • Ibrahim, J.G., Chen, M.H., Sinha, D.: Bayesian Survival Analysis. Springer, New York (2001)

    Book  Google Scholar 

  • Jeffreys, H.: Theory of Probability, 3rd edn. Oxford University Press, Oxford (1961)

    MATH  Google Scholar 

  • Jones, M.C.: Families of distributions arising from distributions of order statistics. Test 13, 1–43 (2004)

    Article  MathSciNet  Google Scholar 

  • Kass, R.E., Raftery, A.E.: Bayes factor. J. Am. Stat. Assoc. 90(430), 773–795 (1995)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • McGilchrist, C.A., Aisbett, C.W.: Regression with frailty in survival analysis. Biometrics 47, 461–466 (1991)

    Article  Google Scholar 

  • Mudholkar, G.S., Srivastava, D.K.: Exponentiated Weibull family for analyzing bathtub failure data. IEEE Trans. Reliab. 42, 299–302 (1993)

    Article  Google Scholar 

  • Mudholkar, G.S., Srivastava, D.K., Freimer, M.: The exponential Weibull family: a re-analysis of the bus-motor-failure data. Technometrics 37, 436–445 (1995)

    Article  Google Scholar 

  • Mudholkar, G.S., Srivastava, D.K., Kollia, G.D.: A generalization of the Weibull distribution with application to the analysis of survival data. J. Am. Stat. Assoc. 91(436), 1575–1583 (1996)

    Article  MathSciNet  Google Scholar 

  • Nelsen, R.B.: An Introduction to Copulas, 2nd edn. Springer, New York (2006)

    Google Scholar 

  • O’Quigley, J., Struthers, L.: Survival models based upon the logistic and loglogistic distributions. Comput. Programs Biomed. Res. 15, 3–12 (1982)

    Article  Google Scholar 

  • Raftey, A.E.: Approximate Bayes factors and accounting for model uncertainty in generalized linear models. Biometrika 83(2), 251–266 (1994)

    Article  Google Scholar 

  • Santos, C.A., Achcar, J.A.: A Bayesian analysis for multivariate survival data in the presence of covariates. J. Stat. Theory Appl. 9, 233–253 (2010)

    MathSciNet  Google Scholar 

  • Schwarz, G.: Estimating the dimension of a model. Ann. Stat. 6(2), 461–464 (1978)

    Article  MathSciNet  Google Scholar 

  • Spiegelhalter, D.J., Best, N.G., Carlin, B.P., Van der Linde, A.: Bayesian measure of model complexity and fit (with discussion). J. R. Stat. Soc. B 64, 583–639 (2002)

    Article  MathSciNet  Google Scholar 

  • Surles, J.G., Padgett, W.J.: Inference for reliability and stress-strength for a scaled Burr Type X distribution. Lifetime Data Anal. 7, 187–200 (2001)

    Article  MathSciNet  Google Scholar 

  • Surles, J.G., Padgett, W.J.: Some properties of a scaled Burr type X distribution. J. Stat. Plan. Inference 18(1), 271–280 (2004)

    Article  MathSciNet  Google Scholar 

  • Xie, M., Tang, Y., Goh, T.N.: A modified Weibull extension with bathtub-shaped failure rate function. Reliab. Eng. Syst. Saf. 76(3), 279–285 (2002)

    Article  Google Scholar 

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Correspondence to David D. Hanagal .

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Hanagal, D.D. (2019). Shared Gamma Frailty Models. In: Modeling Survival Data Using Frailty Models. Industrial and Applied Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-15-1181-3_9

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