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Bivariate Frailty Model and Association Measure

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Mathematical and Statistical Applications in Life Sciences and Engineering

Abstract

In this paper, we present a bivariate frailty model and the association measure. The relationship between the conditional and the unconditional hazard gradients are derived and some examples are provided. A correlated frailty model is presented and its application in the competing risk theory is given. Some applications to real data sets are also pointed out.

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References

  1. Agresti, A., B. Caffo, and P. Ohman-Strickland. 2004. Examples in which misspecification of a random effects distribution reduces efficiency and possible remedies. Compuatational Statistics and Data Analysis 47: 639–653.

    Article  MathSciNet  MATH  Google Scholar 

  2. Clayton, D.G. 1978. A model for association in bivariate life tables and its application in epidemiological studies of family tendancy in chronic disease incidence. Biometrika 65: 141–151.

    Article  MathSciNet  MATH  Google Scholar 

  3. Clayton, D.G., and J. Cuzick. 1985. Multivariate generalization of the proprtional hazard model. Journal of the Royal Statistical Society, Sereis A 148: 82–117.

    Article  MATH  Google Scholar 

  4. Duchateau, L., and P. Janssen. 2008. The Frailty Model. NY: Springer.

    MATH  Google Scholar 

  5. Finkelstein, M.S., and V. Esaulova. 2006. Asymptotic behaviour of a general class of mixture failure rates. Advances in Applied Probability 38 (1): 244–262.

    Article  MathSciNet  MATH  Google Scholar 

  6. Gupta, P.L., and R.C. Gupta. 1996. Ageing charaterstics of the Weibull mixtures. Probability in the Engineering and Informational Sciences 10: 591–600.

    Google Scholar 

  7. Gupta, R.C., and R.D. Gupta. 2009. General frailty model and stochastic orderings. Journal of Statistical Planning and Inference 139: 3277–3287.

    Article  MathSciNet  MATH  Google Scholar 

  8. Gupta, R.C., and R.D. Gupta. 2010. Random effect survival models and stochastic comparisons. Journal of Applied Probability 47: 426–440.

    Article  MathSciNet  MATH  Google Scholar 

  9. Gupta, R.C. 2010. Reliability functions of bivariate distributions in modeling marked point processes. Stochastic Models 26 (2): 195–211.

    Article  MathSciNet  MATH  Google Scholar 

  10. Gupta, R.C., and S.N. Kirmani. 2006. Stochastic comparisons in frailty models. Journal of Statistical Planning and Inference 136: 3647–3658.

    Google Scholar 

  11. Gupta, R.C. 2016. Properties of additive frailty model in survival analysis. Metrika 79: 1–17.

    Article  MathSciNet  MATH  Google Scholar 

  12. Hens, N., A. Wienke, M. Aerts, and G. Molenberghs. 2009. The correlated and shared gamma frailty model for bivariate current status data: an illustration for cross-sectional serological data. Statistics in Medicine 28: 2785–2800.

    Article  MathSciNet  Google Scholar 

  13. Hanagal, D.D. 2011. Modeling Survival Data Using Frailty Models. Boca Raton, FL: Chapman and Hall.

    Book  Google Scholar 

  14. Hanagal, D.D., and A.D. Dabade. 2015. Comparison of shared frailty models for kidney infection data under exponential power baseline distribution. Communications in Statistics, Theory and Methods 44: 5091–5108.

    Article  MathSciNet  MATH  Google Scholar 

  15. Heckman, J.J., and B. Singer. 1984. The identifibility of the proportional hazard model. Review Economic Studies 231–241.

    Google Scholar 

  16. Hougaard, P. 1984. Lifetable methods for heterogeneous populations: distributions describing the heterogeneity. Biometrika 71: 75–83.

    Article  MathSciNet  MATH  Google Scholar 

  17. Hougaard, P. 1991. Modeling hetereogeneity in survival data. Journal of Applied Probability 28: 695–701.

    Article  MathSciNet  MATH  Google Scholar 

  18. Hougaard, P. 1995. Frailty models for survival data. Lifetime Data Analysis 1: 255–273.

    Article  Google Scholar 

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

    Book  MATH  Google Scholar 

  20. Kersey, H., D. Weisdorf, M.E. Nesbit, T.W. Lebien, P.B. McGlave, T. Kim, D.A. Vellera, A.I. Goldman, B. Bostrom, D. Hurd, and N.K.C. Ramsay. 1987. Comparison of autologous and allogenic bone merrow transplantation for treatment of high-risk refractory acute lymphoblastic leukemia. New England Journal of Medicine 317: 461–467.

    Article  Google Scholar 

  21. Keiding, N., and P.K. Andersen. 1997. The role of frailty models and accelerated failure time models in describing heterogeneity due to omitted covariates. Statistics in Medicine 16: 215–224.

    Google Scholar 

  22. Korsgaard, I.R., and A.H. Anderson. 1998. The additive genetic gamma frailty model. Scandinavian Journal of Statistics 25: 255–269.

    Article  MathSciNet  MATH  Google Scholar 

  23. Lambert, P., D. Collett, A. Kimber, and R. Johnson. 2004. Parametric accelerated failure time models with random effects and an application to kidney transplant survival. Statistics in Medicine 23: 3177–3192.

    Article  Google Scholar 

  24. Liang, K.Y., S.G. Self, K.J. Bandeen-Roche, and S. Zeger. 1995. Some recent developments for regression analysis of multivariate failure time data. Lifetime Data Analysis 1: 403–406.

    Google Scholar 

  25. Manatunga, A.K., and D. Oakes. 1996. A measure of association for bivariate frailty ditributions. Journal of Multivariate Analysis 56: 60–74.

    Article  MathSciNet  MATH  Google Scholar 

  26. Missov, T.I., and M.S. Finkelstein. 2011. Admissible mixing distributions for a general class of mixture survival models with known asymptotics. Theoretical Population Biology 80 (1): 64–70.

    Article  Google Scholar 

  27. Rizopoulos, D., G. Verbke, and G. Molenberghs. 2008. Shared parameter models under random effect misspecification. Biometrika 95 (1): 63–74.

    Article  MathSciNet  MATH  Google Scholar 

  28. Sargent, D.J. 1998. A general framework for random effects survival analysis in the Cox proportional hazards setting. Biometrics 54: 1486–1497.

    Article  MATH  Google Scholar 

  29. Oakes, D. 1989. Bivariate survival models induced by frailties. Journal of the American Statistical Association 84: 497–493.

    Article  MathSciNet  MATH  Google Scholar 

  30. Pan, W. 2001. Using frailties in the accelarated failure time model (2001). Lifetime Data Analysis 7: 55–64.

    Article  MathSciNet  MATH  Google Scholar 

  31. Price, D.L., and A.K. Manatunga. 2001. Modelling survival data with a cure fraction using frailty models. Statistics in Medicine 20: 1515–1527.

    Article  Google Scholar 

  32. Vaupel, J.W., K.G. Manton, and E. Sttalard. 1979. The impact of heterogeneity in individual frailty on the dynamics of mortality. Demography 16 (3): 439–454.

    Article  Google Scholar 

  33. Wienke, A., P. Lichtenstein, and A.I. Yashin. 2003. A bivariate frailty model with a cure fraction for modelling familial correlations in diseases. Biometrics 59: 1178–1183.

    Article  MathSciNet  MATH  Google Scholar 

  34. Wienke, A., I. Licantelli, and A.I. Yashin. 2006. the modeling of a cure fraction in bivariate time to event data. Austrian Journal of Statistics 35 (1): 67–76.

    Article  Google Scholar 

  35. Wienke, A. 2010. Frailty Models in Survival Analysis. Boca Raton: Chapman & Hall/CRC.

    Book  Google Scholar 

  36. Xue, X., and A.Y. Ding. 1999. Assesing heterogeneity and correlation of paired failure times with the bivariate frailty model. Statistics in Medicine 18: 907–918.

    Article  Google Scholar 

  37. Yin, G., and J.G. Ibrahim. 2005. A class of Bayesian shared gamma frailty models with multivariate failure time data. Biometrics 61: 208–216.

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author is thankful to the referee for some useful suggestions which enhanced the presentation.

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Authors and Affiliations

  1. University of Maine, Orono, ME, USA

    Ramesh C. Gupta

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  1. Ramesh C. Gupta
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Correspondence to Ramesh C. Gupta .

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Editors and Affiliations

  1. Department of Pure Mathematics, University of Calcutta, Kolkata, West Bengal, India

    Avishek Adhikari

  2. IMBIC, Kolkata, West Bengal, India

    Mahima Ranjan Adhikari

  3. Department of Mathematics and Statistics, Concordia University, Montreal, Québec, Canada

    Yogendra Prasad Chaubey

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Gupta, R.C. (2017). Bivariate Frailty Model and Association Measure. In: Adhikari, A., Adhikari, M., Chaubey, Y. (eds) Mathematical and Statistical Applications in Life Sciences and Engineering. Springer, Singapore. https://doi.org/10.1007/978-981-10-5370-2_10

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