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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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.
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.
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.
Duchateau, L., and P. Janssen. 2008. The Frailty Model. NY: Springer.
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.
Gupta, P.L., and R.C. Gupta. 1996. Ageing charaterstics of the Weibull mixtures. Probability in the Engineering and Informational Sciences 10: 591–600.
Gupta, R.C., and R.D. Gupta. 2009. General frailty model and stochastic orderings. Journal of Statistical Planning and Inference 139: 3277–3287.
Gupta, R.C., and R.D. Gupta. 2010. Random effect survival models and stochastic comparisons. Journal of Applied Probability 47: 426–440.
Gupta, R.C. 2010. Reliability functions of bivariate distributions in modeling marked point processes. Stochastic Models 26 (2): 195–211.
Gupta, R.C., and S.N. Kirmani. 2006. Stochastic comparisons in frailty models. Journal of Statistical Planning and Inference 136: 3647–3658.
Gupta, R.C. 2016. Properties of additive frailty model in survival analysis. Metrika 79: 1–17.
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.
Hanagal, D.D. 2011. Modeling Survival Data Using Frailty Models. Boca Raton, FL: Chapman and Hall.
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.
Heckman, J.J., and B. Singer. 1984. The identifibility of the proportional hazard model. Review Economic Studies 231–241.
Hougaard, P. 1984. Lifetable methods for heterogeneous populations: distributions describing the heterogeneity. Biometrika 71: 75–83.
Hougaard, P. 1991. Modeling hetereogeneity in survival data. Journal of Applied Probability 28: 695–701.
Hougaard, P. 1995. Frailty models for survival data. Lifetime Data Analysis 1: 255–273.
Hougaard, P. 2000. Analysis of Multivariate Survival Data. New York: Springer.
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.
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.
Korsgaard, I.R., and A.H. Anderson. 1998. The additive genetic gamma frailty model. Scandinavian Journal of Statistics 25: 255–269.
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.
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.
Manatunga, A.K., and D. Oakes. 1996. A measure of association for bivariate frailty ditributions. Journal of Multivariate Analysis 56: 60–74.
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.
Rizopoulos, D., G. Verbke, and G. Molenberghs. 2008. Shared parameter models under random effect misspecification. Biometrika 95 (1): 63–74.
Sargent, D.J. 1998. A general framework for random effects survival analysis in the Cox proportional hazards setting. Biometrics 54: 1486–1497.
Oakes, D. 1989. Bivariate survival models induced by frailties. Journal of the American Statistical Association 84: 497–493.
Pan, W. 2001. Using frailties in the accelarated failure time model (2001). Lifetime Data Analysis 7: 55–64.
Price, D.L., and A.K. Manatunga. 2001. Modelling survival data with a cure fraction using frailty models. Statistics in Medicine 20: 1515–1527.
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.
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.
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.
Wienke, A. 2010. Frailty Models in Survival Analysis. Boca Raton: Chapman & Hall/CRC.
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.
Yin, G., and J.G. Ibrahim. 2005. A class of Bayesian shared gamma frailty models with multivariate failure time data. Biometrics 61: 208–216.
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The author is thankful to the referee for some useful suggestions which enhanced the presentation.
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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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DOI: https://doi.org/10.1007/978-981-10-5370-2_10
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