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Random effect bivariate survival models and stochastic comparisons

Part of: Distribution theory - Probability Survival analysis and censored data Applications

Published online by Cambridge University Press:  14 July 2016

Ramesh C. Gupta  and
Rameshwar D. Gupta
Ramesh C. Gupta*
Affiliation:
University of Maine
Rameshwar D. Gupta*
Affiliation:
University of New Brunswick
*
∗Postal address: Department of Mathematics and Statistics, University of Maine, Orono, ME 04469-5752, USA. Email address: ramesh_gupta@umit.maine.edu
∗∗Postal address: Department of Computer Science and Applied Statistics, University of New Brunswick, Saint John, E2L 4L5, Canada.
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Abstract

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In this paper we propose a general bivariate random effect model with special emphasis on frailty models and environmental effect models, and present some stochastic comparisons. The relationship between the conditional and the unconditional hazard gradients are derived and some examples are provided. We investigate how the well-known stochastic orderings between the distributions of two frailties translate into the orderings between the corresponding survival functions. These results are used to obtain the properties of the bivariate multiplicative model and the shared frailty model.

Keywords

Frailty modelenvironmental effect modelhazard gradientbivarate multiplicative modelshared frailty model

MSC classification

Secondary: 62N99: None of the above, but in this section 62P99: None of the above, but in this section 60E15: Inequalities; stochastic orderings
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 2 , June 2010 , pp. 426 - 440
Copyright
Copyright © Applied Probability Trust 2010 

References

Agresti, A., Caffo, B. and Ohman-Strickland, P. (2004). Examples in which misspecification of a random effects distribution reduces efficiency, and possible remedies. Comput. Statist. Data Anal. 47, 639–653.Google Scholar
Clayton, D. G. (1978). 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.Google Scholar
Clayton, D. and Cuzick, J. (1985). Multivariate generalizations of the proportional hazards model. J. R. Statist. Soc. A 148, 82–117.Google Scholar
Deshpande, J. V., Singh, H., Bagai, I. and Jain, K. (1990). Some partial orders describing positive ageing. Commun. Statist. Stoch. Models 6, 471–481.CrossRefGoogle Scholar
Gupta, P. L. and Gupta, R. C. (1996). Ageing characteristics of the Weibull mixtures. Prob. Eng. Inf. Sci. 10, 591–600.Google Scholar
Gupta, R. C. and Gupta, R. D. (2009). General frailty model and stochastic orderings. J. Statist. Planning Infer. 139, 3277–3287.CrossRefGoogle Scholar
Gupta, R. C. and Kirmani, S. N. U. A. (2006). Stochastic comparisons in frailty models. J. Statist. Planning Infer. 136, 3647–3658.Google Scholar
Heckman, J. and Singer, B. (1984). The identifiability of the proportional hazard model. Rev. Econom. Stud. 51, 231–241.Google Scholar
Holland, P. W. and Wang, Y. J. (1987). Dependence function for continuous bivariate densities. Commun. Statist. Theory Meth. 16, 863–876.Google Scholar
Hougaard, P. (1984). Life table methods for heterogeneous populations: distributions describing the heterogeneity. Biometrika 71, 75–83.CrossRefGoogle Scholar
Hougaard, P. (1991). Modelling heterogeneity in survival data. J. Appl. Prob. 28, 695–701.Google Scholar
Hougaard, P. (1995). Frailty models for survival data. Lifetime Data Anal. 1, 255–273.Google Scholar
Hougaard, P. (2000). Analysis of Multivariate Survival Data. Springer, New York.CrossRefGoogle Scholar
Karlin, S. (1968). Total Positivity, Vol. I. Stanford University Press.Google Scholar
Liang, K.-Y., Self, S. G., Bandeen-Roche, K. J. and Zeger, S. L. (1995). Some recent developments for regression analysis of multivariate failure time data. Lifetime Data Anal. 1, 403–415.Google Scholar
Rizopoulos, D., Verbeke, G. and Molenberghs, G. (2008). Shared parameter models under random effects misspecification. Biometrika 95, 63–74.Google Scholar
Sargent, D. J. (1998). A general framework for random effects survival analysis in the Cox proportional hazards setting. Biometrics 54, 1486–1497.CrossRefGoogle ScholarPubMed
Shaked, M. (1977). A family of concepts of dependence for bivariate distributions. J. Amer. Statist. Assoc. 72, 642–650.CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.CrossRefGoogle Scholar
Vaupel, J. W., Manton, K. G. and Stallard, E. (1979). The impact of heterogeneity in individual frailty on the dynamics of mortality. Demography 16, 439–454.Google Scholar