Frailty models for survival data
 Philip Hougaard
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A frailty model is a random effects model for time variables, where the random effect (the frailty) has a multiplicative effect on the hazard. It can be used for univariate (independent) failure times, i.e. to describe the influence of unobserved covariates in a proportional hazards model. More interesting, however, is to consider multivariate (dependent) failure times generated as conditionally independent times given the frailty. This approach can be used both for survival times for individuals, like twins or family members, and for repeated events for the same individual. The standard assumption is to use a gamma distribution for the frailty, but this is a restriction that implies that the dependence is most important for late events. More generally, the distribution can be stable, inverse Gaussian, or follow a power variance function exponential family. Theoretically, large differences are seen between the choices. In practice, using the largest model makes it possible to allow for more general dependence structures, without making the formulas too complicated.
 Aaberge, R., Kravdal, O., Wennemo, T. (1989) Unobserved heterogeneity in models of marriage dissolution. Central Bureau of Statistics, Norway
 Aalen, O. O. (1987) Two examples of modelling heterogeneity in survival analysis. Scand. J. Statist 14: pp. 1925
 Aalen, O. O. (1987) Mixing distributions on a Markov chain. Scand. J. Statist. 14: pp. 2819
 Aalen, O. O. (1988) Heterogeneity in survival analysis. Statist. Med. 7: pp. 112137
 Aalen, O. O. (1992) Modelling heterogeneity in survival analysis by the compound Poisson distribution. Ann. Appl. Prob. 2: pp. 95172
 Aalen, O. O. (1994) Effects of frailty in survival analysis. Statistical Methods in Medical Research 3: pp. 22743
 Andersen, P. K., Borgan, O. (1985) Counting process models for life history data: A review. Scand. J. Statist. 12: pp. 97158
 P. K. Andersen, O. Borgan, R. D. Gill, and N. Keiding,Statistical Models Based on Counting Processes, Springer Verlag, 1993.
 BarLev, S. K., Enis, P. (1986) Reproducibility and natural exponential families with power variance functions. Ann. Statist. 14: pp. 150722
 Burridge, J. (1981) Empirical Bayes analysis of survival time data. J.R. Statist. Soc. B 43: pp. 6575
 Clayton, D. (1978) A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika 65: pp. 14151
 Clayton, D., Cuzick, J. (1985) Multivariate generalizations of the proportional hazards model (with discussion). J.R. Statist. Soc. A 148: pp. 82117
 Cox, D. R. (1972) Regression models and life tables (with discussion). J.R. Statist. Soc. B 34: pp. 187220
 Crowder, M. (1989) A multivariate distribution with Weibull connections. J.R. Statist. Soc. B 51: pp. 93107
 Elbers, C., Ridder, G. (1982) True and spurious duration dependence: the identifiability of the proportional hazard model. Rev. Econ. Stud. XLIX: pp. 4039
 Ellermann, R., Sullo, P., Tien, J. M. (1992) An alternative approach to modeling recidivism using quantile residual life functions. Operations Research 40: pp. 485504
 Freund, J. E. (1961) A bivariate extension of the exponential distribution. J. Am. Statist. Assoc. 56: pp. 9717
 Genest, C., MacKay, (1986) Copules Archimediennes et familles de lois bidimensionnelles dont les marges sont donnees. Canadian J. Statist. 14: pp. 14559
 Gumbel, E. J. (1960) Bivariate exponential distributions. J. Am. Statist. Assoc. 55: pp. 698707
 Guo, G. (1993) Use of sibling data to estimate family mortality effects in Guatemala. Demography 30: pp. 1532
 Guo, G., Rodriguez, G. (1992) Estimating a multivariate proportional hazards model for clustered data using the EM algorithm. With an application to child survival in Guatemala. J. Am. Statist. Assoc. 87: pp. 96976
 Hougaard, P. (1984) Life table methods for heterogeneous populations: Distributions describing the heterogeneity. Biometrika 71: pp. 7584
 Hougaard, P. (1986) Survival models for heterogeneous populations derived from stable distributions. Biometrika 73: pp. 38796
 Hougaard, P. (1986) A class of multivariate failure time distributions. Biometrika 73: pp. 6718
 Hougaard, P. (1987) Modelling multivariate survival. Scand. J. Statist. 14: pp. 291304
 Hougaard, P. (1989) Fitting a multivariate failure time distribution. IEEE Transactions on Reliability 38: pp. 4448
 Hougaard, P. (1991) Modelling heterogeneity in survival data. J. Appl. Prob. 28: pp. 695701
 Hougaard, P., Harvald, B., Holm, N. V. (1992) Measuring the similarities between the lifetimes of adult Danish twins born between 1881–1930. J. Am. Statist. Assoc. 87: pp. 1724
 P. Hougaard, B. Harvald, and N. V. Holm, “Assessment of dependence in the life times of twins,”Survival Analysis: State of the Art (J. P. Klein and P. K. Goel, eds.) pp. 77–97, Kluwer Academic Publishers, 1992b.
 P. Hougaard, B. Harvald, and N. V. Holm, “Models for multivariate failure time data, with application to the survival of twins,”Statistical Modelling (P. G. M. van der Heijden, W. Jansen, B. Francis and G. U. H. Seeber, eds.) pp. 159–173, Elsevier Science Publishers, 1992c.
 Hougaard, P., Myglegaard, P., BorchJohnsen, K. (1994) Heterogeneity models of disease susceptibility, with application to diabetic nephropathy. Biometrics 50: pp. 117888
 Hutchinson, T.P., Lai, C.D. (1991) The Engineering Statistician's Guide to Continuous Bivariate Distributions. Rumsby Scientific Publishing, Adelaide
 Joe, H. (1993) Parametric families of multivariate distributions with given margins. J. Mult. Anal. 46: pp. 26282
 Jørgensen, B. (1981) Statistical properties of the generalized inverse Gaussian distribution. SpringerVerlag, Heidelberg
 Jørgensen, B. (1987) Exponential dispersion models. J.R. Statist. Soc. B 49: pp. 12762
 Klein, J. P. (1992) Semiparametric estimation of random effects using the Cox model based on the EM algorithm. Biometrics 48: pp. 795806
 J. P. Klein, M. Moeschberger, Y. H. Li, and S. T. Wang, “Estimating random effects in the Framingham heart study,”Survival Analysis: State of the Art (J. P. Klein and P. K. Goel, eds.) pp. 99–120, Kluwer Academic Publishers, 1992.
 Lancaster, T. (1979) Econometric methods for the duration of unemployment. Econometrica 47: pp. 93956
 Lee, L. (1979) Multivariate distributions having Weibull properties. J. Mult. Anal. 9: pp. 26777
 Lee, M.L. T., Whitmore, G. A. (1993) Stochastic processes directed by randomized time. J. Appl. Prob. 30: pp. 30214
 Lu, J.C. (1990) Least squares estimation for the multivariate Weibull model of Hougaard based on accelerated life test of system and component. Commun. Statist.Theory Meth. 19: pp. 372539
 Lu, J.C., Bhattacharyya, G. K. (1990) Some new constructions of bivariate Weibull models. Ann. Inst. Statist. Math. 42: pp. 54359
 Marshall, A. W., Olkin, I. (1967) A multivariate exponential distribution. J. Am. Statist. Assoc. 62: pp. 3044
 Murphy, S. A. (1994) Consistency in a proportional hazards model incorporating a random effect. Ann. Statist. 22: pp. 71231
 Nielsen, G. G., Gill, R. D., Andersen, P. K., Sørensen, T. I. A. (1992) A counting process approach to maximum likelihood estimation in frailty models. Scand. J. Statist. 19: pp. 2543
 Oakes, D. (1982) A model for association in bivariate survival data. J.R. Statist. Soc. B 44: pp. 41422
 Oakes, D. (1989) Bivariate survival models induced by frailties. J. Am. Statist. Assoc. 84: pp. 48793
 Oakes, D., Manatunga, A. (1992) Fisher information for a bivariate extreme value distribution. Biometrika 79: pp. 82732
 Pickles, A., Crouchley, R., Simonoff, E., Eaves, L., Meyer, J., Rutter, M., Hewitt, J., Silberg, J. (1994) Survival models for development genetic data: Age of onset of puberty and antisocial behaviour in twins. Genetic Epidemiology 11: pp. 15570
 C. S. Rocha, “Survival models for heterogeneity using the noncentral chisquared distribution with zero degrees of freedom,” Notas e Comunicações, Centro de estatistica e aplicações da universidade de Lisboa, 1994.
 Tawn, J. A. (1988) Bivariate extreme value theory; Models and estimation. Biometrika 75: pp. 397415
 Thomas, D. C., Langholz, B., Mack, W., Floderus, B. (1990) Bivariate survival models for analysis of genetic and environmental effects in twins. Genetic Epidemiology 7: pp. 2135
 M. C. K. Tweedie, “An index which distinguishes between some important exponential families,”Statistics: Applications and New Directions. Proceedings of the Indian Statistical Institute Golden Jubilee International Conference (J. K. Ghosh and J. Roy, eds.) pp. 579–604, 1984.
 Vaupel, J. W., Manton, K. G., Stallard, E. (1979) The impact of heterogeneity in individual frailty on the dynamics of mortality. Demography 16: pp. 43954
 J. W. Vaupel and A. I. Yashin, “The deviant dynamics of death in heterogeneous populations,”Sociological Methodology (N. B. Tuma, ed.) pp. 179–211, JosseyBass Publishers, 1985.
 Wassell, J. T., Moeschberger, M. L. (1993) A bivariate survival model with modified gamma frailty for assessing the impact of interventions. Statist. Med. 12: pp. 2418
 Whitmore, G. A., Lee, M.L. T. (1991) A multivariate survival distribution generated by an inverse Gaussian mixture of exponentials. Technometrics 33: pp. 3950
 Yashin, A. I., Manton, K. G., Stallard, E. (1986) Dependent competing risks: a stochastic process model. J. Math. Biol. 24: pp. 11940
 Yashin, A. I., Manton, K. G., Stallard, E. (1989) The propagation of uncertainty in human mortality processes operating in stochastic environments. Theoretical Population Biology 35: pp. 11941
 Title
 Frailty models for survival data
 Journal

Lifetime Data Analysis
Volume 1, Issue 3 , pp 255273
 Cover Date
 19950901
 DOI
 10.1007/BF00985760
 Print ISSN
 13807870
 Online ISSN
 15729249
 Publisher
 Kluwer Academic Publishers
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 Authors

 Philip Hougaard ^{(1)}
 Author Affiliations

 1. Novo Nordisk, Novo Alle, DK2880, Bagsvaerd, Denmark