Abstract
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.
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R. Aaberge, O. Kravdal, and T. Wennemo, “Unobserved heterogeneity in models of marriage dissolution,” Discussion paper no. 42, Central Bureau of Statistics, Norway, 1989.
O. O. Aalen, “Two examples of modelling heterogeneity in survival analysis,”Scand. J. Statist, vol. 14 pp. 19–25, 1987a.
O. O. Aalen, “Mixing distributions on a Markov chain,”Scand. J. Statist. vol. 14 pp. 281–9, 1987b.
O. O. Aalen, “Heterogeneity in survival analysis,”Statist. Med. vol. 7 pp. 1121–37, 1988.
O. O. Aalen, “Modelling heterogeneity in survival analysis by the compound Poisson distribution,”Ann. Appl. Prob. vol. 2 pp. 951–72, 1992.
O. O. Aalen, “Effects of frailty in survival analysis,”Statistical Methods in Medical Research vol. 3 pp. 227–43, 1994.
P. K. Andersen, and O. Borgan, “Counting process models for life history data: A review,”Scand. J. Statist. vol. 12 pp. 97–158, 1985.
P. K. Andersen, O. Borgan, R. D. Gill, and N. Keiding,Statistical Models Based on Counting Processes, Springer Verlag, 1993.
S. K. Bar-Lev and P. Enis, “Reproducibility and natural exponential families with power variance functions,”Ann. Statist. vol. 14 pp. 1507–22, 1986.
J. Burridge, “Empirical Bayes analysis of survival time data,J.R. Statist. Soc. B vol. 43 pp. 65–75, 1981.
D. Clayton, “A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence,”Biometrika vol. 65 pp. 141–51, 1978.
D. Clayton and J. Cuzick, “Multivariate generalizations of the proportional hazards model (with discussion),”J.R. Statist. Soc. A vol. 148 pp. 82–117, 1985.
D. R. Cox, “Regression models and life tables (with discussion),”J.R. Statist. Soc. B vol. 34 pp. 187–220, 1972.
M. Crowder, “A multivariate distribution with Weibull connections,”J.R. Statist. Soc. B vol. 51 pp. 93–107, 1989.
C. Elbers and G. Ridder, “True and spurious duration dependence: the identifiability of the proportional hazard model.”Rev. Econ. Stud. vol. XLIX pp. 403–9, 1982.
R. Ellermann, P. Sullo, and J. M. Tien, “An alternative approach to modeling recidivism using quantile residual life functions,”Operations Research vol. 40 pp. 485–504, 1992.
J. E. Freund, “A bivariate extension of the exponential distribution,”J. Am. Statist. Assoc. vol. 56 pp. 971–7, 1961.
C. Genest and MacKay, “Copules Archimediennes et familles de lois bidimensionnelles dont les marges sont donnees,”Canadian J. Statist. vol. 14 pp. 145–59, 1986.
E. J. Gumbel, “Bivariate exponential distributions,”J. Am. Statist. Assoc. vol. 55 pp. 698–707, 1960.
G. Guo, “Use of sibling data to estimate family mortality effects in Guatemala,”Demography vol. 30 pp. 15–32, 1993.
G. Guo and G. Rodriguez, “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. vol. 87 pp. 969–76, 1992.
P. Hougaard, “Life table methods for heterogeneous populations: Distributions describing the heterogeneity,”Biometrika vol. 71 pp. 75–84, 1984.
P. Hougaard, “Survival models for heterogeneous populations derived from stable distributions,”Biometrika vol. 73 pp. 387–96, 1986a (Correction, vol. 75 pp. 395).
P. Hougaard, “A class of multivariate failure time distributions,”Biometrika vol. 73 pp. 671–8, 1986b. (Correction, vol. 75 pp. 395).
P. Hougaard, “Modelling multivariate survival,”Scand. J. Statist. vol. 14 pp. 291–304, 1987.
P. Hougaard, “Fitting a multivariate failure time distribution,”IEEE Transactions on Reliability vol. 38 pp. 444–8, 1989.
P. Hougaard, “Modelling heterogeneity in survival data,”J. Appl. Prob. vol. 28 pp. 695–701, 1991.
P. Hougaard, B. Harvald, and N. V. Holm, “Measuring the similarities between the lifetimes of adult Danish twins born between 1881–1930,”J. Am. Statist. Assoc. vol. 87 pp. 17–24, 1992a.
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.
P. Hougaard, P. Myglegaard, and K. Borch-Johnsen, “Heterogeneity models of disease susceptibility, with application to diabetic nephropathy,”Biometrics vol. 50 pp. 1178–88, 1994.
T.P. Hutchinson and C.D. Lai,The Engineering Statistician's Guide to Continuous Bivariate Distributions, Rumsby Scientific Publishing: Adelaide, 1991.
H. Joe, “Parametric families of multivariate distributions with given margins,”J. Mult. Anal. vol. 46 pp. 262–82, 1993.
B. Jørgensen, “Statistical properties of the generalized inverse Gaussian distribution,”Lecture Notes in Statistics vol. 9, Springer-Verlag: Heidelberg, 1981.
B. Jørgensen, “Exponential dispersion models,”J.R. Statist. Soc. B vol. 49 pp. 127–62, 1987.
J. P. Klein, “Semiparametric estimation of random effects using the Cox model based on the EM algorithm,”Biometrics vol. 48 pp. 795–806, 1992.
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.
T. Lancaster, “Econometric methods for the duration of unemployment,”Econometrica vol. 47 pp. 939–56, 1979.
L. Lee, “Multivariate distributions having Weibull properties,”J. Mult. Anal. vol. 9 pp. 267–77, 1979.
M.-L. T. Lee and G. A. Whitmore, “Stochastic processes directed by randomized time,”J. Appl. Prob. vol. 30 pp. 302–14, 1993.
J.-C. Lu, “Least squares estimation for the multivariate Weibull model of Hougaard based on accelerated life test of system and component,”Commun. Statist.-Theory Meth. vol. 19 pp. 3725–39, 1990.
J.-C. Lu and G. K. Bhattacharyya, “Some new constructions of bivariate Weibull models,”Ann. Inst. Statist. Math. vol. 42 pp. 543–59, 1990.
A. W. Marshall and I. Olkin, “A multivariate exponential distribution,”J. Am. Statist. Assoc. vol. 62 pp. 30–44, 1967.
S. A. Murphy, “Consistency in a proportional hazards model incorporating a random effect,”Ann. Statist. vol. 22 pp. 712–31, 1994.
G. G. Nielsen, R. D. Gill, P. K. Andersen, and T. I. A. Sørensen, “A counting process approach to maximum likelihood estimation in frailty models,”Scand. J. Statist. vol. 19 pp. 25–43, 1992.
D. Oakes, “A model for association in bivariate survival data,”J.R. Statist. Soc. B vol. 44 pp. 414–22, 1982.
D. Oakes, “Bivariate survival models induced by frailties,”J. Am. Statist. Assoc. vol. 84 pp. 487–93, 1989.
D. Oakes and A. Manatunga, “Fisher information for a bivariate extreme value distribution,”Biometrika vol. 79 pp. 827–32, 1992.
A. Pickles, R. Crouchley, E. Simonoff, L. Eaves, J. Meyer, M. Rutter, J. Hewitt, and J. Silberg, “Survival models for development genetic data: Age of onset of puberty and antisocial behaviour in twins,”Genetic Epidemiology vol. 11 pp. 155–70, 1994.
C. S. Rocha, “Survival models for heterogeneity using the non-central chi-squared distribution with zero degrees of freedom,” Notas e Comunicações, Centro de estatistica e aplicações da universidade de Lisboa, 1994.
J. A. Tawn, “Bivariate extreme value theory; Models and estimation,”Biometrika vol. 75 pp. 397–415, 1988.
D. C. Thomas, B. Langholz, W. Mack, and B. Floderus, “Bivariate survival models for analysis of genetic and environmental effects in twins,”Genetic Epidemiology vol. 7 pp. 21–35, 1990.
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.
J. W. Vaupel, K. G. Manton, and E. Stallard, “The impact of heterogeneity in individual frailty on the dynamics of mortality,”Demography vol. 16 pp. 439–54, 1979.
J. W. Vaupel and A. I. Yashin, “The deviant dynamics of death in heterogeneous populations,”Sociological Methodology (N. B. Tuma, ed.) pp. 179–211, Jossey-Bass Publishers, 1985.
J. T. Wassell and M. L. Moeschberger, “A bivariate survival model with modified gamma frailty for assessing the impact of interventions,”Statist. Med. vol. 12 pp. 241–8, 1993.
G. A. Whitmore and M.-L. T. Lee, “A multivariate survival distribution generated by an inverse Gaussian mixture of exponentials,”Technometrics vol. 33 pp. 39–50, 1991.
A. I. Yashin, K. G. Manton, and E. Stallard, “Dependent competing risks: a stochastic process model,”J. Math. Biol. vol. 24 pp. 119–40, 1986.
A. I. Yashin, K. G. Manton, and E. Stallard, “The propagation of uncertainty in human mortality processes operating in stochastic environments,”Theoretical Population Biology vol. 35 pp. 119–41, 1989.
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This paper is a revised version of a review, which together with ten papers by the author made up a thesis for a Doctor of Science degree at the University of Copenhagen.
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Hougaard, P. Frailty models for survival data. Lifetime Data Anal 1, 255–273 (1995). https://doi.org/10.1007/BF00985760
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DOI: https://doi.org/10.1007/BF00985760