In studies of survival, the hazard function for each individual may depend on a set of risk factors or explanatory variables but usually not all such variables are known or measurable. This unknown and unobservable risk factor of the hazard function is often termed as the individual’s heterogeneity or frailty—a term coined by Vaupel, Manton, and Stallard (1979). Frailty models are becoming increasing popular in multivariate survival analysis since they allow us to model the association between the individual survival times within subgroups or clusters of subjects. With recent advances in computing technology, Bayesian approaches to frailty models are now computationally feasible, and several approaches have been discussed in the literature. The various approaches differ in the modeling of the baseline hazard or in the distribution of the frailty. Fully parametric approaches to frailty models are examined in Sahu, Dey, Aslanidou, and Sinha (1997), where they consider a frailty model with a Weibull baseline hazard. Semiparametric approaches have also been examined. Clayton (1991) and Sinha (1993, 1997) consider a gamma process prior on the cumulative baseline hazard in the frailty model. Sahu, Dey, Aslanidou, and Sinha (1997), Sinha and Dey (1997), Aslanidou, Dey, and Sinha (1998), and Sinha (1998) discuss frailty models with piecewise exponential baseline hazards. Qiou, Ravishanker, and Dey (1999) examine a positive stable frailty distribution, and Gustafson (1997) and Sargent (1998) examine frailty models using Cox’s partial likelihood. In this chapter, we present an overview of these various approaches to frailty models, and discuss Bayesian inference as well as computational implementation of these methods.
KeywordsBaseline Hazard Frailty Model Partial Likelihood Baseline Hazard Function Gamma Process
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