Clayton (1991) A Monte Carlo Method for Bayesian Inference in Frailty Models

Summary

Clayton gives a fully Bayesian treatment of the frailty (random-effects proportional-hazards) model and shows how to do inference by Monte Carlo. The Clayton-Cuzick (1985) frailty generalization of Cox regression is recast as a hierarchical graphical model — a gamma-process prior on the integrated baseline hazard (after Kalbfleisch 1978), a gamma frailty distribution, and a vague prior on its variance — and the joint posterior is sampled by Gibbs sampling ("stochastic relaxation"/"stochastic substitution"), following Gelfand-Smith (1990). The method is demonstrated on data from an animal carcinogenesis experiment.

Key Claims

• Why frailty is needed. The counting-process/multiplicative-intensity extension of Cox's model assumes transitions are independent given covariates — almost always false because unmeasured variables cluster risk (families, litters, repeated events); a random "frailty" restores correct inference.
• Bayesian graphical formulation. The frailty model is written as a hierarchical Bayesian graphical model: a gamma-process prior on the integrated baseline hazard $\Lambda_0$, a gamma frailty $\nu$ with a $G(\gamma,\gamma)$ distribution, and a (vague, improper) hyperprior on the frailty variance; the Poisson/counting-process likelihood factorization makes the full conditionals tractable.
• Inference by Gibbs sampling. Because the model defines a set of full conditional distributions, the joint and marginal posteriors are obtained by stochastic substitution — iteratively drawing each block given the others (Gelfand-Smith 1990; Geman-Geman) — the Markov chain converging to the posterior. Presented as a Bayesian analogue of the bootstrap that holds data fixed and resamples parameters.
• Application. Illustrated on a litter-matched animal carcinogenesis experiment, where the litter frailty captures shared genetic/environmental risk.

Concepts Introduced or Extended

• Frailty Model

Entities Mentioned

• David Clayton

Quotes

"This paper suggests a Monte Carlo approach suggested by a Bayesian casting of the problem."

"The Bayesian model described above … defines a Gibbs distribution … a stochastic substitution method in which, starting from $(U^0,V^0)$, we generate a sequence … which is Markovian [and] converges to an equilibrium distribution which is the [posterior]."

My Take

An early and influential import of Gibbs sampling into survival analysis — contemporaneous with Gelfand-Smith (1990), and one of the first places a working biostatistician showed that the intractable frailty likelihood, which had defeated frequentist asymptotics, becomes routine once you cast it as a hierarchical Bayesian graphical model and sample it. It is the Bayesian/MCMC counterpart to the penalized-likelihood route to the same models (Therneau-Grambsch-Pankratz): same model, two computational philosophies. It also sits directly downstream of Clayton's own Poisson/counting-process view of survival likelihoods, which is exactly what makes the full conditionals fall out.