Abstract
Frailty models derived from the proportional hazards regression model are frequently used to analyze clustered right-censored survival data. We propose a semiparametric Bayesian methodology for this purpose, modeling both the unknown baseline hazard and density of the random effects using mixtures of B-splines. The posterior distributions for all regression coefficients and spline parameters are obtained using Markov Chain Monte Carlo (MCMC). The methodology permits the use of weighted mixtures of parametric and nonparametric components in modeling the hazard function and frailty distribution; in addition, the spline knots may also be selected adaptively using reversible-jump MCMC. Simulations indicate that the method produces smooth and accurate posterior hazard and frailty density estimates. The Bayesian approach not only produces point estimators that outperform existing approaches in certain circumstances, but also offers a wealth of information about the parameters of interest in the form of MCMC samples from the joint posterior probability distribution. We illustrate the adaptability of the method with data from a study of congestive heart failure.
Highlights
Extending and building upon these frequentist maximum-likelihood methods, Bayesian approaches to the frailty model have emerged
The algorithm used to sample from the joint posterior distribution consists of three types of steps: initialization, parameter updates via Gibbs sampling and Metropolis-Hastings Markov Chain Monte Carlo (MCMC), and, if desired, adaptive knot selection via reversible-jump MCMC
The proposed Bayesian methodology permits the analysis of clustered survival data when the underlying frailty distribution is unknown, helping to mitigate the impact associated with the specification of an incorrect parametric model on the assessment of heterogeneity
Summary
The basic proportional hazards frailty model, described, utilizes B-spline formulations for both the baseline hazard and frailty probability density function, along with prior specifications that may be used to encourage smoothness. The resulting model, which continues to retain the form of a proportional hazards frailty model, permits (but does not force) shrinkage towards a specific parametric hazard function or frailty distribution as well as allows one to incorporate specific prior knowledge. Each of the baseline hazard and frailty density curves can be specified as some combination of a basic model and its optional extensions, indexed by a variety of priors and options. The result is a flexible family of models that allows prior knowledge about the form and smoothness of either curve to be incorporated into the model fit to the desired extent
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