Frailty modeling via the empirical Bayes–Hastings sampler

Studies of ocular disease and analyses of time to disease onset are complicated by the correlation expected between the two eyes from a single patient. We overcome these statistical modeling challenges through a nonparametric Bayesian frailty model. While this model suggests itself as a natural one for such complex data structures, model fitting routines become overwhelmingly complicated and computationally intensive given the nonparametric form assumed for the frailty distribution and baseline hazard function. We consider empirical Bayesian methods to alleviate these difficulties through a routine that iterates between frequentist, data-driven estimation of the cumulative baseline hazard and Markov chain Monte Carlo estimation of the frailty and regression coefficients. We show both in theory and through simulation that this approach yields consistent estimators of the parameters of interest. We then apply the method to the short-wave automated perimetry (SWAP) data set to study risk factors of glaucomatous visual field deficits.