The dependence between subjects in clustered survival data is commonly modeled by means of a frailty, a multiplicative random effect with a distribution that is usually specified in advance. Misspecification of the frailty distribution can lead to error when estimating parameters of interest.

This dissertation contains two distinct approaches to frailty models for the analysis of clustered survival data that do not require the frailty distribution to be known a priori.

The first is a Bayesian method, in which the distribution of both the baseline hazard and frailty are modeled nonparametrically as mixtures of B-splines, and estimated by Markov Chain Monte Carlo. Smooth curve estimates are attained by adaptive selection of the spline knots, or by means of an explicit smoothness penalty. The method is illustrated with data sets from studies of congestive heart failure and diabetic retinopathy.

The second is a method for clustered bivariate recurrent event data, in which the hierarchical bivariate frailty need only be specified through its first two moments. Estimation relies on a correspondence between the modulated renewal process likelihood and an auxiliary Poisson model likelihood, which allows the frailties to be estimated by their best linear unbiased predictors in an iterative algorithm. Data on recurrent basal and squamous cell carcinomas collected during the Nutritional Prevention of Cancer trial serves to illustrate the method.