Bayesian Analysis in Applications of Hierarchical Models: Issues and Methods

In applications of hierarchical models (HMs), a potential weakness of empirical Bayes estimation approaches is that they do not to take into account uncertainty in the estimation of the variance components (see, e.g., Dempster, 1987 ). One possible solution entails employing a fully Bayesian approach, which involves specifying a prior probability distribution for the variance components and then integrating over the variance components as well as other unknowns in the HM to obtain a marginal posterior distribution of interest (see, e.g., Draper, 1995 ; Rubin, 1981 ). Though the required integrations are often exceedingly complex, Markov-chain Monte Carlo techniques (e.g., the Gibbs sampler) provide a viable means of obtaining marginal posteriors of interest in many complex settings. In this article, we fully generalize the Gibbs sampling algorithms presented in Seltzer (1993) to a broad range of settings in which vectors of random regression parameters in the HM (e.g., school means and slopes) are assumed multivariate normally or multivariate t distributed across groups. Through analyses of the data from an innovative mathematics curriculum, we examine when and why it becomes important to employ a fully Bayesian approach and discuss the need to study the sensitivity of results to alternative prior distributional assumptions for the variance components and for the random regression parameters.