Implementation and performance issues in the Bayesian and likelihood fitting of multilevel models

We use simulation studies (a) to compare Bayesian and likelihood fitting methods, in terms of validity of conclusions, in two-level random-slopes regression (RSR) models, and (b) to compare several Bayesian estimation methods based on Markov chain Monte Carlo, in terms of computational efficiency, in random-effects logistic regression (RELR) models. We find (a) that the Bayesian approach with a particular choice of diffuse inverse Wishart prior distribution for the (co)variance parameters performs at least as well-in terms of bias of estimates and actual coverage of nominal 95% intervals-as maximum likelihood methods in RSR models with medium sample sizes (expressed in terms of the number J of level-2 units), but neither approach performs as well as might be hoped with small J; and (b) that an adaptive hybrid Metropolis-Gibbs sampling method we have developed for use in the multilevel modeling package MlwiN outperforms adaptive rejection Gibbs sampling in the RELR models we have considered, sometimes by a wide margin.