ABSTRACT
Bayesian statistical modeling represents a fundamental shift from the frequentist methods of model parameter estimation that we have been using heretofore. This paradigm shift is evident in part through the methodology used to obtain the estimates, Markov Chain Monte Carlo (MCMC) most commonly for the Bayesian approach, and maximum likelihood (ML) and restricted maximum likelihood (REML) in the frequentist case. The reader may rightly question why, or in what situations, Bayesian multilevel modeling might be particularly useful, or even preferable to frequentist methods. One primary advantage of Bayesian methods in some situations, including with multilevel modeling, is that unlike ML and REML, it does not rely on any distributional assumptions about the data. Thus, the determination of Bayesian credibility intervals (corresponding to confidence intervals) can be made without worry even if the data come from a skewed distribution. Although we will not describe the MCMC process in much detail here, it is necessary to discuss conceptually how the methodology works, in part so that the reader might be more comfortable with where the parameter estimates come from, and in part because we will need to diagnose whether the method has worked appropriately so that we can have confidence in the final parameter estimates.
