Multilevel models are regression models that incorporate group-specific effects.
Groups may represent different levels of hierarchy such as hospitals, doctors nested within hospitals, and patients nested within doctors nested within hospitals. Group-specific effects are assumed to vary randomly across groups according to some a priori distribution, commonly a normal distribution. This assumption makes multilevel models natural candidates for Bayesian analysis. Bayesian multilevel models additionally assume that other model parameters such as regression coefficients and variance components—variances of group-specific effects—are also random.
Why use Bayesian multilevel models? In addition to standard reasons for Bayesian analysis, Bayesian multilevel modeling is often used when the number of groups is small or in the presence of many hierarchical levels. Bayesian information criteria such as deviance information criterion (DIC) are also popular for comparing multilevel models. When the comparison of groups is of main interest, Bayesian multilevel modeling can provide entire distributions of group-specific effects.
You can now fit Bayesian multilevel models in Stata and you can do this easily—just prefix your multilevel command with bayes:
Of course, when we say "easily", we refer to the model specification and not the model formulation. Just like any other modeling task, Bayesian multilevel modeling requires careful consideration.