ABSTRACT
Improved software for Bayesian multilevel analysis is exemplified by the rstan based brms package, with an overview provided by Y. Mai and Z. Zhang. This chapter considers the normal linear multilevel model, general linear and conjugate models for multilevel discrete data, crossed factor and multiple member random effect models, and robust multilevel models. A multilevel model typically assumes observations to be independent conditional on fixed regression and random effects defined at one or more levels in the data hierarchy. An alternative fully hierarchical presentation of the normal linear multilevel model is based on the scheme of D. V. Lindley and A. F. Smith. A robust alternative to the normal linear mixed model based on the multivariate t density is proposed by J. Pinheiro et al., and shown to outperform normality assumptions when outliers are present in multilevel data. An alternative mixture prior to reduce the impact of parametric assumptions is the mixture of Dirichlet process approach.
