Full conditional distributions for Bayesian multilevel models with additive or interactive effects and missing data on covariates

Missing data are a common occurrence in analyses of multivariate data, including in multilevel modeling. Bayesian approaches to handling missing data in multilevel modeling have garnered increasing attention, either on their own or in service of multiple imputation. However, these applications are largely confined to specific models or missingness patterns. The current work provides a coherent account of Bayesian analysis of multilevel models in the presence of missing data on the outcomes, level-1 predictors, and level-2 predictors, that covers the main aspects of the models and missingness. In doing so, this work provides a grounding for estimation in fully Bayesian approaches that employ Gibbs sampling, and provides an account of how to generate the imputations in the first phase of a multiple imputation approach.