Bayesian Estimation of Hierarchical Linear Models from Incomplete Data: Cluster-Level Interaction Effects and Small Sample Sizes

We consider Bayesian estimation of a hierarchical linear model (HLM) from small sample sizes where 37 patient-physician encounters are repeatedly measured at four time points. The continuous response $Y$ and continuous covariates $C$ are partially observed and assumed missing at random. With $C$ having linear effects, the HLM may be efficiently estimated by available methods. When $C$ includes cluster-level covariates having interactive or other nonlinear effects given small sample sizes, however, maximum likelihood estimation is suboptimal, and existing Gibbs samplers are based on a Bayesian joint distribution compatible with the HLM, but impute missing values of $C$ by a Metropolis algorithm via a proposal density having a constant variance while the target conditional distribution has a nonconstant variance. Therefore, the samplers are not guaranteed to be compatible with the joint distribution and, thus, not guaranteed to always produce unbiased estimation of the HLM. We introduce a compatible Gibbs sampler that imputes parameters and missing values directly from the exact conditional distributions. We analyze repeated measurements from patient-physician encounters by our sampler, and compare our estimators with those of existing methods by simulation.