Compatible Imputation for Hierarchical Linear Models with Incomplete Data: Interaction Effects of Continuous and Categorical Covariates MAR

TL;DR

This paper introduces a compatible Gibbs sampler for hierarchical linear models with incomplete data, improving estimation accuracy by directly sampling from the exact posterior, especially in small sample and complex interaction scenarios.

Contribution

It proposes a novel Gibbs sampling method that ensures compatibility with the joint distribution, addressing biases in existing imputation techniques for hierarchical models with missing data.

Findings

The new sampler produces unbiased estimates in simulations.

It outperforms existing methods in small sample scenarios.

Application to real patient data demonstrates practical utility.

Abstract

This article focuses on Bayesian estimation of a hierarchical linear model (HLM) from incomplete data assumed missing at random where continuous covariates C and discrete categorical covariates $D$ have interaction effects on a continuous response $R$. Given small sample sizes, 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 $C$ and the underlying latent continuous variables $D^*$ of $D$ by a Metropolis algorithm via proposal normal densities having constant variances while the target conditional distributions of $C$ and $D$ have nonconstant variances. Therefore, the samplers are neither guaranteed to be compatible with the joint distribution nor ensured to always produce unbiased estimation of the HLM. We assume a Bayesian joint distribution of parameters and partially observed variables, including correlated categorical $D$, and introduce a compatible Gibbs sampler that draws parameters and missing values directly from the exact posterior distributions. We apply our sampler to incompletely observed longitudinal data from the small number of patient-physician encounters during office visits, and compare our estimators with those of existing methods by simulation.