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.

翻译：我们考虑在小样本规模下对分层线性模型（HLM）进行贝叶斯估计，其中37组患者-医生交互在四个时间点被重复测量。连续响应变量 $Y$ 与连续协变量 $C$ 存在部分观测缺失，并假设为随机缺失。当 $C$ 具有线性效应时，HLM 可通过现有方法进行有效估计。然而，当 $C$ 包含在给定小样本规模下具有交互效应或其他非线性效应的集群层面协变量时，最大似然估计并非最优，且现有的 Gibbs 采样器基于与 HLM 兼容的贝叶斯联合分布，但通过 Metropolis 算法填补 $C$ 的缺失值，该算法使用具有恒定方差的提议密度，而目标条件分布具有非常数方差。因此，这些采样器不能保证与联合分布兼容，从而不能保证始终产生 HLM 的无偏估计。我们引入了一种兼容的 Gibbs 采样器，可直接从精确条件分布中填补参数和缺失值。我们使用该采样器分析了患者-医生交互的重复测量数据，并通过模拟将我们的估计量与现有方法的估计量进行了比较。