Linear mixed-effects models are a central analytical tool for modeling hierarchical and longitudinal data, as they allow simultaneous representation of fixed and random sources of variation. In practice, inference for such models is most often based on likelihood-based approximations, which are computationally efficient, but rely on numerical integration and may be unreliable example wise in small-sample settings. In this study, the somewhat obscure four-parameter generalized beta density is shown to be usable as a conjugate prior distribution for a simple linear mixed model. This leads to a closed-form Bayesian solution for a balanced mixed-model design, representing a methodological development beyond standard approximate or simulation-based Bayesian approaches. Although the derivation is restricted to a balanced setting, the proposed framework suggests a pathway toward analytically tractable Bayesian inference for more complex mixed-model structures. The method is evaluated through comparison with a standard frequentist solution based on likelihood estimation for linear mixed-effects models. Results indicate that the Bayesian approach performs just as well as the frequentist alternative, while yielding slightly reduced mean squared error. The study further discusses the use of empirical Bayes strategies for hyperparameter specification and outlines potential directions for extending the approach beyond the balanced case.

翻译：线性混合效应模型是处理层次数据和纵向数据的核心分析工具，因其能够同时表征固定效应与随机效应的变异来源。实践中，此类模型的推断通常基于似然近似方法，这些方法计算效率高，但依赖于数值积分，且在小样本情境下可能出现不可靠的案例。本研究表明，相对少见的四参数广义贝塔分布可作为简单线性混合模型的共轭先验分布。这为平衡设计的混合模型推导出了闭式贝叶斯解，代表了在标准近似或基于模拟的贝叶斯方法之外的方法学进展。尽管推导过程限于平衡设计，但所提出的框架为更复杂混合模型结构的解析可处理贝叶斯推断提供了可行路径。通过与基于线性混合效应模型似然估计的标准频率学派方法进行比较，评估了本方法的性能。结果表明，贝叶斯方法与频率学派方法表现相当，同时能产生略低的均方误差。研究进一步探讨了使用经验贝叶斯策略进行超参数设定的方法，并概述了将该方法推广至非平衡设计的潜在方向。