We present a computational motivation for restricted maximum likelihood (REML) estimation in linear mixed models using an expectation--maximization (EM) algorithm. At each iteration, maximum likelihood (ML) and REML solve the same mixed-model equations for the best linear unbiased estimator (BLUE) of the fixed effects and the best linear unbiased predictor (BLUP) of the random effects. They differ only in the trace adjustments used in the variance-component updates: ML uses conditional covariances of the random effects given the data, whereas REML uses prediction-error covariances from Henderson's C-matrix, reflecting uncertainty from estimating the fixed effects. Short R code makes this switch explicit, exposes the key matrices for classroom inspection, and reproduces lme4 ML and REML fits.

翻译：本文通过期望最大化算法为线性混合模型中的限制性最大似然估计提供了计算动机。在每次迭代中，最大似然估计与限制性最大似然估计均通过求解相同的混合模型方程来获得固定效应的最佳线性无偏估计量以及随机效应的最佳线性无偏预测量。两者的差异仅体现在方差分量更新中的迹调整项：最大似然估计采用给定数据条件下随机效应的条件协方差，而限制性最大似然估计则采用基于Henderson C矩阵的预测误差协方差，后者反映了估计固定效应所产生的不确定性。简短的R代码明确实现了这种切换，揭示了可供课堂检验的关键矩阵结构，并复现了lme4包的最大似然估计与限制性最大似然估计拟合结果。