Restricted maximum likelihood (REML) estimation is a widely accepted and frequently used method for fitting linear mixed models, with its principal advantage being that it produces less biased estimates of the variance components. However, the concept of REML does not immediately generalize to the setting of non-normally distributed responses, and it is not always clear the extent to which, either asymptotically or in finite samples, such generalizations reduce the bias of variance component estimates compared to standard unrestricted maximum likelihood estimation. In this article, we review various attempts that have been made over the past four decades to extend REML estimation in generalized linear mixed models. We establish four major classes of approaches, namely approximate linearization, integrated likelihood, modified profile likelihoods, and direct bias correction of the score function, and show that while these four classes may have differing motivations and derivations, they often arrive at a similar if not the same REML estimate. We compare the finite sample performance of these four classes through a numerical study involving binary and count data, with results demonstrating that they perform similarly well in reducing the finite sample bias of variance components.

翻译：限制最大似然（REML）估计是拟合线性混合模型广泛接受且常用的方法，其主要优势在于能产生偏差较小的方差分量估计。然而，REML 的概念并不能直接推广至非正态分布响应变量场景，且尚不明确此类推广在渐近或有限样本条件下，相比于标准无限制最大似然估计，能在多大程度上减少方差分量估计的偏差。本文回顾了过去四十年间将 REML 估计扩展至广义线性混合模型的多种尝试。我们确立了四种主要方法类别，即近似线性化、积分似然、修正剖面似然和评分函数的直接偏差校正，并论证了尽管这四类方法的动机和推导过程可能不同，但它们通常能得到相似甚至相同的 REML 估计。通过一项涉及二分类和计数数据的数值研究，我们比较了这四类方法的有限样本性能，结果表明它们在减少方差分量有限样本偏差方面表现相似。