Two predominant computing methods for generalized linear mixed models (GLMMs) are linearization, e.g., pseudo-likelihood (PL), and integral approximation, e.g., Gauss–Hermite quadrature. The primary GLMM package in R, LME4, only uses integral approximation. The primary GLMM procedure in SAS®, PROC GLIMMIX, was originally developed using linearization, but integral approximation methods were added in the 2008 release. This presents a dilemma for GLMM users: Which method should one use, and why? Linearization methods are more versatile and able to handle both conditional and marginal GLMMs. Linearization can be implemented with REML-like variance component estimation, whereas quadrature is strictly maximum likelihood. However, GLMM software documentation and the literature on which it is based tend to focus on linearization’s limitations. Stroup (Generalized linear mixed models: modern concepts, methods and applications, CRC Press, Boca Raton, 2013) reiterates this theme in his GLMM textbook. As a result, “conventional wisdom” has arisen that integral approximation—quadrature when possible—is always best. Meanwhile, ongoing experience with GLMMs and research about their small sample behavior suggest that “conventional wisdom” circa 2013 is often not true. Above all, it is clear there is no one-size-fits-all best method. The purpose of this paper is to provide an updated look at what we now know about quadrature and PL and to offer some general operating principles for making an informed choice between the two. A series of simulation studies investigating distributions and designs representative of research in agricultural and related disciplines provide an overview of each method with respect to estimation accuracy, type I error control, and robustness (or lack thereof) to model misspecification. Supplementary materials accompanying this paper appear online.

中文翻译：

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伪似然或正交？我们认为我们知道什么，我们认为我们知道什么，以及我们仍在努力弄清楚什么

广义线性混合模型 (GLMM) 的两种主要计算方法是线性化，例如伪似然 (PL) 和积分近似，例如高斯-厄米正交。R 中的主要 GLMM 包 LME4 仅使用积分近似。SAS® 中的主要 GLMM 程序 PROC GLIMMIX 最初是使用线性化开发的，但在 2008 版本中添加了积分近似方法。这给 GLMM 用户带来了一个困境：应该使用哪种方法，为什么？线性化方法更通用，能够处理条件和边际 GLMM。线性化可以用类似 REML 的方差分量估计来实现，而正交是严格的最大似然。然而，GLMM 软件文档和它所基于的文献倾向于关注线性化的局限性。Stroup（广义线性混合模型：现代概念、方法和应用，CRC Press，Boca Raton，2013 年）在他的 GLMM 教科书中重申了这个主题。结果，出现了“传统智慧”，即积分近似——可能时求正交——总是最好的。同时，对 GLMM 的持续经验以及对其小样本行为的研究表明，大约 2013 年的“传统智慧”通常是不正确的。最重要的是，很明显没有一刀切的最佳方法。本文的目的是更新我们现在对正交和 PL 的了解，并提供一些通用的操作原则，以便在两者之间做出明智的选择。一系列模拟研究调查了代表农业和相关学科研究的分布和设计，在估计精度、I 类错误控制和鲁棒性（或缺乏鲁棒性）方面对每种方法进行了概述，以模拟错误指定。本文随附的补充材料出现在网上。