There has been considerable and controversial research over the past two decades into how successfully random effects misspecification in mixed models (i.e. assuming normality for the random effects when the true distribution is non‐normal) can be diagnosed and what its impacts are on estimation and inference. However, much of this research has focused on fixed effects inference in generalised linear mixed models. In this article, motivated by the increasing number of applications of mixed models where interest is on the variance components, we study the effects of random effects misspecification on random effects inference in linear mixed models, for which there is considerably less literature. Our findings are surprising and contrary to general belief: for point estimation, maximum likelihood estimation of the variance components under misspecification is consistent, although in finite samples, both the bias and mean squared error can be substantial. For inference, we show through theory and simulation that under misspecification, standard likelihood ratio tests of truly non‐zero variance components can suffer from severely inflated type I errors, and confidence intervals for the variance components can exhibit considerable under coverage. Furthermore, neither of these problems vanish asymptotically with increasing the number of clusters or cluster size. These results have major implications for random effects inference, especially if the true random effects distribution is heavier tailed than the normal. Fortunately, simple graphical and goodness‐of‐fit measures of the random effects predictions appear to have reasonable power at detecting misspecification. We apply linear mixed models to a survey of more than 4 000 high school students within 100 schools and analyse how mathematics achievement scores vary with student attributes and across different schools. The application demonstrates the sensitivity of mixed model inference to the true but unknown random effects distribution.

中文翻译：

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线性混合模型中随机效应的错误指定可能对随机效应的推断产生严重后果

在过去的二十年中，关于如何成功诊断混合模型中的随机效应错误指定（即假设真实分布为非正态时随机效应的正态性）及其对估计和推论的影响，已经进行了相当多且颇具争议的研究。 。但是，许多研究集中在广义线性混合模型中的固定效应推断上。在本文中，受关注于方差成分的混合模型的应用数量增加的影响，我们研究了线性混合模型中随机效应错误指定对随机效应推论的影响，有关文献很少。我们的发现令人惊讶，并且与普遍的看法背道而驰：对于点估计，尽管在有限样本中，偏差和均方误差都可能很大，但是在错误指定情况下方差分量的最大似然估计是一致的。作为推论，我们通过理论和仿真表明，在错误指定的情况下，真正非零方差分量的标准似然比测试可能会遭受严重夸大的I型错误，并且方差分量的置信区间会显示出相当低的覆盖率。此外，随着簇数或簇尺寸的增加，这些问题都不会渐近消失。这些结果对随机效应的推断具有重要意义，尤其是当真正的随机效应分布比正常分布重尾的时候。幸运的是，对随机效应预测的简单图形化和拟合优度度量在检测错误指定方面似乎具有合理的能力。我们使用线性混合模型对100所学校中的4000多名高中生进行了调查，并分析了数学成就分数如何随学生属性以及不同学校而变化。该应用程序演示了混合模型推断对真实但未知的随机效应分布的敏感性。