Linear mixed models (LMMs) are widely used for regression analysis of data that are assumed to be clustered or correlated. Assessing model fit is important for valid inference but to date no confirmatory tests are available to assess the adequacy of the fixed effects part of LMMs against general alternatives. We therefore propose a class of goodness-of-fit tests for the mean structure of LMMs. Our test statistic is a quadratic form of the difference between observed values and the values expected under the estimated model in cells defined by a partition of the covariate space. We show that this test statistic has an asymptotic chi-squared distribution when model parameters are estimated by maximum likelihood or by least squares and method of moments, and study its power under local alternatives both analytically and in simulations. Data on repeated measurements of thyroglobulin from individuals exposed to the accident at the Chernobyl power plant in 1986 are used to illustrate the proposed test.

中文翻译：

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线性混合模型的拟合优度检验

线性混合模型 (LMM) 广泛用于对假定为聚类或相关的数据进行回归分析。评估模型拟合对于有效推理很重要，但迄今为止还没有验证性测试可用于评估 LMM 的固定效应部分相对于一般替代方案的充分性。因此，我们为 LMM 的平均结构提出了一类拟合优度检验。我们的检验统计量是观测值与估计模型下由协变量空间分区定义的单元格中的预期值之间差异的二次形式。我们表明，当通过最大似然或最小二乘法和矩量法估计模型参数时，该检验统计量具有渐近卡方分布，并在分析和模拟中研究其在局部替代方案下的功效。