For the special case of balanced one-way random effects ANOVA, it has been established that the generalized likelihood ratio test (LRT) and Wald’s test are largely equivalent in testing the variance component. We extend these results to explore the relationships between Wald’s F test, and the LRT for a much broader class of linear mixed models; the generalized split-plot models. In particular, we explore when the two tests are equivalent and prove that when they are not equivalent, Wald’s F test is more powerful, thus making the LRT test inadmissible. We show that inadmissibility arises in realistic situations with common number of degrees of freedom. Further, we derive the statistical distribution of the LRT under both the null and alternative hypotheses $$H_0$$ H 0 and $$H_1$$ H 1 where $$H_0$$ H 0 is the hypothesis that the between variance component is zero. Providing an exact distribution of the test statistic for the LRT in these models will help in calculating a more accurate p-value than the traditionally used p-value derived from the large sample chi-square mixture approximations.

中文翻译：

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关于在一类线性混合模型中检验方差分量的 LRT 和 F 检验之间的等价性

对于平衡单向随机效应方差分析的特殊情况，已确定广义似然比检验 (LRT) 和 Wald 检验在检验方差分量方面在很大程度上是等效的。我们扩展这些结果以探索 Wald 的 F 检验与更广泛的线性混合模型类别的 LRT 之间的关系；广义裂区模型。特别是，我们探索了两个测试何时等价，并证明当它们不等价时，Wald's F 检验更强大，从而使 LRT 测试不可接受。我们表明，在具有常见自由度数的现实情况下，不可受理性会出现。更多，我们在原假设和替代假设 $$H_0$$H 0 和 $$H_1$$ H 1 下推导出 LRT 的统计分布，其中 $$H_0$$ H 0 是方差分量为零的假设。在这些模型中为 LRT 提供测试统计量的精确分布将有助于计算比传统使用的从大样本卡方混合近似得出的 p 值更准确的 p 值。