The likelihood ratio test (LRT) is widely used for comparing the relative fit of nested latent variable models. Following Wilks' theorem, the LRT is conducted by comparing the LRT statistic with its asymptotic distribution under the restricted model, a $\chi^2$-distribution with degrees of freedom equal to the difference in the number of free parameters between the two nested models under comparison. For models with latent variables such as factor analysis, structural equation models and random effects models, however, it is often found that the $\chi^2$ approximation does not hold. In this note, we show how the regularity conditions of Wilks' theorem may be violated using three examples of models with latent variables. In addition, a more general theory for LRT is given that provides the correct asymptotic theory for these LRTs. This general theory was first established in Chernoff (1954) and discussed in both van der Vaart (2000) and Drton (2009), but it does not seem to have received enough attention. We illustrate this general theory with the three examples.
翻译:概率比测试( LRT) 被广泛用于比较嵌套潜伏变量模型的相对适切性。 按照 Wilks 的理论, LRT 的计算方法是将 LRT 统计数据与限制模式下的无症状分布进行比较, 其自由度等于两个嵌套模型之间自由参数数量的差异。 但是,对于具有潜在变量的模型, 如要素分析、结构方程模型和随机效应模型, 通常发现 $\ chi ⁇ 2$近似值无法维持。 在本说明中, 我们用三个具有潜伏变量的模型示例来说明Wilks 的常态性条件是如何被违反的。 此外, LRT 给出了更一般性的理论, 提供了这些 LRT 的正确性参数理论。 这个一般性理论最初在 Chernoff (1954年) 建立, 并在 van der Vaart ( 2000) 和 Drton(2009年) 讨论过, 但似乎没有得到足够重视。 我们用三个示例来说明这一一般性理论。