a University of Notre Dame.
Multivariate Behav Res. 2017 Nov-Dec;52(6):673-698. doi: 10.1080/00273171.2017.1354759. Epub 2017 Sep 11.
Survey data often contain many variables. Structural equation modeling (SEM) is commonly used in analyzing such data. With typical nonnormally distributed data in practice, a rescaled statistic T proposed by Satorra and Bentler was recommended in the literature of SEM. However, T has been shown to be problematic when the sample size N is small and/or the number of variables p is large. There does not exist a reliable test statistic for SEM with small N or large p, especially with nonnormally distributed data. Following the principle of Bartlett correction, this article develops empirical corrections to T so that the mean of the empirically corrected statistics approximately equals the degrees of freedom of the nominal chi-square distribution. Results show that empirically corrected statistics control type I errors reasonably well even when N is smaller than 2p, where T may reject the correct model 100% even for normally distributed data. The application of the empirically corrected statistics is illustrated via a real data example.
调查数据通常包含许多变量。结构方程模型（SEM）常用于分析此类数据。由于实际中存在典型的非正态分布数据，文献中推荐使用 Satorra 和 Bentler 提出的重标统计量 T。然而，当样本量 N 较小和/或变量数 p 较大时，T 已被证明存在问题。对于具有小 N 或大 p 的 SEM，特别是对于非正态分布数据，不存在可靠的检验统计量。本文遵循 Bartlett 校正原理，对 T 进行了经验校正，以使经验校正统计量的均值近似等于名义卡方分布的自由度。结果表明，即使在 N 小于 2p 的情况下，经验校正统计量也能合理地控制Ⅰ类错误，即使对于正态分布数据，T 也可能 100%拒绝正确的模型。通过一个实际数据示例说明了经验校正统计量的应用。
