Abstract: Structural equation modeling is an important method for analyzing multivariate data in the studies of psychology, behavior, management, marketing, etc. The usual regression models are simple cases of structural equation models. Compared with regression models from the view of covariance structural analysis, structural equation models are easier to be understood why they need model-fit testing by using goodness of fit indexes. We introduce the source of fit indexes, methods of model-fit testing and criteria of fit indexes in structural equation models. Our main purpose is to reveal the essence of model-fit testing using some popular fit-indices, including NNFI (Non-normed Fit Index), CFI (Comparative Fit Index), Mc (Measure of Centrality), and RMSEA (Root Mean Square Error of Approximation). It is shown that the criterion of an absolute fit index based on chi-square (e.g., RMSEA, Mc) is to set significance levels (might be much lower than traditional level of 0.05) for chi-square test. According to these comparable significance levels we could know which criterion is harsher to accept the theoretical model. When the degree of freedom is not larger than 32, the theoretical model will be accepted under the criteria of Mc > 0.9 if the model is accepted under the criteria of RMSEA < 0.08. When the degree of freedom is not less than 33, conversely, the theoretical model will be accepted under the criteria of RMSEA < 0.08 if the model is accepted under the criteria of Mc > 0.9. Thus RMSEA < 0.08 and Mc > 0.9 are compensatory criteria for model-fit testing in structural equation modeling. It is also shown that the criterion of a relative fit index is to set a proportion of reduced mean square (the ratio of chi-square to its degree of freedom) from the null model to the theoretical model. The null model is the worst model-fit because all indicators in the null model are set to be uncorrelated each other. Thus, the null model has the largest chi-square and the largest degree of freedom. For any given cutoff value, if the theoretical model is accepted under the criterion of NNFI, the model will be accepted definitively under the criterion of CFI. In other words, CFI is always not less than NNFI. Therefore, the criterion of CFI is covered by the criterion of NNFI. It is recommended that applied researchers should report and test at least one absolute fit index and one relative fit index. For absolute fit index, both Mc (cutoff value 0.9) and RMSEA Mc (cutoff value 0.08) are recommended. For relative fit index, however, only NNFI (cutoff value 0.9) is recommended. In addition, SRMR (Standardized Root Mean square Residual, cutoff value 0.08) is deserved of reference because it is only one that is not defined based on chi-square in the popular fit indexes.

Key words: structural equation model, fit index, fit testing, chi-square, mean square, cutoff value

摘要： 拟合指数检验是评价结构方程模型(SEM)的重要环节。从协方差结构分析的角度将SEM与传统的回归模型比较，容易理解为什么SEM需要拟合指数。揭示了目前几种流行的拟合指数检验的实质：基于卡方的绝对拟合指数(如RMSEA和Mc)检验的实质是重新设定卡方检验的显著性水平(不同于通常的0.05)，相对拟合指数(如NNFI和CFI)检验的实质是基于虚模型设定均方(卡方与自由度之比)降低到的比例；RMSEA和Mc两个检验有互补的作用，但CFI检验被NNFI检验覆盖。根据研究结果提出了一些方便实用的拟合检验建议。

关键词: 结构方程模型, 拟合指数, 拟合检验, 卡方, 均方, 临界值