I have a dataset (n = 200) with several observed variables (answers to a 5-point Likert scale) and would like to perform CFA on that dataset (let's say it's a one-factor model with reflective indicators).
Before the actual evaluation, I assessed univariate normality of every variable (using the tests of Shapiro-Wilk and Anderson-Darling) and had to deny the hypothesis of normality in every single case. I also denied multivariate normality on the ground of Mardia's test with a high z-value.
Consequently, I checked several references and found (at least for me) divergent answers regarding the "requirements" for Satorra-Bentler test-statistics. One paper reported the following: "The variables in the data were multivariately nonnormally distributed, with normalized Mardia’s multivariate kurtosis of 6.03 (p < 001). We employed a maximum likelihood estimation method with robust standard errors together with the Satorra-Bentler rescaled chisquare statistic (Satorra and Bentler 1994), which compensates for nonnormality of variables." (Stern, Katz-Navon, Naveh, 2008, p. 1558)
I also checked Kline (2010, pp. 176-177) in which it is stated that in the case of non-normality of continuous endogeneous variables (the section is also termed "Corrected Normal Theory Methods for Continuous but Non-normal Outcomes"), corrected test statistics, such as Satorra-Bentler, are applicable.
At least from my understanding, both answers differ as Kline argues on the ground of the endogenous variables while Stern et al. simply analyze the raw data. Which of the two is correct and on the ground of which test would I normally argue for the use of Satorra-Bentler in a paper?
Kline, R. B. (2010). Principles and practice of structural equation modeling (3rd ed.). Guilford publications
Stern, Z., Katz-Navon, T., & Naveh, E. (2008). The influence of situational learning orientation, autonomy, and voice on error making: The case of resident physicians. Management Science, 54(9), 1553-1564.
Best Answer
If you are doing CFA, all your variables are endogenous.
The Satorra-Bentler correction is a sandwich estimator in regression (also called Huber-White, and confusingly also called robust). Hence it doesn't just account for non-normality, but is also corrects for heteroscedasticity.
But, like a sandwich estimator in regression, if you didn't violate any assumptions, it has almost no effect. And if you did violate any assumptions, it gives you better standard errors.
There is almost no cost to using it. So go ahead and use it.
(Although the Yung-Bentler T2*, which is more commonly known as MLR, is extremely similar and can handle missing data - so I almost always use that instead.)