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.


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.)

  • $\begingroup$ I assume that I have a misunderstanding here; could you please give me a hint why the variables are considered endogenous? Is it because they are reflective indicators of the latent variable and thus, "the presumed causes of endogenous variables are explicitly represented in the model" (Kline, 2011, p. 96)? $\endgroup$ – user150538 Feb 24 '17 at 16:16
  • $\begingroup$ Exogenous measured variables don't have arrows pointing at them. $\endgroup$ – Jeremy Miles Feb 24 '17 at 19:07
  • $\begingroup$ Alright, just to make (really) sure that I understood it correctly (as you mentioned "if you are doing CFA" in your answer): is it correct that the same would hold true for the measured variables even if "their" reflective latent variable would predict another factor (as in the example here? I remember that I once read a question that was also answered by you. The questioner mentioned that he had hoped that you would respond to the question - I do totally understand why. $\endgroup$ – user150538 Feb 24 '17 at 22:50
  • $\begingroup$ Yes. From the page: davidakenny.net/cm/basics.htm: Exogenous Variable - A variable that is not caused by another variable in the model. Usually this variable causes one or more variables in the model. $\endgroup$ – Jeremy Miles Feb 25 '17 at 0:04
  • $\begingroup$ Your measured vars are effects, not causes. $\endgroup$ – Jeremy Miles Feb 25 '17 at 0:05

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