# Good Multivariate normality coefficient but suspicious univariate indices in AMOS, what to do?

We are conducting a research with 15 observed variables. Before starting a SEM model based on our data-set (of 531 cases), we tried to check for uni-variate & multivariate normality problems. The attached image shows the results from AMOS. A discussion between team members started from this point: are these results acceptable for the data-set to be analyzed in SEM or not?

Some researchers argued that while we have a good multivariate normality indices we can continue analyzing with ML estimation method. While other researches said that we should look, not only at the multivariate Mardia's coefficient, but also at the univariate indices which may indicate a univariate non-normality problem with some observed variables.

Also, a debate between the members about the acceptable c.r. values of Skew & Kurtosis was launched without access to a convincing opinion.

So, first, are the results acceptable for starting the SEM analyze with ML estimation method?

Second, if not, where is the major problem and how to resolve it?

Third, What is the acceptable range of c.r. values for Skew & Kurtosis coefficients?

I'm paraphrasing your questions a litte bit, in order to still answer them even though I don't think you have the data problems that you think you do:

Question 1: Are The Observed Skewness/Kurtosis values acceptable for ML-based SEM?

I would say yes. Based on suggested cutoffs for normality that I am familiar with (Skewness > 2, Kurtosis > 7; from Cohen, Cohen, West, & Aiken, 2002), your data actually do not violate univariate normality assumptions. And I think under most circumstances, it is quite unusual (if not impossible) to come across data that meet multivariate but not univariate normality assumptions.

Question 2: What To Do When Data Do Not Meet Normality Assumptions?

Let's say that you had data that did, in fact, have clear Skewness/Kurtosis problems. A likely consequence of this non-normality would be that your $\chi^2$ statistic (and therefore other indexes) of model fit will be biased, as would your standard errors for your model parameter estimates (Finney & DiStefano, 2008). One potential solution is the use of "robust" maximum-likelihood estimators (e.g., MLR, MLM) that will produce a Satorra-Bentler scaled $\chi^2$ test statistic and standard errors (Satorra & Bentler, 2010), which will be less biased. Finney and DiStefano's review (2008) suggests that this approach performs relatively well compared to alternatives, both under cases of moderately and severely non-normal data. If you go this route, just be mindful that you will need to use an additional correction factor in the course of performing nested-model comparisons with competing $\chi^2$ values (the Mplus folks have a good discussion of this issue here)

PS: sadly, I'm not certain what "c.r." values for Skewness and Kurtosis represent.

References

Cohen, J., Cohen, P., & Stephen, G. (2002). West, and Leona S. Aiken. Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences (3rd edition). Mahwah, NJ: Lawrence Erlbaum.

Finney, S. J., & DiStefano, C. (2008). Non-normal and categorical data in structural equation modeling. In G. R. Hancock & R. D. Mueller (Eds.), Structural Equation Modeling: A Second Course (pp. 269-314). Information Age Publishing.

Satorra, A., & Bentler, P.M. (2010). Ensuring positiveness of the scaled difference chi-square test statistic. Psychometrika, 75, 243-248.

• cr is critical ratio - it's the estimate divide by SE. Not sure why they call it cr, not z (or t). Dec 11, 2015 at 22:02

C. r. values (critical ratio) values equal to 1.96 or less point to the existence of non-significant Skewness or kurtosis. Value of 0.679 < 1.96 mean there isn't significant non-normality, for the correspondent Mardia's coefficient value (2.55). I would next check the Mahalanobis d-squared distance, verify the highest possible outliers and might delete them. Re-run and check the values of sk and ku. Them possibility I would go with the ML estimator, and then check the loadings of the itens with "bad" sk and ku values. The second approach it's using the ADF estimator.