Testing Normality in Structural Equation Modeling


Posted February 5, 2020

When conducting a structural equation model (SEM) or confirmatory factor analysis (CFA), it is often recommended to test for multivariate normality. Some popular SEM software packages (such as AMOS) assume your variables are continuous and produce the best results when your data are normally distributed. Here we discuss a few options for testing normality in SEM.

First, it is possible to test for multivariate normality using a quantile (Q-Q) or probability (P-P) plot, which can be done though the Analyze > Descriptive Statistics menu in SPSS (see our previous blog on this topic for more details). Similarly, you can conduct a quantile plot of Mahalanobis distances to test for normality (the steps for calculating Mahalanobis distances in SPSS are outlined here). If you use Intellectus Statistics to conduct your analysis, the Mahalanobis distances method will automatically be performed for you. For all of these methods, a plot is produced, and the points on that plot should follow a relatively straight line. Marked deviations from a straight line suggest that the data are not multivariate normal.

If you are conducting your analysis in AMOS, the built-in test for normality involves the calculation of Mardia’s coefficient, which is a multivariate measure of kurtosis. AMOS will provide this coefficient and a corresponding “critical value” which can be interpreted as a significance test (a critical value of 1.96 corresponds to a p-value of .05). If Mardia’s coefficient is significant, (i.e., the critical ratio is greater than 1.96 in magnitude) the data may not be normally distributed. However, this significance test on its own is not a practical assessment of normality, especially in SEM. This is because tests such as these are highly sensitive to sample size, with larger sample sizes being more likely to produce significant (non-normal) results. In SEM, where your sample size is expected to be very large, this means that Mardia’s coefficient is almost always guaranteed to be significant. Thus, the significance test on its own does not provide very useful information. In light of this, it is recommended that the significance tests be used in conjunction with descriptive statistics, namely the kurtosis values for individual variables (Stevens, 2009). Kurtosis values greater than 3.00 in magnitude may indicate that a variable is not normally distributed (Westfall & Henning, 2013).

There are many ways to test for normality, and these are just a few of the most popular methods used in support of SEM analysis.

References

Stevens, J. P. (2009). Applied multivariate statistics for the social sciences (5th ed.). Mahwah, NJ: Routledge Academic.

Westfall, P. H., & Henning, K. S. S. (2013). Texts in statistical science: Understanding advanced statistical methods. Boca Raton, FL: Taylor & Francis.


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