I'm using covariance-based SEM software. Do I need to normalize the data before I run the SEM model? The SEM model is fine and the data is large (744).
1. Do I need to normalize the data before I run SEM
2. Would the central limit theorem not apply and so I need not normalize the data
3. Normalizing would change the basic characteristics.Would the findings still be valid
You might find the following helpful:
https://centerstat.org/can-i-estimate-an-sem-if-the-sample-data-are-not-normally-distributed/#:~:text=First%2C%20the%20assumption%20of%20normality,likelihood%20(ML)%20estimator%20does.
I would add, the usual maximum-likelihood estimation of a structural equation model assumes multivariate normality, and even if you can successfully normalize each variable separately (through log transformation, or whatever), you would not guarantee multivariate normality.
In CFA and SEM, the default estimator in most software programs is maximum likelihood (ML). ML estimation is based on the assumption of multivariate normality.
Ignoring multivariate non-normality and using regular (uncorrected) ML estimation can lead to biased tests of model fit (e.g., chi-square), biased standard errors, and incorrect tests of statistical significance. The parameter estimates (e.g., factor loadings, regression and path coefficients) are relatively unaffected by non-normality.
If your data are non-normal, rather than applying data transformations, you can simply use robust ML estimation (e.g., Satorra-Bentler correction; Bollen-Stine bootstrap) to obtain corrected standard errors and test statistics. Alternatively, you can use an estimation method that does not require normality (e.g., weighted least squares or WLS estimation). However, WLS requires very large samples to provide valid results, so most researchers choose use the first option (robust ML estimation).
You can also check out my Youtube videos on this topic here:
https://youtu.be/ny30wHaX0qA
It is not necessary to normalize the variables before conducting the analyses. Besides multivariate normality, another issue to consider is the measurement scales of the observed variables involved in the analysis. If the variables are "metric" (continuous) but do not meet the assumption of multivariate normality, one can follow the advice of Christian Geiser and use robust ML estimators (e.g., Satorra-Bentler correction). If the variables are ordinal categorical (typical Likert scales), it all depends on the number of scale categories. If the number of categories is high (e.g., 6 or more), it can be assumed that they are continuous and the previous method can be applied. If the number of categories is low (e.g., less than 5), it is usually recommended to use estimation methods that do not require assuming normality (e.g., WLS, DWLS, WLSMV). In any case, the minimum sample size required will be higher in the latter scenario.
You can find a brief introduction to the topic at https://web.pdx.edu/~newsomj/ho_estimate2.pdf.
DISCLAIMER: I claim no expertise in SEM. I just have a question about the multivariate normality assumption for SEM.
Christian Geiser wrote: ML estimation is based on the assumption of multivariate normality. One can find similar statements in many resources. But the question that always comes to mind for me is this: Multivariate normality of what?
If we back up for a moment and consider OLS regression, the necessary normality assumption is that the sampling distributions of the regression coefficients are approximately normal. Approximate normality of the model errors, which is often described as if it is necessary--but it is not, it is a sufficient normality assumption. I.e., if the error distribution is approximately normal, the sampling distributions of the coefficients will be approximately normal too.
Now, suppose I decided to estimate the same linear regression model using MLE rather than OLS. Would that change anything about the necessary normality assumption for the model? I am not mathematical statistician, but I don't see why it would.
Now, what about SEM? Would the necessary normality assumption for SEM be about "the data" rather than sampling distributions of something or other? Again, I cannot understand why that would be the case. But I have had a devil of a time finding reputable resources that answer my question, multivariate normality of what? Thus far, I have found only one resource, this article by John Fox:
- https://socialsciences.mcmaster.ca/jfox/Misc/sem/SEM-paper.pdf
On p. 475, Fox says that the latent variables and the errors are assumed to be "multinormally distributed"--see the attached image. And that makes much more sense to me than claims that "the data" must be multivariate normal, which is what many resources either say or imply.
Finally, Christian Geiser and other folks who are far more knowledgeable about SEM than I am: What say you about Fox's statement? Has he got it right? If he does, why is it so difficult to find resources that say clearly that the multivariate normality assumption applies to the latent variables and the errors? Thanks.
Dear Bruce,
interesting question.
To the best of my knowledge, the normality assumption is about the distribution of the observed variables (if the exogenous variables of the model are normally distributed, also the observed variables are normally distributed, so Fox is not doing wrong). There are (at least) two derivations of the ML fit function in this case. First, directly via the density function of the multivariate normal distribution (see e.g., Bollen 1989, Appendix 4A). Second, via the distribution of the sample variance-covariance matrix, see e.g., Hayduk (1998, Chapter 5.2.1).
Best regards,
Florian
Reference
Bollen, K. A. (1989). Structural Equations with Latent Variables. John Wiley & Sons.
Hayduk, L. A. (1988). Structural Equation Modeling with LISREL: Essentials and Advances. John Hopkins University Press.