# How robust is the maximum likelihood estimator in structural equation modelling to a lack of multivariate normality?

In a Structural Equation Model, one often uses the ML estimator. In the case where the variables are not multivariate normal, can ML be used?

Often times the indicators you have available to work with are not multivariate normal. I am not sure how to proceed in that case.

• Does this question appear unclear? I see two close votes without even a comment asking for clarification... Dec 16 '15 at 21:52

There is a nicely written and highly-cited chapter by Finney & DiStefano (2008) that speaks to your questions (you can view most of it on Google Books). In summary, multivariate normality is typically evaluated using univariate skewness and kurtosis, and multivariate kurtosis--values less than 2, 7 and 3, respectively, are generally considered acceptable, though as of their writing, no simulation work had thoroughly vetted these cutoffs.

If your variables do not meet those criteria, could you still use ML estimation? Sure, and your parameters estimates (factor loadings, factor variances and covariances, etc.,) would be pretty accurate. Your standard errors and $\chi^2$ test of model fit (and therefore your other typical indexes of model fit), however, would be biased; the greater the departure from multivariate normality, the greater the amount of bias you could expect.

In most cases, and as Finney & DiStefano's (2008) review suggests, the most straightforward way to handle to non-normality is to use a robust ML estimator, that corrects for non-normality-induced bias in the standard errors, and produces a Satorra-Bentler (S-B) $\chi^2$ (and associated model fit indexes) that more accurately captures the appropriate amount of misfit in your model than the standard $\chi^2$ test of perfect fit (Satorra & Bentler, 2010).

lavaanhas a few robust ML estimators, though only the MLM estimator produces the S-B $\chi^2$. I'm not familiar with simulation work comparing the S-B $\chi^2$ to other corrections like the Yuan-Bentler (Y-B) $\chi^2$ produced by the MLR estimator, or their technical differences from one another. However, I have used both MLM and MLR in other SEM software (e.g., Mplus) and they usually produce very similar results. You might also want to consider MLR over MLM if you have some missing data to deal with (MLM is for complete cases only), and then read up on how the Y-B $\chi^2$ is different from the S-B $\chi^2$.

References

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.

• Yuan-Bentler and Satorra-Bentler tend to be very similar. YB has the advantage that you can use full information estimators for missing data. Dec 17 '15 at 6:21