Robust transformation with applications to structural equation modelling
Robust transformation with applications to structural equation modelling (DOI: https://doi.org/10.1348/000711000159169) was written by Ke-Hai Yuan, Wai Chan, and Peter M. Bentler in 2000. It was published in the British Journal of Mathematical and Statistical Psychology (vol. 53).
The authors discuss the assumption of normality in SEM.
One way to address nonnormality is a GLS approach which is asymptotically distribution-free. However, they note that "the sample covariance matrix S is efficient only when data are normal." Any method based on a sample covariance matrix can be inefficient or biased.
Another approach is transforming variables to achieve normality. Transformations are straightforward to implement and reason about, but the conceptual challenge is relating the statistics (esp. variance) which were fit on transformed data back to the original sample.
The final approach is robust statistics. Robust estimators are also more efficient than standard statistics when the sample distribution has fat tails.
The authors proceed with a comparison of these two approaches. They demonstrate that the sample covariance of data that is transformed to normality is the same as the robust sample covariance. Therefore, given that robust statistics also gain efficiency in some cases, they are generally preferable.