Non-normality of Data in Structural Equation Models
Non-normality of Data in Structural Equation Models was written by Shengyi Gao, Patricia L. Mokhtarian, and Robert A. Johnston in 2008. It was published in Transportation Research Record (vol. 2082).
A problem for SEM is the assumption that data has a multivariate normal distribution. The authors document that some analysts proceed by
- deleting outlier observations to 'force' normality
- transforming variables "(e.g. square root, logarithm, Box-Cox)"
Either way, goal is to reduce skewness and kurtosis. These are generally evaluated according to Mardia's test, i.e. delete outliers identified by Mahalanobis distance until the test statistics indicate that the distribution is multivariate normal.
The authors test these approaches using Census data for Sacramento County.
The authors fund that deleting observations significantly changes the patterns of covariance. They found that 137 observations had to be deleted to achieve a critical ratio below 1.96. Compare to deleting the 6 extreme outliers, which lowered the critical ratio by more than 86%. They also note that when Muthén and Kaplan indicated a kurtosis of over 21, the bias from non-normality was less than 5%. Altogether, deleting until a critical ratio is achieved is not recommended.