= 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 [[Statistics/StructuralEquationModeling|SEM]] is the assumption that data has a [[Analysis/NormalDistribution|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 [[Statistics/MardiasTest|Mardia's test]], i.e. delete outliers identified by [[Statistics/MahalanobisDistance|Mahalanobis distance]] until the [[Statistics/TestStatistic|test statistics]] indicate that the distribution is multivariate normal.

The authors test these approaches using [[UnitedStates/CensusBureau/Census|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 [[AComparisonOfMethodologiesForTheFactorAnalysisOfNonnormalLikertVariables|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.



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