Measures of Multivariate Kurtosis Spreadsheet

This spreadsheet is available when you click the Multivariate Kurtosis button on the Assumptions tab of the Structural Equation Modeling Results dialog box. The statistics described here enable you to analyze raw data to examine whether assumptions of multivariate normality have been violated.

Mardia Coefficient of Multivariate Kurtosis. If the sample comes from a multivariate normal distribution, this coefficient should be close to zero.

Normalized Multivariate Kurtosis. This normalized version of the Mardia coefficient has a distribution that is approximately standard normal at large samples. The measure may be evaluated roughly like any other standard normal statistic.

Additional Measures of Kurtosis. The following additional measures of kurtosis are provided for compatibility with other structural modeling programs.

Mardia-Based Kappa. The elliptical distribution family includes the multivariate normal distribution as a special case.  In this distribution family, all variables have a common kurtosis parameter

k = (σiiii/3sii2) - 1

This parameter can be used to rescale the Chi-square statistic if the assumption of an elliptical distribution is valid. The Mardia-based kappa is an estimate of kappa obtained by rescaling Mardia's coefficient of multivariate kurtosis. This number should be close to zero if the population distribution is multivariate normal

Mean Scaled Univariate Kurtosis. This is an alternate estimate of kappa, obtained simply by averaging the rescaled univariate kurtoses.

Adjusted Mean Scaled Univariate Kurtosis. Distribution theory provides a lower bound for kappa. It must never be less than -2/(p+2), where p is the number of variables. The preceding two estimates do not always obey this bound.  This estimate averages the scaled univariate kurtosis, but adjusts each one that falls below this bound to be at the lower bound point. This coefficient should be close to zero if the distribution is multivariate normal

Relative Multivariate Kurtosis. This measure is the Mardia-based kappa, rescaled to have a mean of 1. It should be close to 1 in value if the distribution is multivariate normal.