Multivariate Normality testing

A useful statistic for checking multivariate Normality, Mardia's (1970,1974) multivariate kurtosis coefficient, which can be normalised and compared to a Standard Normal Distribution may be evaluated [:FAQ/Rmardia: using MATLAB code], [:FAQ/Rcodeg2: using R code] or the statistical software package EQS (1995) which is available for use in CBSU. Most other structural equation modelling software should also routinely compute Mardia's kurtosis coefficient.

For N cases with p variables and a sample covariance matrix, S, we have

$$ g_text{2,p} = N^{text{-1} \sum_text{i=1}}text{N}[ (z_text{t} - \bar{z})^text{T}Stext{-1}(z_text{t} - \bar{z}) ]^text{2} - p(p+2) $$

where vectors, $$z_text{t}$$ and $$\bar{z}$$ are individual case score and mean vectors.

and the normalised estimate

$$ g_text{2,p} / \sqrt{(8p(p+2)/N)} $$

The hypothesis of multivariate Normality should be rejected for both large and small of the normalised estimate values when using very large samples ie values above +1.96 or below -1.96.

A Fortran 77 program for evaluating Kant's method for testing multivariate normality is also available together with test data sets and references for other methods. It is contained in this zip file [attachment:mvnorm.zip here.]

References

Bentler P.M. (1995) EQS Structural Equations Program Manual. Multivariate software Inc. :Enchino, CA.

Mardia, K.V. (1970) Measures of multivariate skewness and kurtosis with applications. Biometrika, 57, 519-530.

Mardia, K.V. (1974) Applications of some measures of multivariate skewness and kurtosis in testing normality and robustness studies. Sankhya, B36, 115-128.