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| 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] 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. | 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. |
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| $$ g_text{2,p} = N^text{-1} \sum_text{i=1}^text{N}[ (z_text{t} - \bar{z})^text{T}S^text{-1}(z_text{t} - \bar{z}) ]^text{2} - p(p+2) $$ | g(2,p) = N^-1^ Sum(i=1^N) [ (z(t) - zbar})^T^} S^-1^ (z(t) - zbar) ]^2^ - p(p+2) |
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| where vectors, $$z_text{t}$$ and $$\bar{z}$$ are individual case score and mean vectors. | where vectors, z(t) and zbar are individual case score and mean vectors. |
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| $$ g_text{2,p} / \sqrt{(8p(p+2)/N)} $$ | g(2,p) / Sqrt{(8p(p+2)/N)} |
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| MANOVA is robust to modest violations of multivariate Normality for equal sample sizes, df of 20 in the univariate analyses or at least 20 observations in cells when dealing with unequal samples (Tabachnick and Fidell, 2007). | |
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| 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.]] | |
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| 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.] | A SPSS macro from DeCarlo (1997) for evaluating Mardia's g2 test of kurtosis and skewness (g1) are available from [[http://www.columbia.edu/~ld208/|here.]] In this article DeCarlo points out that a lack of ''univariate'' skewness and kurtosis are necessary but not sufficient conditions for multivariate skewness, kurtosis and normality. |
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| __Note__: A slightly edited version of DeCarlo's SPSS macro is reproduced [[FAQ/mvnmacro| here]] where full stops have been added to the comment lines (asterisked) and the DO IF statement to allow the program to run. | |
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| * [[FAQ/Simon| More on testing univariate skew and kurtosis (which DeCarlo(1997) suggests as necessary but not sufficient for testing multivariate Normality).]] | |
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| DeCarlo L.T. (1997) [[http://www.columbia.edu/~ld208/psymeth97.pdf|On the meaning and use of kurtosis.]] ''Psychological Methods'', '''2''', 292-307. |
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Tabachnick, B. G. and Fidell, L. S. (2007) Using multivariate statistics. Pearson International:Boston,USA. |
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 using MATLAB code, 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(2,p) = N-1 Sum(i=1^N) [ (z(t) - zbar})T} S-1 (z(t) - zbar) ]2 - p(p+2)
where vectors, z(t) and zbar are individual case score and mean vectors.
and the normalised estimate
g(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.
MANOVA is robust to modest violations of multivariate Normality for equal sample sizes, df of 20 in the univariate analyses or at least 20 observations in cells when dealing with unequal samples (Tabachnick and Fidell, 2007).
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 here.
A SPSS macro from DeCarlo (1997) for evaluating Mardia's g2 test of kurtosis and skewness (g1) are available from here. In this article DeCarlo points out that a lack of univariate skewness and kurtosis are necessary but not sufficient conditions for multivariate skewness, kurtosis and normality.
Note: A slightly edited version of DeCarlo's SPSS macro is reproduced here where full stops have been added to the comment lines (asterisked) and the DO IF statement to allow the program to run.
References
Bentler P.M. (1995) EQS Structural Equations Program Manual. Multivariate software Inc. :Enchino, CA.
DeCarlo L.T. (1997) On the meaning and use of kurtosis. Psychological Methods, 2, 292-307.
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.
Tabachnick, B. G. and Fidell, L. S. (2007) Using multivariate statistics. Pearson International:Boston,USA.