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MATLAB code for computing Mardia's multivariate kurtosis coefficient in Multivariate Normality testing Normality testing
David Graham (2006) has written MATLAB code based upon the work of A. Trujillo-Oriz and R. Hernandez-Walls (2003) from the Universidad Autonoma de Baja California who compute Mardia's multivariate kurtosis coefficient which, including an example, is located here.
Reference
Trujillo-Ortiz, A. and R. Hernandez-Walls. (2003). Mskekur: Mardia's multivariate skewness and kurtosis coefficients and its hypotheses testing. A MATLAB file. [WWW document]. URL http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?
This code is also reproduced below:
function [H stats] = mardiatest(X,alpha)
% MARDIATEST Mardia multivariate normality test using skewness & kurtosis.
% H = MARDIATEST(X,ALPHA) performs Mardia's test of multivariate
% normality to determine if the null hypothesis of multivariate
% normality is a reasonable assumption regarding the population
% distributions of a random samples contained within the columns of X. X
% must be a N-values by M-samples array. The desired significance level,
% ALPHA, is an optional scalar input (default = 0.05).
%
% The function performs three tests: of the multivariate skewness; the
% multivariate skewness corrected for small samples; and the multivariate
% kurtosis. H is a 1-by-3 array indicating the results of the hypothesis
% tests:
% H(i) = 0 => Do not reject the null hypothesis at significance level
% ALPHA.
% H(i) = 1 => Reject the null hypothesis at significance level ALPHA.
%
% [H STATS] = MARDIATEST(...) also returns a structure array with the
% following fields:
% stats.Hs - logical scalar indicating whether to reject the null
% hypothesis that the multivariate skewness is consistent
% with a multivariate normal distribution.
% stats.Ps - the P value for the multivariate skewness.
% stats.Ms - the Mardia test statistic for the multivariate skewness.
% stats.CVs - the critical value for the multivariate skewness.
% stats.Hsc - logical scalar indicating whether to reject the null
% hypothesis that the multivariate skewness (corrected for
% small samples) is consistent with a multivariate normal
% distribution.
% stats.Psc - the P value for the multivariate skewness (corrected for
% small samples).
% stats.Msc - the Mardia test statistic for the multivariate skewness
% (corrected for small samples).
% stats.Hk - logical scalar indicating whether to reject the null
% hypothesis that the multivariate kurtosis is consistent
% with a multivariate normal distribution.
% stats.Pk - the P value for the multivariate kurtosis.
% stats.Mk - the Mardia test statistic for the multivariate kurtosis.
% stats.CVk - the critical value for the multivariate kurtosis.
%
% Notes:
% For multivariate data, tests of normality of the individual variables
% are not sufficent to determine multivariate normality. Even if every
% variable in a set is normally distributed, it is still possible that
% the combined distribution is not multivariate normal. Mardia's test is
% one means testing for multivariate normality. The test is based on
% independent tests of multivariate skewness and kurtosis. Data can be
% assumed to conform to multivariate normality only if the null
% hypothesis is not rejected for both tests.
%
% Example:
% >> X = [2.4 2.1 2.4; ...
% 3.5 1.8 3.9; ...
% 6.7 3.6 5.9; ...
% 5.3 3.3 6.1; ...
% 5.2 4.1 6.4; ...
% 3.2 2.7 4.0; ...
% 4.5 4.9 5.7; ...
% 3.9 4.7 4.7; ...
% 4.0 3.6 2.9; ...
% 5.7 5.5 6.2; ...
% 2.4 2.9 3.2; ...
% 2.7 2.6 4.1];
%
% >> [H stats] = mardiatest(X, 0.05)
%
% H =
% 0 0 1
%
% stats =
% Hs: 0
% Ps: 0.9660
% Ms: 3.5319
% CVs: 18.3070
% Hsc: 0
% Psc: 0.8918
% Msc: 4.9908
% Hk: 1
% Pk: 0.0452
% Mk: -1.6936
% CVk: 1.6449
%
% The magnitude of the Mardia kurtosis test statistic (Mk) is greater
% than critical value (CVk) for a 5% level test, so the hypothesis of
% multivariate normality is rejected.
%
% See also JBTEST, KSTEST, KSTEST2, LILLIETEST.
%
% This version:
% Copyright 2006 David Graham, Loughborough University
% (D.J.Graham@lboro.ac.uk)
% This function is based on MSKEKUR by A. Trujillo-Ortiz and R.
% Hernandez-Walls. Modifications include:
% - additional checking of inputs
% - modification of help text for consistency with Matlab conventions
% - the output of the test results to variables
% - removal of the code sending test results as a string to the command
% window
% - removed call to 'eval'.
%
% Statements from the original code:
%
% % Created by A. Trujillo-Ortiz and R. Hernandez-Walls
% % Facultad de Ciencias Marinas
% % Universidad Autonoma de Baja California
% % Apdo. Postal 453
% % Ensenada, Baja California
% % Mexico
% % atrujo@uabc.mx
% %
% % May 22, 2003.
% %
% % To cite this file, this would be an appropriate format:
% % Trujillo-Ortiz, A. and R. Hernandez-Walls. (2003). Mskekur: Mardia's multivariate skewness
% % and kurtosis coefficients and its hypotheses testing. A MATLAB file. [WWW document]. URL
% % http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=3519&objectType=FILE
% %
% % References:
% %
% % Mardia, K. V. (1970), Measures of multivariate skewnees and kurtosis with
% % applications. Biometrika, 57(3):519-530.
% % Mardia, K. V. (1974), Applications of some measures of multivariate skewness
% % and kurtosis for testing normality and robustness studies. Sankhyâ A,
% % 36:115-128
% % Stevens, J. (1992), Applied Multivariate Statistics for Social Sciences. 2nd. ed.
% % New-Jersey:Lawrance Erlbaum Associates Publishers. pp. 247-248.
% Check number of input arguements
if (nargin < 1) || (nargin > 2)
error('Requires one or two input arguments.')
end
% Define default ALPHA if only one input is provided
if nargin == 1,
alpha = 0.05;
end
% Check for validity of ALPHA
if ~isscalar(alpha) || alpha>0.5 || alpha <0
error('Input ALPHA must be a scalar between 0 and 0.5.')
end
[n,p] = size(X);
% Check for validity of X
if p < 2 || ~isnumeric(X)
error('Input X must be a numeric array with at least 2 columns.')
end
difT = [];
for j = 1:p
difT = [difT,(X(:,j) - mean(X(:,j)))];
end;
S = cov(X); % Variance-covariance matrix
D = difT * inv(S) * difT'; % Mahalanobis' distances matrix
b1p = (sum(sum(D.^3))) / n^2; % Multivariate skewness coefficient
b2p = trace(D.^2) / n; % Multivariate kurtosis coefficient
k = ((p+1)*(n+1)*(n+3)) / ...
(n*(((n+1)*(p+1))-6)); % Small sample correction
v = (p*(p+1)*(p+2)) / 6; % Degrees of freedom
g1c = (n*b1p*k) / 6; % Skewness test statistic corrected for small sample (approximates to a chi-square distribution)
g1 = (n*b1p) / 6; % Skewness test statistic (approximates to a chi-square distribution)
P1 = 1 - chi2cdf(g1,v); % Significance value of skewness
P1c = 1 - chi2cdf(g1c,v); % Significance value of skewness corrected for small sample
g2 = (b2p-(p*(p+2))) / ...
(sqrt((8*p*(p+2))/n)); % Kurtosis test statistic (approximates to a unit-normal distribution)
P2 = 1-normcdf(abs(g2)); % Significance value of kurtosis
% Prepare outputs
stats.Hs = (P1 < alpha);
stats.Ps = P1;
stats.Ms = g1;
stats.CVs = chi2inv(1-alpha,v);
stats.Hsc = (P1c < alpha);
stats.Psc = P1c;
stats.Msc = g1c;
stats.Hk = (P2 < alpha);
stats.Pk = P2;
stats.Mk = g2;
stats.CVk = norminv(1-alpha,0,1);
H = [stats.Hs stats.Hsc stats.Hk];