function MBoxtstwod(X,n,alpha)
% Multivariate Statistical Testing for the Homogeneity of Covariance Matrices Without Data by
% the Box's M.
%
% Syntax: function MBoxtstwod(X,n,alpha)
%
% Inputs:
% X - covariances matrix (covariances can be input in a rows or columns
% arrangement.)
% n - vector of groups-size.
% alpha - significance level (default = 0.05).
% Outputs:
% - MBox - the Box's M statistic.
% - Chi-sqr. or F - the approximation statistic test.
% - g - number of groups.
% - p - number of variables.
% - df's - degrees' of freedom of the approximation statistic test.
% - P-value.
%
% If the groups sample-size is at least 20 (sufficiently large), Box's M test
% takes a Chi-square approximation; otherwise it takes an F approximation.
%
% Example: For a two groups (g = 2) with three independent variables (p = 3), we
% are interested to test the homogeneity of covariances matrices without
% data with a significance level = 0.05. The two groups have the same
% sample-size n1 = n2 = 5. The variance-covariance matrix (S) for each
% group are:
%
% S1 S2
% -------------------- --------------------
% 5909 7573 1474 5474 8228 2619
% 7573 10401 1717 8228 19205 6204
% 1474 1717 412 2619 6204 3486
% -------------------- --------------------
%
% Data matrix must be:
% X = [5909 7573 1474 5474 8228 2619;7573 10401 1717 8228 19205 6204;
% 1474 1717 412 2619 6204 3486];
%
% n = [5 5];
%
% Calling on Matlab the function:
% mboxtstwod(X,n)
%
% Answer is:
%
% Box's M test for homogeneity of covariance matrices without data.
% --------------------------------------------------------------------------------
% MBox F g p df1 df2 P
% --------------------------------------------------------------------------------
% 27.4339 2.6556 2 3 6 463 0.0153
% --------------------------------------------------------------------------------
% With a given significance level of: 0.05
% Covariance matrices are significantly different.
%
% Created by A. Trujillo-Ortiz, R. Hernandez-Walls and A. Castro-Perez
% Facultad de Ciencias Marinas
% Universidad Autonoma de Baja California
% Apdo. Postal 453
% Ensenada, Baja California
% Mexico.
% atrujo@uabc.mx
% And the special collaboration of the post-graduate students of the 2004:2
% Multivariate Statistics Course: Laura Rodriguez-Cardoso, Rene Garcia-Sanchez,
% Norma A. Ramos-Delgado.
% Copyright. December 8, 2004.
%
% To cite this file, this would be an appropriate format:
% Trujillo-Ortiz, A. R. Hernandez-Walls and A. Castro-Perez (2204). MBoxtstwod:
% Multivariate Statistical Testing for the Homogeneity of Covariance Matrices
% Without Data by the Box's M. A MATLAB file. [WWW document]. URL http://
% www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=6548
%
% References:
%
% Stevens, J. (1992), Applied Multivariate Statistics for Social Sciences. 2nd. ed.
% New-Jersey:Lawrance Erlbaum Associates Publishers. pp. 260-269.
%
if nargin < 2,
error('Requires at least two input arguments.');
end;
if nargin < 3,
alpha = 0.05; %(default)
end;
if (alpha <= 0 | alpha >= 1)
fprintf('Warning: significance level must be between 0 and 1\n');
return;
end;
[r c] = size(X);
if c > r;
X = X';
else
X = X;
end;
p = size(X,2); %Number of variables.
ks = size(X,1);
g = ks/p; %Number of groups.
if g ~= fix(g);
error('Some of the covariance matrix has different size.');
end;
r = 1;
r1 = n(1);
bandera = 2;
for k = 1:g
if n(k) >= 20;
bandera = 1;
end;
end;
ma = 'S';
rr = 1;
for k = 1:g
b = X(rr:rr+p-1,:);
eval([ma num2str(k) '=b;']);
rr = rr+p;
end;
deno = sum(n)-g;
suma = zeros(size(S1));
tl = [];tu = [];
for k = 1:g
eval(['l =tril(S' num2str(k) ');']);
eval(['u =triu(S' num2str(k) ');']);
tl = [tl;l'];tu = [tu;u];
d = tl(:)-tu(:);
if any(d ~= 0);
disp('Warning:Some of the covariance matrix it is not symmetric.');
return;
end;
end;
for k = 1:g
eval(['suma =suma + (n(k)-1)*S' num2str(k) ';']);
end;
Sp = suma/deno; %Pooled covariance matrix.
Falta = 0;
for k = 1:g
eval(['Falta =Falta + ((n(k)-1)*log(det(S' num2str(k) ')));']);
end;
MB = (sum(n)-g)*log(det(Sp))-Falta; %Box's M statistic.
suma1 = sum(1./(n(1:g)-1));
suma2 = sum(1./((n(1:g)-1).^2));
C = (((2*p^2)+(3*p)-1)/(6*(p+1)*(g-1)))*(suma1-(1/deno)); %Computing of correction factor.
if bandera == 1
X2 = MB*(1-C); %Chi-square approximation.
v = (p*(p+1)*(g-1))/2; %Degrees of freedom.
P = 1-chi2cdf(X2,v); %Significance value associated to the observed Chi-square statistic.
disp(' ')
;
disp('Box''s M test for homogeneity of covariance matrices without data.')
fprintf('--------------------------------------------------------------\n');
disp(' MBox Chi-sqr. g p df P')
fprintf('--------------------------------------------------------------\n');
fprintf('%10.4f%11.4f%8.i%8.i%11.i%13.4f\n',MB,X2,g,p,v,P);
fprintf('--------------------------------------------------------------\n');
fprintf('With a given significance level of:% 3.2f\n', alpha );
if P >= alpha
disp('Covariance matrices are not significantly different.');
else
disp('Covariance matrices are significantly different.');
end;
else
%To obtain the F approximation we first define Co, which combined to the before C value
%are used to estimate the denominator degrees of freedom (v2); resulting two possible cases.
Co = (((p-1)*(p+2))/(6*(g-1)))*(suma2-(1/(deno^2)));
if Co-(C^2) >= 0;
v1 = (p*(p+1)*(g-1))/2; %Numerator degrees of freedom.
v21 = fix((v1+2)/(Co-(C^2))); %Denominator degrees of freedom.
F1 = MB*((1-C-(v1/v21))/v1); %F approximation.
P1 = 1-fcdf(F1,v1,v21); %Significance value associated to the observed F statistic.
disp(' ')
;
disp('Box''s M test for homogeneity of covariance matrices without data.')
fprintf('--------------------------------------------------------------------------------\n');
disp(' MBox F g p df1 df2 P')
fprintf('--------------------------------------------------------------------------------\n');
fprintf('%10.4f%11.4f%8.i%10.i%13.i%14.i%13.4f\n',MB,F1,g,p,v1,v21,P1);
fprintf('--------------------------------------------------------------------------------\n');
fprintf('With a given significance level of:% 3.2f\n', alpha );
if P1 >= alpha
disp('Covariance matrices are not significantly different.');
else
disp('Covariance matrices are significantly different.');
end;
else
v1 = (p*(p+1)*(g-1))/2; %Numerator degrees of freedom.
v22 = fix((v1+2)/((C^2)-Co)); %Denominator degrees of freedom.
b = v22/(1-C-(2/v22));
F2 = (v22*MB)/(v1*(b-MB)); %F approximation.
P2 = 1-fcdf(F2,v1,v22); %Significance value associated to the observed F statistic.
disp(' ')
;
disp('Box''s M test for homogeneity of covariance matrices without data.')
fprintf('--------------------------------------------------------------------------------\n');
disp(' MBox F g p df1 df2 P')
fprintf('--------------------------------------------------------------------------------\n');
fprintf('%10.4f%11.4f%8.i%10.i%13.i%14.i%13.4f\n',MB,F2,g,p,v1,v22,P2);
fprintf('--------------------------------------------------------------------------------\n');
fprintf('With a given significance level of:% 3.2f\n', alpha );
if P1 >= alpha
disp('Covariance matrices are not significantly different.');
else
disp('Covariance matrices are significantly different.');
end;
end;
end;
