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T2Hot1.m
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T2Hot1.m
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function [T2,P] = T2Hot1(X,alpha,mu)
%Hotelling's T-Squared test for one multivariate sample.
%
% Syntax: function [T2Hot1] = T2Hot1(X,alpha)
%
% Inputs:
% X - multivariate data matrix.
% alpha - significance level (default = 0.05).
%
% Output:
% n - sample-size.
% p - variables.
% T2 - Hotelling's T-Squared statistic.
% Chi-sqr. or F - the approximation statistic test.
% df's - degrees' of freedom of the approximation statistic test.
% P - probability that null Ho: is true.
%
%
% Example: For the example given by Johnson and Wichern (1992, p. 183),
% with 20 cases (n = 20) and three variables (p = 3). We are interested
% to test if there is any difference against the expected mean vector
% [4 50 10] at a significance level = 0.10.
% -------------- --------------
% x1 x2 x3 x1 x2 x3
% -------------- --------------
% 3.7 48.5 9.3 3.9 36.9 12.7
% 5.7 65.1 8.0 4.5 58.8 12.3
% 3.8 47.2 10.9 3.5 27.8 9.8
% 3.2 53.2 12.0 4.5 40.2 8.4
% 3.1 55.5 9.7 1.5 13.5 10.1
% 4.6 36.1 7.9 8.5 56.4 7.1
% 2.4 24.8 14.0 4.5 71.6 8.2
% 7.2 33.1 7.6 6.5 52.8 10.9
% 6.7 47.4 8.5 4.1 44.1 11.2
% 5.4 54.1 11.3 5.5 40.9 9.4
% -------------- --------------
%
% Total data matrix must be:
% X=[3.7 48.5 9.3;5.7 65.1 8.0;3.8 47.2 10.9;3.2 53.2 12.0;3.1 55.5 9.7;
% 4.6 36.1 7.9;2.4 24.8 14.0;7.2 33.1 7.6;6.7 47.4 8.5;5.4 54.1 11.3;
% 3.9 36.9 12.7;4.5 58.8 12.3;3.5 27.8 9.8;4.5 40.2 8.4;1.5 13.5 10.1;
% 8.5 56.4 7.1;4.5 71.6 8.2;6.5 52.8 10.9;4.1 44.1 11.2;5.5 40.9 9.4];
%
% Calling on Matlab the function:
% T2Hot1(X)
% Immediately it ask:
% -Do you have an expected mean vector? (y/n):
% For this example we must to put:
% y (meaning 'yes')
% Then it ask:
% -Give me the expected mean vector:
% Giving the mean vector:
% [4 50 10]
% Otherwise (n; meaning 'no') it consider a zero mean vector.
%
% Answer is:
% ------------------------------------------------------------------------------------
% Sample-size Variables T2 F df1 df2 P
% ------------------------------------------------------------------------------------
% 20 3 9.7388 2.9045 3 17 0.0649
% ------------------------------------------------------------------------------------
% Mean vectors results significant.
%
%
% 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.
% And the special collaboration of the post-graduate students of the 2002:2
% Multivariate Statistics Course: Karel Castro-Morales, Alejandro Espinoza-Tenorio,
% Andrea Guia-Ramirez.
%
% Copyright (C) December 2002
%
% References:
%
% Johnson, R. A. and Wichern, D. W. (1992), Applied Multivariate Statistical Analysis.
% 3rd. ed. New-Jersey:Prentice Hall. pp. 180-181,199-200.
%
if nargin < 1,
error('Requires at least one input argument.');
end;
if nargin < 2,
alpha = 0.05;
end;
if (alpha <= 0 | alpha >= 1)
fprintf('Warning: significance level must be between 0 and 1\n');
return;
end;
[n,p]=size(X);
if nargin < 3,
mu=zeros([1,p]);
end;
if n <= p,
error('Warning: requires that sample-size (n) must be greater than the number of variables (p).');
return;
else
m=mean(X); %Mean vector from data matrix X.
S=cov(X); %Covariance matrix from data matrix X.
T2=n*(m-mu)*inv(S)*(m-mu)'; %Hotelling's T-Squared statistic.
if n >= 50 %Chi-square approximation.
X2=T2;
v=p; %Degrees of freedom.
P=1-chi2cdf(X2,v); %Probability that null Ho: is true.
% disp(' ')
% fprintf('----------------------------------------------------------------------------\n');
% disp(' Sample-size Variables T2 Chi-sqr. df P')
% fprintf('----------------------------------------------------------------------------\n');
% fprintf('%8.i%13.i%15.4f%14.4f%11.i%14.4f\n\n',n,p,T2,X2,v,P);
% fprintf('----------------------------------------------------------------------------\n');
% if P >= alpha;
% disp('Mean vectors results not significant.');
% else
% disp('Mean vectors results significant.');
% end;
else %F approximation.
F=(n-p)/((n-1)*p)*T2;
v1=p; %Numerator degrees of freedom.
v2=n-p; %Denominator degrees of freedom.
P=1-fcdf(F,v1,v2); %Probability that null Ho: is true.
% disp(' ')
% fprintf('-------------------------------------------------------------------------------------\n');
% disp(' Sample-size Variables T2 F df1 df2 P')
% fprintf('-------------------------------------------------------------------------------------\n');
% fprintf('%8.i%13.i%15.4f%11.4f%9.i%14.i%14.4f\n\n',n,p,T2,F,v1,v2,P);
% fprintf('-------------------------------------------------------------------------------------\n');
% if P >= alpha;
% disp('Mean vectors results not significant.');
% else
% disp('Mean vectors results significant.');
% end;
end;
end;
return;