function [MAOV1] = MAOV1(X,alpha)
%Single-Factor Multivariate Analysis of Variances test.
%Computes a Multivariate Analysis of Variance for equal or unequal sample sizes.
%[Testing of the mean differences in several variables among several samples (groups).
%It consider the matrices of sum-of-squares between (H) and within (E) to compute the
%Wilks'L (lambda) and according to the number of data it chooses to approximate to the
%Chi-square distribution or through several multiple F-approximations such as Rao,
%Pillai, Lawley-Hotelling and Roy tests.]
%
% Syntax: function [MAOV1] = MAOV1(X,alpha)
%
% Inputs:
% X - data matrix (Size of matrix must be n-by-(1+p); sample=column 1, variables=column 2:p).
% alpha - significance (default = 0.05).
% Outputs:
% - Complete multivariate analysis of variance table.
%
% Example: From the example 6.8 of Johnson and Wichern (1992, pp. 249-251), to test the
% multivariate analysis of variance of data with a significance of 0.05.
%
% Data
% ----------------------
% Sample X1 X2
% ----------------------
% 1 9 3
% 1 6 2
% 1 9 7
% 2 0 4
% 2 2 0
% 3 3 8
% 3 1 9
% 3 2 7
% ----------------------
%
% Data matrix must be:
% X=[1 9 3;1 6 2;1 9 7;2 0 4;2 2 0;3 3 8;3 1 9;3 2 7];
%
% Calling on Matlab the function:
% maov1(X)
%
% Answer is:
%
% It is considering as a small sample problem (n < 25).
%
% Multivariate Analysis of Variance Table.
% --------------------------------------------
% No. data Samples Variables L
% --------------------------------------------
% 8 3 2 0.0385
% --------------------------------------------
%
% ------------------------------------------------------------------------------
% Test Statistic df1 df2 F P Conclusion
% ------------------------------------------------------------------------------
% Rao 0.038 4 8 8.20 0.0062 S
% Pillai 1.541 4 10 8.39 0.0031 S
% Lawley-Hotelling 9.941 4 6 7.46 0.0164 S
% Roy 8.076 2.0 5.0 20.19 0.0040 S
% ------------------------------------------------------------------------------
% With a given significance of: 0.05
% According to the P-value, the sample mean vectors could be significant (S) or
% not significant (NS).
%
% 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
%
% August 10, 2003.
%
% To cite this file, this would be an appropriate format:
% Trujillo-Ortiz, A. and R. Hernandez-Walls. (2003). MAOV1: Single-Factor Multivariate Anaysis of
% Variance test. A MATLAB file. [WWW document]. URL http://www.mathworks.com/matlabcentral/fileexchange/
% loadFile.do?objectId=3863&objectType=FILE
%
% References:
%
% Boik, J. R. (2002). Lecture Notes: Statistics 537. Classical Multivariate Analysis.
% Spring 2002. pp. 38-41. [WWW document]. URL http://math.montana.edu/~rjboik/
% classes/537/notes_537.pdf
% Johnson, D. E. (1998), Applied Multivariate Methods for Data Analysis.
% New-York:Brooks Cole Publishing Co., an ITP Company. Chapter 11.
% Johnson, R. A. and Wichern, D. W. (1992), Applied Multivariate Statistical Analysis.
% 3rd. ed. New-Jersey:Prentice Hall. pp. 246-251.
% SAS/Stat User's Guide. Chap. 3. Introduction to Regression Procedures: Statistical
% Background: Multivariate Tests. Software-Kategorie Statistik und Mathematik:
% Datenanalyse mit SAS mit SAS Online Doc. Universitt Zrich. [WWW document].
% URL http://www.id.unizh.ch/software/unix/statmath/sas/sasdoc/stat/chap3/sect10.htm
%
P = [];PP = [];PLH = [];PR = [];
if nargin < 2,
alpha = 0.05;
end
g = max(X(:,1));
N = [];
indice = X(:,1);
for i = 1:g
Xe = find(indice==i);
eval(['X' num2str(i) '= X(Xe,2);']);
eval(['n' num2str(i) '= length(X' num2str(i) ') ;'])
eval(['xn= n' num2str(i) ';'])
N = [N,xn];
end
[f,c] = size(X);
X = X(:,2:c);
[n,p] = size(X);
r = 1;
r2 = N(1);
for k = 1:g
eval(['M' num2str(k) '= mean(X(r:r2,:));']);
if k < g
r = r+N(k);
r2 = r2+N(k+1);
end
end
M = mean(X);
dT = [];
for k = 1:p
eval(['dT = [dT,(X(:,k) - mean(X(:,k)))];']);
end
T = dT'*dT; %total sum of squares
r = 1;
r2 = N(1);
g = length(N);
for k = 1:g
Md(k,:) = (mean(X(r:r2,:)) - mean(X));
if k < g
r = r+N(k);
r2 = r2+N(k+1);
end
end
H = []; %between samples sum of squares
for k = 1:g
h = N(k)*(Md(k,:)'*Md(k,:));
if k == 1
H = h;
else
H = H+h;
end
end
E = T-H; %within samples sum of squares
% Wilks'test (Wilks'U)
LW = det(E)/det(T); %Wilks'lambda
if f >= 25; %approximation to chi-square statistic
disp('It is considering as a large sample problem (n >= 25).')
v = p*(g-1);
X2 = (-1)*(((f-1)-(.5*(p+g)))*log(LW));
P = 1-chi2cdf(X2,v);
disp(' ')
disp('Multivariate Analysis of Variance Table.')
fprintf('-----------------------------------------------------------------------------------\n');
disp('No. data Samples Variables L Chi-sqr. df P')
fprintf('-----------------------------------------------------------------------------------\n');
fprintf('%5.i%11.i%13.i%14.4f%15.4f%12.i%13.4f\n',f,g,p,LW,X2,v,P);
fprintf('-----------------------------------------------------------------------------------\n');
fprintf('With a given significance of: %.2f\n', alpha);
if P >= alpha;
disp('Sample mean vectors results not significant.');
else
disp('Sample mean vectors results significant.');
end
pn2 = 1-sqrt(LW);
fprintf('The strength of association between groups and averaged dependent variables are% 3.2f\n', pn2);
else %approximation to Rao's F-statistic
disp('It is considering as a small sample problem (n < 25).')
if p == 2 | (g-1) == 2
s = 2;
else
s = sqrt((p^2*(g-1)^2-4)/(p^2+((g-1)^2)-5));
end
v1 = p*(g-1);
v2 = (s*((f-1)-((p+g)/2)))-(((p*(g-1))-2)/2);
v = LW^(1/s);
F = ((1-v)/v)*(v2/v1);
P = 1-fcdf(F,v1,v2);
%Pillai's test (Trace test)
V = trace(inv(T)*H);
q = g-1;
s = min(p,q);
v = f-g;
m = (abs(p-q)-1)/2;
n = (v-p-1)/2;
FP = (((2*n)+s+1)/((2*m)+s+1))*(V/(s-V));
v1P = s*((2*m)+s+1);
v2P = s*((2*n)+s+1);
PP = 1-fcdf(FP,v1P,v2P);
%Lawley-Hotelling's trace test
U = trace(inv(E)*H);
FLH = (2*((s*n)+1)*U)/(s^2*((2*m)+s+1));
v1LH = s*((2*m)+s+1);
v2LH = 2*((s*n)+1);
PLH = 1-fcdf(FLH,v1LH,v2LH);
%Roy's Union_intersection test (Largest root test)
R = max(eig(inv(E)*H));
r = max(p,q);
FR = R*((v-r+q)/r);
v1R = r;
v2R = v-r+q;
% Because the numerator and denominator degrees of freedom could results a fraction,
% the probability function associated to the F statistic is resolved by the Simpson's
% 1/3 numerical integration method.
x = linspace(.000001,FR,100001);
DF = x(2)-x(1);
y = ((v1R/v2R)^(.5*v1R)/(beta((.5*v1R),(.5*v2R))));
y = y*(x.^((.5*v1R)-1)).*(((x.*(v1R/v2R))+1).^(-.5*(v1R+v2R)));
N2 = length(x);
PR = 1-(DF.*(y(1)+y(N2) + 4*sum(y(2:2:N2-1))+2*sum(y(3:2:N2-2)))/3.0);
if P >= alpha;
ds1 ='NS';
else
ds1 =' S';
end
if PP >= alpha;
ds2 ='NS';
else
ds2 =' S';
end
if PLH >= alpha;
ds3 ='NS';
else
ds3 =' S';
end
if PR >= alpha;
ds4 ='NS';
else
ds4 =' S';
end
;
disp(' ')
disp('Multivariate Analysis of Variance Table.')
fprintf('--------------------------------------------\n');
disp('No. data Samples Variables L')
fprintf('--------------------------------------------\n');
fprintf('%5.i%11.i%13.i%14.4f\n',f,g,p,LW);
fprintf('--------------------------------------------\n');
disp(' ')
fprintf('------------------------------------------------------------------------------\n');
disp('Test Statistic df1 df2 F P Conclusion')
fprintf('------------------------------------------------------------------------------\n');
fprintf('Rao %13.3f%10i%8i%11.2f%9.4f %s\n',LW,v1,v2,F,P,ds1);
fprintf('Pillai %13.3f%10i%8i%11.2f%9.4f %s\n',V,v1P,v2P,FP,PP,ds2);
fprintf('Lawley-Hotelling %6.3f%10i%8i%11.2f%9.4f %s\n',U,v1LH,v2LH,FLH,PLH,ds3);
fprintf('Roy %13.3f%10.1f%8.1f%11.2f%9.4f %s\n',R,v1R,v2R,FR,PR,ds4);
fprintf('------------------------------------------------------------------------------\n');
fprintf('With a given significance of:% 3.2f\n', alpha);
disp('According to the P-value, the sample mean vectors could be significant (S) or');
disp('not significant (NS).');
pn2 = 1-sqrt(LW);
fprintf('The strength of association between groups and averaged dependent variables are% 3.2f\n', pn2);
end
