function PROFANA(X,alpha)
%PROFANA Profile Analysis of Multivariate Repeated Measures.
% Profile analysis is a special application of multivariate analysis of
% variance (MANOVA) in which several dependent variables are measured and
% they are all measured on the same scale. Is also used in research where
% subjects are measured repeatedly on the same dependent variable. In this
% case, profile analysis is an alternative to univariate repeated measures
% analysis of variance. The major question to be answered by profile
% analysis is whether profiles of groups differ on a set of measures. To
% apply profile analysis, all measures must have the same range of possible
% scores, with the same score value having the same meaning on all the
% measures. The restriction on scaling of the measures is present because
% in two of the major tests of profile analysis (parallelism and flatness)
% the numbers that are actually tested are difference scores between
% dependent variables measured on adjacent occasions. Difference scores are
% called segments in profile analysis.
% There are three main questions to be answered on profile analysis:
% --1. Do different groups have parallel profiles?: test of parallelism.
% When profile analysis is used as a substitute for univariate repeated
% measures ANOVA, the parallelism test is the test of interaction.
% --2. Whether or not groups produce parallel profiles, does one group, on
% average, score higher on the collected set of measures than another?:
% test of difference in levels. In regular ANOVA, this question is
% answered by test of the groups hypothesis. In repeated-measures ANOVA,
% it address as the between-subjects main effects quastion.
% --3. Do the dependent variables all elicit the same average response?:
% test of flatness. This is relevant only if the profiles are parallel.
% If not, the at least one of them is necessarily not flat. This test
% evaluates the within-subjects main effect hypothesis in repeated-
% measures ANOVA.
%
% Syntax: function PROFANA(X,alpha)
%
% Inputs:
% X - data matrix (Size of matrix must be n-by-(1+p); sample=column 1,
% variables=column 2:p).
% alpha - significance level (default = 0.05).
%
% Outputs:
% - Complete analysis of variance for difference levels test.
% - Complete analysis of variance for parallelism test.
% - Complete Hotelling's T-squared analysis for flatness test.
% - Figure of profiles of dependent variables for the studied groups.
%
% Example: From the data Table 10.3 of Tabachnick and Fidell (1996, p. 448).
% Three groups (independent variables) whose profiles are compared are
% belly dancers (1), politicians (2), and administrators (3). The
% five respondents in each of these occupational groups participate in
% four leisure activities (dependent variables) and, during each, are
% asked to rate their satisfaction on a 10-points scale. Dependent
% variables are read(1), dance(2), TV(3), and ski(4). Profile analysis
% of difference in levels, parallelism, and flatness are tested with a
% significance of 0.10.
%
% Activity
% ------------------------------------------
% Group 1 2 3 4
% ------------------------------------------
% 1 7 10 6 5
% 8 9 5 7
% 5 10 5 8
% 6 10 6 8
% 7 8 7 9
% 2 4 4 4 4
% 6 4 5 3
% 5 5 5 6
% 6 6 6 7
% 4 5 6 5
% 3 3 1 1 2
% 5 3 1 5
% 4 2 2 5
% 7 1 2 4
% 6 3 3 3
% ------------------------------------------
%
% Data matrix must be:
% X=[1 7 10 6 5;1 8 9 5 7;1 5 10 5 8;1 6 10 6 8;1 7 8 7 9;2 4 4 4 4;2 6 4 5 3;
% 2 5 5 5 6;2 6 6 6 7;2 4 5 6 5;3 3 1 1 2;3 5 3 1 5;3 4 2 2 5;3 7 1 2 4;3 6 3 3 3];
%
% Calling on Matlab the function:
% PROFANA(X,0.1)
%
% Answer is:
%
% The number of groups are: 3
% The number of dependent variables are: 4
%
% Difference in Levels Test.
%
% Analysis of Variance Summary Table for Test of Levels Effect.
% ----------------------------------------------------------------------------
% SOV SS df MS F P
% ----------------------------------------------------------------------------
% Between Groups 172.900 2 86.450 44.145 0.0000
% Within Groups 23.500 12 1.958
% Total 196.400 14
% ----------------------------------------------------------------------------
% The associated probability for the F test is smaller than 0.10
% So, the assumption that differences between means of the groups over the dependent
% variables are equal was met.
% The strength of association between groups and averaged dependent variables are 0.88
%
% Parallelism Test.
% It is considering as a small sample problem (n < 25).
%
% Multivariate Analysis of Variance Table.
% --------------------------------------------
% No. data Samples Variables L
% --------------------------------------------
% 15 3 4 0.0763
% --------------------------------------------
%
% ------------------------------------------------------------------------------
% Test Statistic df1 df2 F P Conclusion
% ------------------------------------------------------------------------------
% Rao 0.076 6 20 8.74 0.0001 S
% Pillai 1.433 6 22 9.28 0.0000 S
% Lawley-Hotelling 5.428 6 18 8.14 0.0002 S
% Roy 3.541 3.0 11.0 12.98 0.0006 S
% ------------------------------------------------------------------------------
% With a given significance of: 0.10
% According to the P-value, the sample mean vectors could be significant (S) or
% not significant (NS).
% The strength of association between groups and averaged dependent variables are 0.72
% Here, if significant, this leading to rejection of the hypothesis of parallelism.
%
% Flatness Test.
% As n is less than 50: 15
% We use the F approximation.
%
% -------------------------------------------------------------------------------------
% Sample-size Variables T2 F df1 df2 P
% -------------------------------------------------------------------------------------
% 15 3 2.5825 8.6082 3 10 0.0040
% -------------------------------------------------------------------------------------
% The associated probability for the F test is smaller than 0.10
% There is significant deviation from flatness.
% The strength of association between groups and averaged dependent variables are 0.72
%
% Created by A. Trujillo-Ortiz, R. Hernandez-Walls, A. Castro-Perez
% and K. Barba-Rojo
% Facultad de Ciencias Marinas
% Universidad Autonoma de Baja California
% Apdo. Postal 453
% Ensenada, Baja California
% Mexico.
% atrujo@uabc.mx
%
% Copyright (C) December 24, 2006.
%
% To cite this file, this would be an appropriate format:
% Trujillo-Ortiz, A., R. Hernandez-Walls, A. Castro-Perez and K. Barba-Rojo. (2006).
% PROFANA:Profile Analysis of Multivariate Repeated Measures. A MATLAB file.
% [WWW document]. URL http://www.mathworks.com/matlabcentral/fileexchange/
% loadFile.do?objectId=13483
%
% Reference:
% Tabachnick, B. G. and Fidell, L. S. (1996), Using Multivariate Statistics.
% 3rd. ed. NY: HarperCollins College Pub, p. 441-505.
%
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;
if nargin < 1,
error('Requires at least one input argument.');
return;
end;
g = max(X(:,1));
p = size(X(:,2:end),2);
fprintf('The number of groups are:%2i\n', g);
fprintf('The number of dependent variables are:%2i\n\n', p);
%%%Difference in Levels.
disp('Difference in Levels Test.')
MGL = [];M = [];n = [];
indice=X(:,1);
for i = 1:g
Xe=find(indice==i);
eval(['XG' num2str(i) '= X(Xe,2:end);']);
eval(['mG' num2str(i) '= mean(XG' num2str(i) ',2);'])
eval(['mGl' num2str(i) '= mean(XG' num2str(i) ',1);'])
eval(['mmG' num2str(i) '= mean(mG' num2str(i) ');'])
eval(['nG' num2str(i) '= length(XG' num2str(i) ') ;'])
eval(['xm = mmG' num2str(i) ';'])
eval(['xmgl = mGl' num2str(i) ';'])
eval(['xn = nG' num2str(i) ';'])
MGL = [MGL;xmgl];M = [M;xm];n = [n;xn];
end
S1 = [];
for i = 1:g
eval(['S1' num2str(i) '= nG' num2str(i) '*(mmG' num2str(i) ' - mean(M)).^2;'])
eval(['xS1 = S1' num2str(i) ';'])
S1 = [S1;xS1];
end
SSBG = p*sum(S1); %sum of squares between groups
v1 = g-1;
MSBG = SSBG/v1;
S2 = [];
for i = 1:g
eval(['S2' num2str(i) '= (mG' num2str(i) ' - mmG' num2str(i) ').^2;'])
eval(['xS2 = S2' num2str(i) ';'])
S2 = [S2;xS2];
end
SSWG = p*sum(S2); %sum of squares within groups
v2 = sum(n)-g;
MSWG = SSWG/v2;
SST = SSBG+SSWG;
v3 = v1+v2;
FL = MSBG/MSWG;
P = 1 - fcdf(FL,v1,v2); %probability associated to the F-statistic.
disp(' ')
disp('Analysis of Variance Summary Table for Test of Levels Effect.')
fprintf('----------------------------------------------------------------------------\n');
disp('SOV SS df MS F P')
fprintf('----------------------------------------------------------------------------\n');
fprintf('Between Groups %15.3f%10i%13.3f%12.3f%9.4f\n',SSBG,v1,MSBG,FL,P);
fprintf('Within Groups%18.3f%10i%13.3f\n',SSWG,v2,MSWG);
fprintf('Total%26.3f%10i\n',SST,v3);
fprintf('----------------------------------------------------------------------------\n');
if P >= alpha;
fprintf('The associated probability for the F test is equal or larger than% 3.2f\n', alpha);
disp('So, the assumption that differences between means of the groups over the dependent')
disp('variables are equal was met.');
else
fprintf('The associated probability for the F test is smaller than% 3.2f\n', alpha);
disp('So, the assumption that differences between means of the groups over the dependent')
disp('variables are equal was met.');
end
n2 = SSBG/SST;
fprintf('The strength of association between groups and averaged dependent variables are% 3.2f\n', n2);
%%%Parallelism.
disp(' ')
disp('Parallelism Test.')
Xi = X(:,2:p);
Xf = X(:,3:p+1);
X = [X(:,1) [Xi-Xf]];
MAOV1(X,alpha);
disp('Here, if significant, this leading to rejection of the hypothesis of parallelism.')
%%%Flatness.
disp(' ')
disp('Flatness Test.')
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
T2 = sum(N)*M*inv(E)*M'; %observed Hotelling's T-squared
LW = 1/(1+T2);
if n >= 50 %Chi-square approximation.
X2 = T2;
v = p+1; %Degrees of freedom.
P = 1-chi2cdf(X2,v); %Probability that null Ho: is true.
fprintf('As n is equal or larger than 50:% i\n', sum(n));
disp('We use the Chi-square approximation.')
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;
fprintf('The associated probability for the Chi-squared test is equal or larger than% 3.2f\n', alpha);
disp('There is not significant deviation from flatness.');
else
fprintf('The associated probability for the Chi-squared test is smaller than% 3.2f\n', alpha);
disp('There is significant deviation from flatness.');
end
n2 = 1-LW;
fprintf('The strength of association between groups and averaged dependent variables are% 3.2f\n', n2);
else %F approximation.
F = T2*(sum(N)-g-p+1)/p; %F approximation
v1 = p; %Numerator degrees of freedom.
v2 = sum(N)-g-p+1; %Denominator degrees of freedom.
P = 1-fcdf(F,v1,v2); %Probability that null Ho: is true.
fprintf('As n is less than 50:% i\n', sum(n));
disp('We use the F approximation.')
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,p,T2,F,v1,v2,P);
fprintf('-------------------------------------------------------------------------------------\n');
if P >= alpha;
fprintf('The associated probability for the F test is equal or larger than% 3.2f\n', alpha);
disp('There is not significant deviation from flatness.');
else
fprintf('The associated probability for the F test is smaller than% 3.2f\n', alpha);
disp('There is significant deviation from flatness.');
end
n2 = 1-LW;
fprintf('The strength of association between groups and averaged dependent variables are% 3.2f\n', n2);
end
G = [];
for k = 1:g
G = [G;k];
end
sc = [G MGL];
figure;
hold on;
lg = [];
for k = 1:g
plot(1:p+1,sc(k,2:end),DCSYMB0(k),'LineStyle','--','Color',DCRGB0(k),'MarkerFaceColor',DCRGB0(k),'MarkerEdgeColor',DCRGB0(k));
lg = [lg,['''Group ' num2str(k) ''',']];
axis([0 p+2 min(min((sc(:,2:end))))-1 max(max((sc(:,2:end))))+1])
%line('LineStyle','--','Color',DCRGB0(k));
end
lg(end)=' ';
eval(['legend(' lg ')']);
title('Profiles of Dependent Variables for the Studied Groups.');
xlabel('Dependent Variables');
ylabel('Mean of Dependent Variable');
set(gca,'XTick',1:1:p+1)
hold off;
return,
