function [RMA1multap] = RMA1multap(X,alpha)
%RM1MULTAP One-Sample Repeated Measures Analysis by the Multivariate Approach.
% One-sample repeated measures is used to analyze the relationship between
% the independent variable and dependent variable when:(1) the dependent
% variable is quantitative in nature and is measured on a level that at
% least approximates interval characteristics, (2) the independent variable
% is within-subjects in nature, and (3) the independent variable has three
% or more levels. It is an extension of the correlated-groups t test, where
% the main advantage is controlling the disturbance variables or individual
% differences that could influence the dependent variable.
% Another approach to repeated measures analyses is through using multivariate
% statistical techniques. This requires a paradigm shift. When considering
% the univariate analysis techniques, the experimental design was subjects
% as a random factor crossed with treatments or repeated measures as a fixed
% factor. To shift to the multivariate techniques, the repeated measures become
% a series of dependent variables and subjects are considered as replications
% in a single-cell design (Lewis, 1993). The most common approach is to
% transform the p dependent variables into p-1 linearly independent pairwise
% difference scores. Analysis is performed on these p-1 new dependent variables.
% The null hypothesis that is most often tested in this situation is that the
% difference scores have population means of zero, using an F transformation of
% Hotelling's T2 (Lewis, 1993;Stevens, 1996).
% There are advantages and disadvantages to using the multivariate approach.
% The multivariate approach does not require the sphericity assumption. However,
% researchers have not come to an agreement as to the best multivariate approach
% to take when considering power and robustness against assumption violations.
% There are serious concerns about power when the number of subjects is less than
% or equal to the degrees of freedom for a repeated measures main effect or
% interaction; in fact, the test statistic could not be computed. When the number
% of subjects is greater than, but still close to the degrees of freedom, the test
% has little power. But, power increases rapidly as the number of subjects
% increases (Lewis, 1993;Stevens, 1996).
% In general, it is recommended that both the univariate and the multivariate
% approaches be run since the two approaches evaluate different aspects of the
% data. The only safeguard if this approach is taken is to decrease the alpha-
% level for each approach by half, in order to control for experiment-wise Type
% I error (Barcikowski and Robey, 1984;Lewis, 1993;Stevens, 1996).
% Applied statisticians tend to prefer the multivariate test to the standard or
% the alternative univariate test because the multivariate test and follow-up
% tests have a close conceptual link to each other.
% According to Box (1954) if sphericity (circularity) assumption is not met, then
% the F ratio is positively biased and we are rejecting falsely too often. So, if
% sphericity holds the univariate approach is more powerful. When small but
% reliable effects are present with the effects being highly variable, the
% multivariate test is far more powerful than the univariate test.
% The exact F transformation of T2 is multiply it by the ratio (n-p+1)/(n-1)(p-1).
% The F-citical value is with alpha-value and (p-1) and (n-p+1) degrees of
% freedom.
%
% Syntax: function [RM1multap] = RMA1multap(X,alpha)
%
% Inputs:
% X - data matrix (Size of matrix must be n subjects-by-p correlated variables).
% alpha - significance level (default = 0.05).
% Output:
% - Complete Multivariate Analysis Table.
%
% Example: From the example given by Dr. Matthias Winkel* (http://www.stats.ox.ac.uk/~winkel/phs.html)
% on the relaxation therapy against migrane. Nine subjects participated in a relaxation therapy
% with several weeks baseline frequency/duration recording (w1 and w2) and several weeks
% therapy (w3 to w5). Its is of interest to test if there exist differences on the relaxation
% therapy and within subjects with a significance level = 0.05.
%
% Weeks
% ------------------------------------------------------
% Subject 1 2 3 4 5
% ------------------------------------------------------
% 1 21 22 8 6 6
% 2 20 19 10 4 4
% 3 17 15 5 4 5
% 4 25 30 13 12 17
% 5 30 27 13 8 6
% 6 19 27 8 7 4
% 7 26 16 5 2 5
% 8 17 18 8 1 5
% 9 26 24 14 8 9
% ------------------------------------------------------
%
% *Note: Due to a typing error, on the data table given by Dr. Winkel the value of subject 6 on
% week 4 must be 7, not 6.
%
% Data matrix must be:
% X=[21 22 8 6 6;20 19 10 4 4;17 15 5 4 5;25 30 13 12 17;30 27 13 8 6;
% ;19 27 8 7 4;26 16 5 2 5;17 18 8 1 5;26 24 14 8 9];
%
% Calling on Matlab the function:
% RMA1multap(X)
%
% Answer is:
%
% One-Sample Repeated Measures Analysis by the Multivariate Approach Table.
% -------------------------------------------------------------------------------------
% Sample-size Variables T2 F df1 df2 P
% -------------------------------------------------------------------------------------
% 9 5 552.9009 86.3908 4 5 0.0001
% -------------------------------------------------------------------------------------
% With a given significance = 0.050
% Mean vectors results significant.
%
% Created by A. Trujillo-Ortiz, R. Hernandez-Walls, K. Barba-Rojo and A. Castro-Perez
% Facultad de Ciencias Marinas
% Universidad Autonoma de Baja California
% Apdo. Postal 453
% Ensenada, Baja California
% Mexico.
% atrujo@uabc.mx
%
% Copyright.September 24, 2007.
%
% To cite this file, this would be an appropriate format:
% Trujillo-Ortiz, A., R. Hernandez-Walls, K. Barba-Rojo and A. Castro-Perez (2007).
% RM1MULTAP:One-Sample Repeated Measures Analysis by the Multivariate Approach.
% A MATLAB file. [WWW document]. URL http://www.mathworks.com/matlabcentral/
% fileexchange/loadFile.do?objectId=16575
%
% References:
% Barcikowski, R.S. and Robey, R.R. (1984). Decisions in single group repeated measures analysis:
% Statistical tests and three computer packages. The American Statistician. 38:148-150.
% Box, G.E.P. (1954). Some theorems on quadratic forms applied in the study of analysis of
% variance problems, II. Effects of inequality of variance and of correlation between
% errors in the two-way classification. The Annals of Mathematical Statistics. 25:484-498.
% Lewis, C. (1993). Analyzing means from repeated measures data. In G. Keren and C. Lewis (eds.),
% A Handbook for data analysis in the behavioral sciences (pp.73-94). Hillsdale, NJ:Erlbaum.
% Stevens, J. (1996). Applied multivariate statistics for the social sciences (3rd ed.).
% Hillsdale, NJ:Erlbaum.
%
if nargin < 2,
alpha = 0.05; %(default)
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
[n,p] = size(X); %n=No. of subjects;p=No. of correlated variables (=p repeated
%measures on n subjacts)
M = mean(X); %vector of means of the p-correlated variables
S = cov(X); %covariance matrix of the p-correlated variables
C = [eye(p-1,p-1) -ones(p-1,1)]; %(p-1) x p matrix whose rows are orthonormal
%contrasts normalized to unit length
%(rank = p-1)
T2 = n*(C*M')'*inv(C*S*C')*(C*M'); %Hotelling's T2 statistic
F = ((n-1)-(p-1)+1)/((n-1)*(p-1))*T2; %exact F transformation of T2
v1 = p-1; %Numerator degrees of freedom
v2 = n-p+1; %Denominator degrees of freedom.
P = 1-fcdf(F,v1,v2); %Probability that null Ho: is true.
disp(' ')
disp('One-Sample Repeated Measures by the Multivariate Approach Table.')
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');
fprintf('With a given significance = %3.3f\n', alpha);
if P >= alpha;
disp('Mean vectors results not significant.');
else
disp('Mean vectors results significant.');
end
return,