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RMAOV1.m
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RMAOV1.m
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function [RMAOV1] = RMAOV1(X,alpha)
% RMAOV1 Repeated Measures Single-Factor Analysis of Variance Test.
% One-way repeated measures ANOVA 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.
% In contrast to the independent groups ANOVA (between-subjects), the
% repeated measures procedure is generally more powerful (ie. leads to
% a greater likelihood of rejecting a false null hypothesis). Statistically,
% this is the case because unwanted variability (error) due to individual
% differences among participants can be quantified and removed from the
% denominator of the F-ratio. This procedure requires fewer participants,
% but it can give rise to research confounds such as order or practice effects
% and participant bias.
% As with the independent groups ANOVA, the total variability of a set of scores
% is subdivided into sources that provide estimates of population variability.
% However with the repeated measures ANOVA, the process of subdividing the
% variability is a two step process outlined below.
%
% Total variability broken down into two components:
% -Between subjects. Variability in scores due to individual differences
% among participants.
% -Within subjects.
% The within subjects variability is subdivided into the following components:
% -Treatment. Variance among the treatment means (same as MS between in the
% independent groups ANOVA).
% -Residual. Leftover or unwanted error variability (can be thought of an
% inconsistencies in the effects of the treatments across the participants).
%
% The F-ratio is calculated by dividing the treatment variance (Mean Square
% treatment) by the residual variance (Mean Square residual). The numerator
% variance is the same as the MS between groups in the independent groups ANOVA
% while the denominator variance (residual) is the leftover or unwanted variability
% (error) but without variability due to individual differences among participants
% which has been separated from it.
%
% At first glance, type of experiment resambles a randomized complete block design.
% However, the block design considering groups or sets of subjects (blocks), whereas
% the within-subjects design each of the n subjects are exposed to different experimental
% conditions or trials.
%
% Syntax: function [RMAOV1] = RMAOV1(X,alpha)
%
% Inputs:
% X - data matrix (Size of matrix must be n-by-3;dependent variable=column 1,
% independent variable=column 2;subject=column 3).
% alpha - significance level (default = 0.05).
% Outputs:
% - Complete Analysis of Variance Table.
% - Strength of the relationship.
%
% 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 1 1;20 1 2;17 1 3;25 1 4;30 1 5;19 1 6;26 1 7;17 1 8;26 1 9;
% 22 2 1;19 2 2;15 2 3;30 2 4;27 2 5;27 2 6;16 2 7;18 2 8;24 2 9;
% 8 3 1;10 3 2;5 3 3;13 3 4;13 3 5;8 3 6;5 3 7;8 3 8;14 3 9;
% 6 4 1;4 4 2;4 4 3;12 4 4;8 4 5;7 4 6;2 4 7;1 4 8;8 4 9;
% 6 5 1;4 5 2;5 5 3;17 5 4;6 5 5;4 5 6;5 5 7;5 5 8;9 5 9];
%
% Calling on Matlab the function:
% RMAOV1(X)
%
% Answer is:
%
% The number of IV levels are: 5
%
% The number of subjects are: 9
%
% Analysis of Variance Table.
% --------------------------------------------------------------
% SOV SS df MS F P
% --------------------------------------------------------------
% Subjects 486.711 8 60.839[ 8.450 0.0000]
% IV 2449.200 4 612.300 85.042 0.0000
% Error 230.400 32 7.200
% Total 3166.311 44
% --------------------------------------------------------------
% If the P result is smaller than 0.05
% the Ho tested results statistically significant. Otherwise, it is not significative.
% [Generally speaking, no Mean Square is computed for the variable "subjects" since it is assumed
% that subjects differ from one another thus making a significance test of "subjects" superfluous.
% However, for all the interested people we are given it anyway].
%
% The percentage of the variability in the DV associated with the IV is 91.40
% (After the effects of individual differences have been removed).
%
% Created by A. Trujillo-Ortiz, R. Hernandez-Walls and R.A. Trujillo-Perez
% Facultad de Ciencias Marinas
% Universidad Autonoma de Baja California
% Apdo. Postal 453
% Ensenada, Baja California
% Mexico.
% atrujo@uabc.mx
%
% Copyright.July 21, 2004.
%
% To cite this file, this would be an appropriate format:
% Trujillo-Ortiz, A., R. Hernandez-Walls and R.A. Trujillo-Perez. (2004). RMAOV1:One-way repeated
% measures ANOVA. A MATLAB file. [WWW document]. URL http://www.mathworks.com/matlabcentral/
% fileexchange/loadFile.do?objectId=5576
%
% References:
% Huck, S. W. (2000), Reading Statistics and Research. 3rd. ed.
% New-York:Allyn&Bacon/Longman Pub. Chapter 16.
% Winkel, M. http://www.stats.ox.ac.uk/~winkel/phs.html
% Zar, J. H. (1999), Biostatistical Analysis. 4th. ed.
% New-Jersey:Upper Saddle River. p. 255-259.
%
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;
k = max(X(:,2));
s = max(X(:,3));
fprintf('The number of IV levels are:%2i\n\n', k);
fprintf('The number of subjects are:%2i\n\n', s);
%Analysis of Variance Procedure.
m=[];n=[];nn=[];A=[];
indice = X(:,2);
for i = 1:k
Xe = find(indice==i);
eval(['X' num2str(i) '=X(Xe,1);']);
eval(['m' num2str(i) '=mean(X' num2str(i) ');'])
eval(['n' num2str(i) '=length(X' num2str(i) ') ;'])
eval(['nn' num2str(i) '=(length(X' num2str(i) ').^2);'])
eval(['xm = m' num2str(i) ';'])
eval(['xn = n' num2str(i) ';'])
eval(['xnn = nn' num2str(i) ';'])
eval(['x =(sum(X' num2str(i) ').^2)/(n' num2str(i) ');']);
m=[m;xm];n=[n;xn];nn=[nn,xnn];A=[A,x];
end;
S=[];
indice=X(:,3);
for j=1:s
Xe=find(indice==j);
eval(['S' num2str(j) '=X(Xe,1);']);
eval(['x =((sum(S' num2str(j) ').^2)/length(S' num2str(j) '));']);
S=[S,x];
end;
C = (sum(X(:,1)))^2/length(X(:,1)); %correction term
SST = sum(X(:,1).^2)-C; %total sum of squares
dfT = length(X(:,1))-1; %total degrees of freedom
SSA = sum(A)-C; %IV sum of squares
v1 = k-1; %IV degrees of freedom
SSS = sum(S)-C; %within-subjects sum of squares
v2 = s-1; %within-subjects degrees of freedom
SSE = SST-SSA-SSS; %error sum of squares
v3 = v1*v2; %error degrees of freedom
MSA = SSA/v1; %IV mean squares
MSS = SSS/v2; %within-subjects mean squares
MSE = SSE/v3; %error mean squares
F1 = MSA/MSE; %IV F-statistic
F2 = MSS/MSE; %within-subjects F-statistic
%Probability associated to the F-statistics.
P1 = 1 - fcdf(F1,v1,v3);
P2 = 1 - fcdf(F2,v2,v3);
eta2 = SSA/(SSA+SSE)*100;
disp('Repeated Measures One-Way Analysis of Variance Table.')
fprintf('---------------------------------------------------------------------------\n');
disp('SOV SS df MS F P')
fprintf('---------------------------------------------------------------------------\n');
fprintf('Subjects %11.3f%10i%15.3f[%13.3f%9.4f]\n\n',SSS,v2,MSS,F2,P2);
fprintf('IV %11.3f%10i%15.3f%14.3f%9.4f\n\n',SSA,v1,MSA,F1,P1);
fprintf('Error %11.3f%10i%15.3f\n\n',SSE,v3,MSE);
fprintf('Total %11.3f%10i\n\n',SST,dfT);
fprintf('---------------------------------------------------------------------------\n');
fprintf('If the P result is smaller than% 3.2f\n', alpha );
disp('the Ho tested results statistically significant. Otherwise, it is not significative.');
disp('[Generally speaking, no Mean Square is computed for the variable "subjects" since it is assumed');
disp('that subjects differ from one another thus making a significance test of "subjects" superfluous.');
disp('However, for all the interested people we are given it anyway].');
disp(' ');
fprintf('The percentage of the variability in the DV associated with the IV (eta squared) is% 3.2f\n', eta2);
disp('(After the effects of individual differences have been removed).');
return;