function [x] = epsGG(X)
%EPSGG Greenhouse-Geisser epsilon.
% The Greenhouse-Geisser epsilon value measures by how much the sphericity
% assumption is violated. Epsilon is then used to adjust for the potential
% bias in the F statistic. Epsilon can be 1, which means that the sphericity
% assumption is met perfectly. An epsilon smaller than 1 means that the
% sphericity assumption is violated. The further it deviates from 1, the worse
% the violation; it can be as low as epsilon = 1/(k - 1), which produces
% the lower bound of epsilon (the worst case scenario). The worst case scenario
% depends on k, the number of levels in the repeated measure factor. In real
% life epsilon is rarely exactly 1. If it is not much smaller than 1, then we
% feel comfortable with the results of repeated measure ANOVA.
% The Greenhouse-Geisser epsilon is derived from the variance-covariance matrix
% of the data. For its evaluation we need to first calculate the variance-
% covariance matrix of the variables (S). The diagonal entries are the variances
% and the off diagonal entries are the covariances. From this variance-covariance
% matrix, the epsilon statistic can be estimated. Also we need the mean of the
% entries on the main diagonal of S, the mean of all entries, the mean of all
% entries in row i of S, and the individual entries in the variance-covariance
% matrix. There are three important values of epsilon. It can be 1 when the
% sphericity is met perfectly. This epsilon procedure was proposed by Greenhouse
% and Geisser (1959).
%
% Syntax: function epsGG(X)
%
% Inputs:
% X - Input matrix can be a data matrix (size n-data x k-treatments)
% Output:
% x - Greenhouse-Geisser epsilon value.
%
% $$We suggest you could take-a-look to the PDF document ''This Week's
% Citation Classics'' CCNumber 28, July 12, 1982, web-page [http://
% garfield.library.upenn.edu/classics1982/A1982NW45700001.pdf]$$
%
% Example 2 of Maxwell and Delaney (p.497). This is a repeated measures example
% with two within and a subject effect. We have one dependent variable:reaction
% time, two independent variables: visual stimuli are tilted at 0, 4, and 8
% degrees; with noise absent or present. Each subject responded to 3 tilt and 2
% noise given 6 trials. Data are,
%
% 0 4 8
% -----------------------------------
% Subject A P A P A P
% --------------------------------------------
% 1 420 480 420 600 480 780
% 2 420 360 480 480 480 600
% 3 480 660 480 780 540 780
% 4 420 480 540 780 540 900
% 5 540 480 660 660 540 720
% 6 360 360 420 480 360 540
% 7 480 540 480 720 600 840
% 8 480 540 600 720 660 900
% 9 540 480 600 720 540 780
% 10 480 540 420 660 540 780
% --------------------------------------------
%
% The three measurements of reaction time were averaging across noise
% ausent/present. Given,
%
% Tilt
% -----------------
% Subject 0 4 8
% ---------------------------
% 1 450 510 630
% 2 390 480 540
% 3 570 630 660
% 4 450 660 720
% 5 510 660 630
% 6 360 450 450
% 7 510 600 720
% 8 510 660 780
% 9 510 660 660
% 10 510 540 660
% ---------------------------
%
% We need to estimate the Greenhouse-Geisser epsilon associated with the angle
% of rotation of the stimulii.
%
% Data matrix must be:
% X=[450 510 630;390 480 540;570 630 660;450 660 720;510 660 630;
% 360 450 450;510 600 720;510 660 780;510 660 660;510 540 660];
%
% Calling on Matlab the function:
% x=epsGG(X)
%
% Answer is:
%
% x = 0.9616
%
% 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. October 31, 2006.
%
% --Special thanks are given to Sren Andersen, Universitt Leipzig, Institut
% Psychologie I, Professur Allgemeine Psychologie & Methodenlehre, Seeburgstr
% 14-20, D-04103 Leipzig, Deutchland, for encouraging us to create this m-file--
%
% To cite this file, this would be an appropriate format:
% Trujillo-Ortiz, A., R. Hernandez-Walls, A. Castro-Perez and K. Barba-Rojo. (2006).
% epsGG:Greenhouse-Geisser epsilon. A MATLAB file. [WWW document]. URL http://
% www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=12839
%
% Reference:
% Greenhouse, S.W. and Geisser, S. (1959), On methods in the analysis
% of profile data. Psychometrika, 24:95-112.
% Maxwell, S.E. and Delaney, H.D. (1990), Designing Experiments and
% Analyzing Data: A model comparison perspective. Pacific Grove,
% CA: Brooks/Cole.
%
error(nargchk(1,1,nargin));
k = size(X,2); %number of treatments
S = cov(X); %variance-covariance matrix
mds = mean(diag(S)); %mean of the entries on the main diagonal of S
ms = mean(mean(S)); %the mean of all entries of S
msr = mean(S,2); %mean of all entries in row i of S
N = k^2*(mds-ms)^2;
D = (k-1)*(sum(sum(S.^2))-2*k*sum(msr.^2)+k^2*ms^2);
epsGG = N/D; %Greenhouse-Geisser epsilon estimation
x = epsGG;
return,
