function [x] = epsHF(X)
%EPSHF Huynh-Feldt epsilon.
% The Huynh-Feldt epsilon its a correction of the Greenhouse-Geisser epsilon.
% This due that the Greenhouse-Geisser epsilon tends to underestimate epsilon
% when epsilon is greater than 0.70 (Stevens, 1990). An estimated epsilon
% = 0.96 may be actually 1. Huynh-Feldt correction is less conservative. The
% Huynh-Feldt epsilon is calculated from the Greenhouse-Geisser epsilon.
% As the Greenhouse-Geisser epsilon, Huynh-Feldt epsilon measures how much
% the sphericity assumption or compound symmetry is violated. The idea of both
% corrections its analogous to pooled vs. unpooled variance Student's t-test:
% if we have to estimate more things because variances/covariances are not
% equal, then we lose some degrees of freedom and P-value increases. These
% epsilons should be 1.0 if sphericity holds. If not sphericity assumption
% appears violated. We must to have in mind that the greater the number of
% repeated measures, the greater the likelihood of violating assumptions of
% sphericity and normality (Keselman et al, 1996) . Therefore, we nedd to have
% the most conservative F values. These are obtained by setting epsilon to its
% lower bound, which represents the maximum violation of these assumptions.
% When a significant result is obtained, it is assumed to be robust. However,
% since this test may be overly conservative, Greenhouse and Geisser (1958,
% 1959) recommend that when the lower-bound epsilon gives a nonsignificant
% result, it should be followed by an approximate test (based on a sample
% estimate of epsilon).
%
% Syntax: function epsHF(X)
%
% Inputs:
% X - Input matrix can be a data matrix (size n-data x k-treatments)
% Output:
% x - Huynh-Feldt epsilon value.
%
% 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=epsHF(X)
%
% Answer is:
%
% x = 1.2176
%
% 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. November 01, 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).
% epsHF:Huynh-Feldt epsilon. A MATLAB file. [WWW document]. URL http://
% www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=12853
%
% --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--
%
% References:
% Geisser, S, and Greenhouse, S.W. (1958), An extension of Boxs results on
% the use of the F distribution in multivariate analysis. Annals of
% Mathematical Statistics, 29:885891.
% Greenhouse, S.W. and Geisser, S. (1959), On methods in the analysis of
% profile data. Psychometrika, 24:95-112.
% Huynh, M. and Feldt, L.S. (1970), Conditions under which mean square rate
% in repeated measures designs have exact-F distributions. Journal of the
% American Statistical Association, 65:1982-1989
% Keselman, J.C, Lix, L.M. and Keselman, H.J. (1996), The analysis of repeated
% measurements: a quantitative research synthesis. British Journal of
% Mathematical and Statistical Psychology, 49:275298.
% 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));
[n k] = size(X);
eGG = epsGG(X); %call to the zipped Greenhouse-Geisser epsilon function
epsHF = (n*(k-1)*eGG-2)/((k-1)*((n-1)-(k-1)*eGG)); %Huynh-Feldt epsilon estimation
x = epsHF;
return,
