function [BFtest] = BFtest(X,alpha)
%Brown-Forsythe's Test for Homogeneity of Variances.
%[In the Brown-Forsythe's test the data are transforming to yij = abs[xij - median(xj)]
%and uses the F distribution performing an one-way ANOVA using y as the
%dependent variable. The Brown-Frosythe statistic is corrected for artificial zeros
%occurring in odd sized samples.
%
% Syntax: function [BFtest] = BFtest(X,alpha)
%
% Inputs:
% X - data matrix (Size of matrix must be n-by-2; data=column 1, sample=column 2).
% alpha - significance level (default = 0.05).
% Outputs:
% - Sample variances vector.
% - Whether or not the homoscedasticity was met.
%
% Example: From the example 10.1 of Zar (1999, p.180), to test the Brown-Forsythe's
% homoscedasticity of data with a significance level = 0.05.
%
% Diet
% ---------------------------------
% 1 2 3 4
% ---------------------------------
% 60.8 68.7 102.6 87.9
% 57.0 67.7 102.1 84.2
% 65.0 74.0 100.2 83.1
% 58.6 66.3 96.5 85.7
% 61.7 69.8 90.3
% ---------------------------------
%
% Data matrix must be:
% X=[60.8 1;57.0 1;65.0 1;58.6 1;61.7 1;68.7 2;67.7 2;74.0 2;66.3 2;69.8 2;
% 102.6 3;102.1 3;100.2 3;96.5 3;87.9 4;84.2 4;83.1 4;85.7 4;90.3 4];
%
% Calling on Matlab the function:
% BFtest(X)
%
% Answer is:
%
% The number of samples are: 4
%
% ----------------------------
% Sample Size Variance
% ----------------------------
% 1 5 9.3920
% 2 5 8.5650
% 3 4 7.6567
% 4 5 8.3880
% ----------------------------
%
% Brown-Forsythe's Test for Equality of Variances F=0.0831, df1= 3, df2=15
% Probability associated to the F statistic = 0.9682
% The associated probability for the F test is larger than 0.05
% So, the assumption of homoscedasticity was met.
%
% Created by A. Trujillo-Ortiz and R. Hernandez-Walls
% Facultad de Ciencias Marinas
% Universidad Autonoma de Baja California
% Apdo. Postal 453
% Ensenada, Baja California
% Mexico.
% atrujo@uabc.mx
%
% April 19, 2003.
%
% To cite this file, this would be an appropriate format:
% Trujillo-Ortiz, A. and R. Hernandez-Walls. (2003). BFtest: Brown-Forsythe's test for homogeneity of
% variances. A MATLAB file. [WWW document]. URL http://www.mathworks.com/matlabcentral/fileexchange/
% loadFile.do?objectId=3412&objectType=FILE
%
% References:
%
% Brown, M. B. and Forsythe, A. B. (1974), Robust Tests for
% the Equality of Variances. Journal of the American
% Statistical Association, 69:364-367.
% Zar, J. H. (1999), Biostatistical Analysis (2nd ed.).
% NJ: Prentice-Hall, Englewood Cliffs. p. 180.
%
if nargin < 2,
alpha = 0.05;
end
Y=X;
k=max(Y(:,2));
fprintf('The number of samples are:%2i\n\n', k);
%Brown-Forsythe's Procedure.
n=[];s2=[];Z=[];
indice=Y(:,2);
for i=1:k
Ye=find(indice==i);
eval(['Y' num2str(i) '=Y(Ye,1);']);
eval(['mdY' num2str(i) '=median(Y(Ye,1));']);
eval(['n' num2str(i) '=length(Y' num2str(i) ') ;']);
eval(['s2' num2str(i) '=(std(Y' num2str(i) ').^2) ;']);
eval(['Z' num2str(i) '= abs((Y' num2str(i) ') - mdY' num2str(i) ');']);
eval(['xn= n' num2str(i) ';']);
eval(['xs2= s2' num2str(i) ';']);
eval(['x= Z' num2str(i) ';']);
n=[n;xn];s2=[s2;xs2];Z=[Z;x];
end
fprintf('-----------------------------\n');
disp(' Sample Size Variance')
fprintf('-----------------------------\n');
for i=1:k
fprintf(' %d %2i %.4f\n',i,n(i),s2(i))
end
fprintf('-----------------------------\n');
disp(' ')
Y=[Z Y(:,2)];
%Correction for artificial zeros occurring in odd sized samples.
for i=1:k
Ye=find(Y(:,2)==i);
ncero=find((Y(:,1)==0)&(Y(:,2)==i));
Y(ncero,1)=999;
Y(ncero,1)=min(Y(Ye,1));
end
%Analysis of variance procedure.
C=(sum(Y(:,1)))^2/length(Y(:,1)); %correction term.
SST=sum(Y(:,1).^2)-C; %total sum of squares.
dfT=length(Y(:,1))-1; %total degrees of freedom.
indice=Y(:,2);
for i=1:k
Ye=find(indice==i);
eval(['A' num2str(i) '=Y(Ye,1);']);
end
A=[];
for i=1:k
eval(['x =((sum(A' num2str(i) ').^2)/length(A' num2str(i) '));']);
A=[A,x];
end
SSA=sum(A)-C; %sample sum of squares.
dfA=k-1; %sample degrees of freedom.
SSE=SST-SSA; %error sum of squares.
dfE=dfT-dfA; %error degrees of freedom.
MSA=SSA/dfA; %sample mean squares.
MSE=SSE/dfE; %error mean squares.
F=MSA/MSE; %Brown-Forsythe's F-statistic.
v1=dfA;df1=v1;
v2=dfE;df2=v2;
P = 1 - fcdf(F,v1,v2); %probability associated to the F-statistic.
fprintf('Brown-Forsythe''s Test for Equality of Variances F=%3.4f, df1=%2i, df2=%2i\n', F,df1,df2);
fprintf('Probability associated to the F statistic = %3.4f\n', P);
if P >= alpha;
fprintf('The associated probability for the F test is equal or larger than% 3.2f\n', alpha);
fprintf('So, the assumption of homoscedasticity was met.\n');
else
fprintf('The associated probability for the F test is smaller than% 3.2f\n', alpha);
fprintf('So, the assumption of homoscedasticity was not met.\n');
end
