function [p, Q]= chi2bintest(x, n)
% Usage: [p, Q]= chi2bintest(x, n)
%
% The chi-squared binary test.
%
% Given a number of samples with a binary outcome ("is or is not something"),
% this function tests the hypothesis that the samples are independent.
% If Q > chi2(p, nu), the hypothesis is rejected.
%
% If x is a vector, each item is the result of a sample.
% n must be a scalar (same total for all samples) or a vector of the same length as x.
%
% If x is a matrix, each row is the result of a sample. Each column is an independent
% sample series. n must be a scalar (same total for all samples) or
% a vector of the same length as the number of samples in x (same totals for all sample
% series) or a matrix of the same size as x.
%
% If you find any errors, please let me know: .
%
% ARGUMENTS:
% x Relative frequencies of the outcome, if 0<=x<=1.
% Absolut number count, if integer x>=0.
% n Number of tries.
% p The prob ability value, calculated from Q.
% Q The resulting Q-value.
%
% EXAMPLE 1
% In region A, 324 of 556 cows were red, whereas in region B 98 of 260 were red.
% [p, Q]= chi2bintest([324; 98], [556; 260])
% p=
% 4.2073e-08
% Q=
% 30.0515
% With an error risk of about 4e-08, we can claim that the samples are independent.
%
% EXAMPLE 2
% In three regions, reindeers were cheched for four various deseases.
% p= chi2bintest([28,39,8,4782; 17,7,5,903; 21,14,6,2322], [15996; 10127; 12476])
% p =
% 0.9869 0.0007 0.9973 0
% It seems that the first and third deseases are region independent, whereas the others
% have regional dependencies (are independent to each other).
%
% SEE ALSO: chi2test, soon to be uploaded to
% http://www.mathworks.com/matlabcentral/fileexchange/
%
% HISTORY: v.1.0, first working version, 2007-08-30.
%
% COPYRIGHT: (c) 2007 Peder Axensten. Use at own risk.
% KEYWORDS: chi-squared test, chi-squared, chi2, binary, test
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Check the arguments.
if(nargin ~= 2), error('Two and only two arguments required!'); end
if(ndims(x) > 2), error( 'First argument (x) must be a vector or a 2d matrix!'); end
if(ndims(n) > 2), error('Second argument (n) must be a vector or a 2d matrix!'); end
if(size(x, 1) == 1), x= x'; end
if(size(n, 1) == 1), n= n'; end
if(size(n,1) == 1), n= repmat(n, size(x, 1), 1); end
if(size(n,2) == 1), n= repmat(n, 1, size(x, 2)); end
if(any(size(x) ~= size(n))),error('Size of second argument (n) doesn''t match size of first argument (x)!'); end
if(any(~isreal(x))), error('All values of first argument (x) must be real values!'); end
if(any(~isreal(n))), error('All values of second argument (n) must be real values!'); end
if(any(n<1 | (n ~= floor(n))))
error('All values of second argument (n) must be positive integers');
end
% Calculate Q = sum( (a-np*)^2/(np*(1-p*)) )
s= size(x, 1);
ep= repmat((sum(x)./sum(n)), s, 1);
Q= sum((x-n .* ep).^2./(n.*ep.*(1 - ep)));
% Calculate cdf of chi-squared to Q. Degrees of freedom, v, is s-1.
p= 1 - gammainc(Q/2, (s-1)/2);
end
