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function [CI, r, p] = cond_indep_fisher_z(X, Y, S, C, N, alpha)
% COND_INDEP_FISHER_Z Test if X indep Y given Z using Fisher's Z test
% CI = cond_indep_fisher_z(X, Y, S, C, N, alpha)
%
% C is the covariance (or correlation) matrix
% N is the sample size
% alpha is the significance level (default: 0.05)
%
% See p133 of T. Anderson, "An Intro. to Multivariate Statistical Analysis", 1984
if nargin < 6, alpha = 0.05; end
r = partial_corr_coef(C, X, Y, S);
z = 0.5*log( (1+r)/(1-r) );
z0 = 0;
W = sqrt(N - length(S) - 3)*(z-z0); % W ~ N(0,1)
cutoff = norminv(1 - 0.5*alpha); % P(|W| <= cutoff) = 0.95
%cutoff = mynorminv(1 - 0.5*alpha); % P(|W| <= cutoff) = 0.95
if abs(W) < cutoff
CI = 1;
else % reject the null hypothesis that rho = 0
CI = 0;
end
p = normcdf(W);
%p = mynormcdf(W);
%%%%%%%%%
function p = normcdf(x,mu,sigma)
%NORMCDF Normal cumulative distribution function (cdf).
% P = NORMCDF(X,MU,SIGMA) computes the normal cdf with mean MU and
% standard deviation SIGMA at the values in X.
%
% The size of P is the common size of X, MU and SIGMA. A scalar input
% functions as a constant matrix of the same size as the other inputs.
%
% Default values for MU and SIGMA are 0 and 1 respectively.
% References:
% [1] M. Abramowitz and I. A. Stegun, "Handbook of Mathematical
% Functions", Government Printing Office, 1964, 26.2.
% Copyright (c) 1993-98 by The MathWorks, Inc.
% $Revision: 1.1.1.1 $ $Date: 2005/04/26 02:29:18 $
if nargin < 3,
sigma = 1;
end
if nargin < 2;
mu = 0;
end
[errorcode x mu sigma] = distchck(3,x,mu,sigma);
if errorcode > 0
error('Requires non-scalar arguments to match in size.');
end
% Initialize P to zero.
p = zeros(size(x));
% Return NaN if SIGMA is not positive.
k1 = find(sigma <= 0);
if any(k1)
tmp = NaN;
p(k1) = tmp(ones(size(k1)));
end
% Express normal CDF in terms of the error function.
k = find(sigma > 0);
if any(k)
p(k) = 0.5 * erfc( - (x(k) - mu(k)) ./ (sigma(k) * sqrt(2)));
end
% Make sure that round-off errors never make P greater than 1.
k2 = find(p > 1);
if any(k2)
p(k2) = ones(size(k2));
end
%%%%%%%%
function x = norminv(p,mu,sigma);
%NORMINV Inverse of the normal cumulative distribution function (cdf).
% X = NORMINV(P,MU,SIGMA) finds the inverse of the normal cdf with
% mean, MU, and standard deviation, SIGMA.
%
% The size of X is the common size of the input arguments. A scalar input
% functions as a constant matrix of the same size as the other inputs.
%
% Default values for MU and SIGMA are 0 and 1 respectively.
% References:
% [1] M. Abramowitz and I. A. Stegun, "Handbook of Mathematical
% Functions", Government Printing Office, 1964, 7.1.1 and 26.2.2
% Copyright (c) 1993-98 by The MathWorks, Inc.
% $Revision: 1.1.1.1 $ $Date: 2005/04/26 02:29:18 $
if nargin < 3,
sigma = 1;
end
if nargin < 2;
mu = 0;
end
[errorcode p mu sigma] = distchck(3,p,mu,sigma);
if errorcode > 0
error('Requires non-scalar arguments to match in size.');
end
% Allocate space for x.
x = zeros(size(p));
% Return NaN if the arguments are outside their respective limits.
k = find(sigma <= 0 | p < 0 | p > 1);
if any(k)
tmp = NaN;
x(k) = tmp(ones(size(k)));
end
% Put in the correct values when P is either 0 or 1.
k = find(p == 0);
if any(k)
tmp = Inf;
x(k) = -tmp(ones(size(k)));
end
k = find(p == 1);
if any(k)
tmp = Inf;
x(k) = tmp(ones(size(k)));
end
% Compute the inverse function for the intermediate values.
k = find(p > 0 & p < 1 & sigma > 0);
if any(k),
x(k) = sqrt(2) * sigma(k) .* erfinv(2 * p(k) - 1) + mu(k);
end
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