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## Copyright (C) 1995-2013 Kurt Hornik
##
## This file is part of Octave.
##
## Octave is free software; you can redistribute it and/or modify it
## under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 3 of the License, or (at
## your option) any later version.
##
## Octave is distributed in the hope that it will be useful, but
## WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
## General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with Octave; see the file COPYING. If not, see
## <http://www.gnu.org/licenses/>.
## -*- texinfo -*-
## @deftypefn {Function File} {[@var{pval}, @var{z}] =} u_test (@var{x}, @var{y}, @var{alt})
## For two samples @var{x} and @var{y}, perform a Mann-Whitney U-test of
## the null hypothesis PROB (@var{x} > @var{y}) == 1/2 == PROB (@var{x}
## < @var{y}). Under the null, the test statistic @var{z} approximately
## follows a standard normal distribution. Note that this test is
## equivalent to the Wilcoxon rank-sum test.
##
## With the optional argument string @var{alt}, the alternative of
## interest can be selected. If @var{alt} is @qcode{"!="} or
## @qcode{"<>"}, the null is tested against the two-sided alternative
## PROB (@var{x} > @var{y}) != 1/2. If @var{alt} is @qcode{">"}, the
## one-sided alternative PROB (@var{x} > @var{y}) > 1/2 is considered.
## Similarly for @qcode{"<"}, the one-sided alternative PROB (@var{x} >
## @var{y}) < 1/2 is considered. The default is the two-sided case.
##
## The p-value of the test is returned in @var{pval}.
##
## If no output argument is given, the p-value of the test is displayed.
## @end deftypefn
## This implementation is still incomplete---for small sample sizes,
## the normal approximation is rather bad ...
## Author: KH <Kurt.Hornik@wu-wien.ac.at>
## Description: Mann-Whitney U-test
function [pval, z] = u_test (x, y, alt)
if ((nargin < 2) || (nargin > 3))
print_usage ();
endif
if (! (isvector (x) && isvector (y)))
error ("u_test: both X and Y must be vectors");
endif
n_x = length (x);
n_y = length (y);
r = ranks ([(reshape (x, 1, n_x)), (reshape (y, 1, n_y))]);
z = (sum (r(1 : n_x)) - n_x * (n_x + n_y + 1) / 2) ...
/ sqrt (n_x * n_y * (n_x + n_y + 1) / 12);
cdf = stdnormal_cdf (z);
if (nargin == 2)
alt = "!=";
endif
if (! ischar (alt))
error ("u_test: ALT must be a string");
endif
if (strcmp (alt, "!=") || strcmp (alt, "<>"))
pval = 2 * min (cdf, 1 - cdf);
elseif (strcmp (alt, ">"))
pval = cdf;
elseif (strcmp (alt, "<"))
pval = 1 - cdf;
else
error ("u_test: option %s not recognized", alt);
endif
if (nargout == 0)
printf (" pval: %g\n", pval);
endif
endfunction
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