## Copyright (C) 1995-2017 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 {} {[@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
