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welch_test.m

## Copyright (C) 1995, 1996, 1997  Kurt Hornik
## 
## This program 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 2, or (at your option)
## any later version.
## 
## This program 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 this file.  If not, write to the Free Software Foundation,
## 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.

## usage:  [pval, t, df] = welch_test (x, y [, alt])
##
## For two samples x and y from normal distributions with unknown means
## and unknown and not necessarily equal variances, perform a Welch test
## of the null hypothesis of equal means.
## Under the null, the test statistic t approximately follows a Student
## distribution with df degrees of freedom.
##
## With the optional argument string alt, the alternative of interest
## can be selected.  
## If alt is "!=" or "<>", the null is tested against the two-sided
## alternative mean(x) != m.
## If alt is ">", the one-sided alternative mean(x) > m is considered,
## similarly for "<".  
## The default is the two-sided case.
##
## pval is the p-value of the test.
##  
## If no output argument is given, the p-value of the test is displayed.

## Author:  KH <Kurt.Hornik@ci.tuwien.ac.at>
## Description:  Welch two-sample t test

function [pval, t, df] = welch_test (x, y, alt)
  
  if ((nargin < 2) || (nargin > 3))
    usage ("[pval, t, df] = welch_test (x, y [, alt])");
  endif
    
  if (! (is_vector (x) && is_vector (y)))
    error ("welch_test:  both x and y must be vectors");
  endif

  n_x  = length (x);
  n_y  = length (y);
  mu_x = sum (x) / n_x;
  mu_y = sum (y) / n_y;
  v_x  = sumsq (x - mu_x) / (n_x * (n_x - 1));
  v_y  = sumsq (x - mu_y) / (n_y * (n_y - 1));
  c    = v_x / (v_x + v_y);
  df   = 1 / ( c^2 / (n_x - 1) + (1 - c)^2 / (n_y - 1));
  t    = (mu_x - mu_y) / sqrt (v_x + v_y);
  cdf  = t_cdf (t, df);

  if (nargin == 2)
    alt  = "!=";
  endif
  
  if (! isstr (alt))
    error ("welch_test:  alt must be a string");
  endif
  if (strcmp (alt, "!=") || strcmp (alt, "<>"))
    pval = 2 * min (cdf, 1 - cdf);
  elseif (strcmp (alt, ">"))
    pval = 1 - cdf;
  elseif (strcmp (alt, "<"))
    pval = cdf;
  else
    error (sprintf ("welch_test:  option %s not recognized", alt));
  endif
  
  if (nargout == 0)
    printf ("  pval:  %g\n", pval);
  endif

endfunction

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