octave-statistics 1.3.0-4 / usr / share / octave / packages / statistics-1.3.0 / ttest.m

## Copyright (C) 2014 Tony Richardson
##
## 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 3 of the License, 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 program; if not, see <http://www.gnu.org/licenses/>.

## -*- texinfo -*-
## @deftypefn  {Function File} {[@var{h}, @var{pval}, @var{ci}, @var{stats}] =} ttest (@var{x})
## @deftypefnx {Function File} {[@var{h}, @var{pval}, @var{ci}, @var{stats}] =} ttest (@var{x}, @var{m})
## @deftypefnx {Function File} {[@var{h}, @var{pval}, @var{ci}, @var{stats}] =} ttest (@var{x}, @var{y})
## @deftypefnx {Function File} {[@var{h}, @var{pval}, @var{ci}, @var{stats}] =} ttest (@var{x}, @var{m}, @var{Name}, @var{Value})
## @deftypefnx {Function File} {[@var{h}, @var{pval}, @var{ci}, @var{stats}] =} ttest (@var{x}, @var{y}, @var{Name}, @var{Value})
## Test for mean of a normal sample with known variance.
##
## Perform a T-test of the null hypothesis @code{mean (@var{x}) ==
## @var{m}} for a sample @var{x} from a normal distribution with unknown
## mean and unknown std deviation.  Under the null, the test statistic
## @var{t} has a Student's t distribution.  The default value of
## @var{m} is 0.
##
## If the second argument @var{y} is a vector, a paired-t test of the
## hypothesis @code{mean (@var{x}) = mean (@var{y})} is performed.
##
## Name-Value pair arguments can be used to set various options.
## @qcode{"alpha"} can be used to specify the significance level
## of the test (the default value is 0.05).  @qcode{"tail"}, can be used
## to select the desired alternative hypotheses.  If the value is
## @qcode{"both"} (default) the null is tested against the two-sided 
## alternative @code{mean (@var{x}) != @var{m}}.
## If it is @qcode{"right"} the one-sided alternative @code{mean (@var{x})
## > @var{m}} is considered.  Similarly for @qcode{"left"}, the one-sided 
## alternative @code{mean (@var{x}) < @var{m}} is considered.
## When argument @var{x} is a matrix, @qcode{"dim"} can be used to selection
## the dimension over which to perform the test.  (The default is the 
## first non-singleton dimension).
##
## If @var{h} is 0 the null hypothesis is accepted, if it is 1 the null
## hypothesis is rejected. The p-value of the test is returned in @var{pval}.
## A 100(1-alpha)% confidence interval is returned in @var{ci}. @var{stats}
## is a structure containing the value of the test statistic (@var{tstat}),
## the degrees of freedom (@var{df}) and the sample standard deviation
## (@var{sd}).
##
## @end deftypefn

## Author: Tony Richardson <richardson.tony@gmail.com>

function [h, p, ci, stats] = ttest(x, my, varargin)
  
  % Set default arguments
  my_default = 0;
  alpha = 0.05;
  tail  = 'both';

  % Find the first non-singleton dimension of x
  dim = min(find(size(x)~=1));
  if isempty(dim), dim = 1; end

  if (nargin == 1)
    my = my_default;
  end 
  
  i = 1;
  while ( i <= length(varargin) )
    switch lower(varargin{i})
      case 'alpha'
        i = i + 1;
        alpha = varargin{i};
      case 'tail'
        i = i + 1;
        tail = varargin{i};
      case 'dim'
        i = i + 1;
        dim = varargin{i};
      otherwise
        error('Invalid Name argument.',[]);
    end
    i = i + 1;
  end
  
  if ~isa(tail, 'char')
    error('tail argument to vartest2 must be a string\n',[]);
  end
    
  if any(and(~isscalar(my),size(x)~=size(my)))
    error('Arrays in paired test must be the same size.');
  end

  % Set default values if arguments are present but empty
  if isempty(my)
    my = my_default;
  end
    
  % This adjustment allows everything else to remain the
  % same for both the one-sample t test and paired tests.
  x = x - my;
  
  % Calculate the test statistic value (tval)
  n = size(x, dim);
  x_bar = mean(x, dim);
  stats.tstat = 0;
  stats.df = n-1;
  stats.sd = std(x, 0, dim);
  x_bar_std = stats.sd/sqrt(n);
  tval = (x_bar)./x_bar_std;
  stats.tstat = tval;

  % Based on the "tail" argument determine the P-value, the critical values,
  % and the confidence interval.
  switch lower(tail)
    case 'both'
      p = 2*(1 - tcdf(abs(tval),n-1));
      tcrit = -tinv(alpha/2,n-1);
      ci = [x_bar-tcrit*x_bar_std; x_bar+tcrit*x_bar_std] + my;
    case 'left'
      p = tcdf(tval,n-1);
      tcrit = -tinv(alpha,n-1);
      ci = [-inf*ones(size(x_bar)); my+x_bar+tcrit*x_bar_std];
    case 'right'
      p = 1 - tcdf(tval,n-1);
      tcrit = -tinv(alpha,n-1);
      ci = [my+x_bar-tcrit*x_bar_std; inf*ones(size(x_bar))];
    otherwise
      error('Invalid fifth (tail) argument to ttest\n',[]);
  end

  % Reshape the ci array to match MATLAB shaping
  if and(isscalar(x_bar), dim==2)
    ci = ci(:)';
  elseif size(x_bar,2)<size(x_bar,1)
    ci = reshape(ci(:),length(x_bar),2);
  end

  % Determine the test outcome
  % MATLAB returns this a double instead of a logical array
  h = double(p < alpha);  
end

%!test
%! x = 8:0.1:12;
%! [h, pval, ci] = ttest (x, 10);
%! assert (h, 0)
%! assert (pval, 1, eps)
%! assert (ci, [9.6219 10.3781], 1E-5)
%! [h, pval, ci0] = ttest (x, 0);
%! assert (h, 1)
%! assert (pval, 0)
%! assert (ci0, ci)