## Copyright (C) 2014 Tony Richardson
##
## 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
## .
## -*- texinfo -*-
## @deftypefn {Function File}
## {[@var{h}, @var{pval}, @var{ci}, @var{stats} ] =}
## ttest2 (@var{x}, @var{y})
##
## {[@var{h}, @var{pval}, @var{ci}, @var{stats} ] =}
## ttest2 (@var{x}, @var{y}, @var{Name}, @var{Value})
##
## 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.
##
## If the second argument @var{y} is a vector, a paired-t test of the
## hypothesis mean(x) = mean(y) is performed.
##
## The argument @var{alpha} can be used to specify the significance level
## of the test (the default value is 0.05). The string
## argument @var{tail}, can be used to select the desired alternative
## hypotheses. If @var{alt} is @qcode{"both"} (default) the null is
## tested against the two-sided alternative @code{mean (@var{x}) != @var{m}}.
## If @var{alt} 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 @var{vartype} is 'equal'
## the variances are assumed to be equal (this is the default). When
## @var{vartype} is 'unequal' the variances are not assumed equal.
## When argument @var{x} is a matrix the @var{dim} argument 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
## Description: Test for mean of a normal sample with known variance
function [h, p, ci, stats] = ttest2(x, y, varargin)
alpha = 0.05;
tail = 'both';
vartype = 'equal';
% Find the first non-singleton dimension of x
dim = min(find(size(x)~=1));
if isempty(dim), dim = 1; 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 'vartype'
i = i + 1;
vartype = 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 ttest2 must be a string\n',[]);
end
m = size(x, dim);
n = size(y, dim);
x_bar = mean(x,dim)-mean(y,dim);
s1_var = var(x, 0, dim);
s2_var = var(y, 0, dim);
switch lower(vartype)
case 'equal'
stats.tstat = 0;
stats.df = (m + n - 2)*ones(size(x_bar));
sp_var = ((m-1)*s1_var + (n-1)*s2_var)./stats.df;
stats.sd = sqrt(sp_var);
x_bar_std = sqrt(sp_var*(1/m+1/n));
case 'unequal'
stats.tstat = 0;
se1 = sqrt(s1_var/m);
se2 = sqrt(s2_var/n);
sp_var = s1_var/m + s2_var/n;
stats.df = ((se1.^2+se2.^2).^2 ./ (se1.^4/(m-1) + se2.^4/(n-1)));
stats.sd = [sqrt(s1_var); sqrt(s2_var)];
x_bar_std = sqrt(sp_var);
otherwise
error('Invalid fifth (vartype) argument to ttest2\n',[]);
end
stats.tstat = x_bar./x_bar_std;
% 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(stats.tstat),stats.df));
tcrit = -tinv(alpha/2,stats.df);
%ci = [x_bar-tcrit*stats.sd; x_bar+tcrit*stats.sd];
ci = [x_bar-tcrit.*x_bar_std; x_bar+tcrit.*x_bar_std];
case 'left'
p = tcdf(stats.tstat,stats.df);
tcrit = -tinv(alpha,stats.df);
ci = [-inf*ones(size(x_bar)); x_bar+tcrit.*x_bar_std];
case 'right'
p = 1 - tcdf(stats.tstat,stats.df);
tcrit = -tinv(alpha,stats.df);
ci = [x_bar-tcrit.*x_bar_std; inf*ones(size(x_bar))];
otherwise
error('Invalid fourth (tail) argument to ttest2\n',[]);
end
% Reshape the ci array to match MATLAB shaping
if and(isscalar(x_bar), dim==2)
ci = ci(:)';
stats.sd = stats.sd(:)';
elseif size(x_bar,2)