## 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} ] =} ## ttest (@var{x}) ## ## {[@var{h}, @var{pval}, @var{ci}, @var{stats} ] =} ## ttest (@var{x}, @var{m}) ## ## {[@var{h}, @var{pval}, @var{ci}, @var{stats} ] =} ## ttest (@var{x}, @var{y}) ## ## {[@var{h}, @var{pval}, @var{ci}, @var{stats} ] =} ## ttest (@var{x}, @var{m}, @var{Name}, @var{Value}) ## ## {[@var{h}, @var{pval}, @var{ci}, @var{stats} ] =} ## ttest (@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. The default value of ## @var{m} is 0. ## ## If the second argument @var{y} is a vector, a paired-t test of the ## hypothesis mean(x) = mean(y) is performed. ## ## Name-Value pair arguments can be used to set various options. ## @var{'alpha'} can be used to specify the significance level ## of the test (the default value is 0.05). @var{'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, @var{'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 ## Description: Test for mean of a normal sample with known variance 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]; case 'left' p = tcdf(tval,n-1); tcrit = -tinv(alpha,n-1); ci = [-inf*ones(size(x_bar)); x_bar+tcrit*x_bar_std]; case 'right' p = 1 - tcdf(tval,n-1); tcrit = -tinv(alpha,n-1); ci = [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)