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## Copyright (C) 1996-2015 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 {Function File} {} manova (@var{x}, @var{g})
## Perform a one-way multivariate analysis of variance (MANOVA).
##
## The goal is to test whether the p-dimensional population means of data
## taken from @var{k} different groups are all equal. All data are assumed
## drawn independently from p-dimensional normal distributions with the same
## covariance matrix.
##
## The data matrix is given by @var{x}. As usual, rows are observations and
## columns are variables. The vector @var{g} specifies the corresponding
## group labels (e.g., numbers from 1 to @var{k}).
##
## The LR test statistic (@nospell{Wilks' Lambda}) and approximate p-values are
## computed and displayed.
## @seealso{anova}
## @end deftypefn
## The Hotelling-Lawley and Pillai-Bartlett test statistics are coded.
## However, they are currently disabled until they can be verified by someone
## with sufficient understanding of the algorithms. Please feel free to
## improve this.
## Author: TF <Thomas.Fuereder@ci.tuwien.ac.at>
## Adapted-By: KH <Kurt.Hornik@wu-wien.ac.at>
## Description: One-way multivariate analysis of variance (MANOVA)
function manova (x, g)
if (nargin != 2)
print_usage ();
endif
if (isvector (x))
error ("manova: X must not be a vector");
endif
[n, p] = size (x);
if (! isvector (g) || (length (g) != n))
error ("manova: G must be a vector of length rows (X)");
endif
s = sort (g);
i = find (s (2:n) > s(1:(n-1)));
k = length (i) + 1;
if (k == 1)
error ("manova: there should be at least 2 groups");
else
group_label = s ([1, (reshape (i, 1, k - 1) + 1)]);
endif
x = x - ones (n, 1) * mean (x);
SST = x' * x;
s = zeros (1, p);
SSB = zeros (p, p);
for i = 1 : k;
v = x (find (g == group_label (i)), :);
s = sum (v);
SSB = SSB + s' * s / rows (v);
endfor
n_b = k - 1;
SSW = SST - SSB;
n_w = n - k;
l = real (eig (SSB / SSW));
if (isa (l, "single"))
l(l < eps ("single")) = 0;
else
l(l < eps) = 0;
endif
## Wilks' Lambda
## =============
Lambda = prod (1 ./ (1 + l));
delta = n_w + n_b - (p + n_b + 1) / 2;
df_num = p * n_b;
W_pval_1 = 1 - chi2cdf (- delta * log (Lambda), df_num);
if (p < 3)
eta = p;
else
eta = sqrt ((p^2 * n_b^2 - 4) / (p^2 + n_b^2 - 5));
endif
df_den = delta * eta - df_num / 2 + 1;
WT = exp (- log (Lambda) / eta) - 1;
W_pval_2 = 1 - fcdf (WT * df_den / df_num, df_num, df_den);
if (0)
## Hotelling-Lawley Test
## =====================
HL = sum (l);
theta = min (p, n_b);
u = (abs (p - n_b) - 1) / 2;
v = (n_w - p - 1) / 2;
df_num = theta * (2 * u + theta + 1);
df_den = 2 * (theta * v + 1);
HL_pval = 1 - fcdf (HL * df_den / df_num, df_num, df_den);
## Pillai-Bartlett
## ===============
PB = sum (l ./ (1 + l));
df_den = theta * (2 * v + theta + 1);
PB_pval = 1 - fcdf (PB * df_den / df_num, df_num, df_den);
printf ("\n");
printf ("One-way MANOVA Table:\n");
printf ("\n");
printf ("Test Test Statistic Approximate p\n");
printf ("**************************************************\n");
printf ("Wilks %10.4f %10.9f \n", Lambda, W_pval_1);
printf (" %10.9f \n", W_pval_2);
printf ("Hotelling-Lawley %10.4f %10.9f \n", HL, HL_pval);
printf ("Pillai-Bartlett %10.4f %10.9f \n", PB, PB_pval);
printf ("\n");
endif
printf ("\n");
printf ("MANOVA Results:\n");
printf ("\n");
printf ("# of groups: %d\n", k);
printf ("# of samples: %d\n", n);
printf ("# of variables: %d\n", p);
printf ("\n");
printf ("Wilks' Lambda: %5.4f\n", Lambda);
printf ("Approximate p: %10.9f (chisquare approximation)\n", W_pval_1);
printf (" %10.9f (F approximation)\n", W_pval_2);
printf ("\n");
endfunction
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