manova.m

GNU Octave is a high-level language, primarily intended for numerical computations. It provides a c

## Copyright (C) 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:  manova (Y, g)
##
## Performs a one-way multivariate analysis of variance (MANOVA). The
## goal is to test whether the p-dimensional population means of data
## taken from k different groups are all equal.  All data are assumed
## drawn independently from p-dimensional normal distributions with the
## same covariance matrix.
##
## Y is the data matrix.  As usual, rows are observations and columns
## are variables.  g is the vector of corresponding group labels (e.g.,
## numbers from 1 to k), so that necessarily, length (g) must be the
## same as rows (Y).
##
## The LR test statistic (Wilks' Lambda) and approximate p-values are
## computed and displayed.

## Three test statistics (Wilks, Hotelling-Lawley, and Pillai-Bartlett)
## and corresponding approximate p-values are calculated and displayed.
## (Currently NOT because the f_cdf respectively betai code is too bad.)
  
## Author:  TF <[e-mail]>
## Adapted-By:  KH <[e-mail]>
## Description:  One-way multivariate analysis of variance (MANOVA)

function manova (Y, g)

  if (nargin != 2)
    usage ("manova (Y, g)");
  endif

  if (is_vector (Y))
    error ("manova:  Y must not be a vector");
  endif

  [n, p] = size (Y);

  if (!is_vector (g) || (length (g) != n))
    error ("manova:  g must be a vector of length rows (Y)");
  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

  Y = Y - ones (n, 1) * mean (Y);
  SST = Y' * Y;

  s = zeros (1, p);
  SSB = zeros (p, p);
  for i = 1 : k;
    v = Y (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));
  l (l < eps) = 0;

  ## 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 - chisquare_cdf (- 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 - f_cdf (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 - f_cdf (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 - f_cdf (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