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anovan.m

## Copyright (C) 2003-2005 Andy Adler
##
## 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 2, or (at your option)
## any later version.
##
## This software 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 software; see the file COPYING.  If not, see
## <http://www.gnu.org/licenses/>.

## -*- texinfo -*-
## @deftypefn {Function File} {[@var{pval}, @var{f}, @var{df_b}, @var{df_e}] =} anovan (@var{data}, @var{grps})
## @deftypefnx {Function File} {[@var{pval}, @var{f}, @var{df_b}, @var{df_e}] =} anovan (@var{data}, @var{grps}, 'param1', @var{value1})
## Perform a multi-way analysis of variance (ANOVA).  The goal is to test
## whether the population means of data taken from @var{k} different
## groups are all equal.
##
## Data is a single vector @var{data} with groups specified by
## a corresponding matrix of group labels @var{grps}, where @var{grps}
## has the same number of rows as @var{data}. For example, if
## @var{data} = [1.1;1.2]; @var{grps}= [1,2,1; 1,5,2];
## then data point 1.1 was measured under conditions 1,2,1 and
## data point 1.2 was measured under conditions 1,5,2.
## Note that groups do not need to be sequentially numbered.
##
## By default, a 'linear' model is used, computing the N main effects
## with no interactions. this may be modified by param 'model'
##
## p= anovan(data,groups, 'model', modeltype)
## - modeltype = 'linear': compute N main effects
## - modeltype = 'interaction': compute N effects and
##                               N*(N-1) two-factor interactions
## - modeltype = 'full': compute interactions at all levels
##
## Under the null of constant means, the statistic @var{f} follows an F
## distribution with @var{df_b} and @var{df_e} degrees of freedom.
##
## The p-value (1 minus the CDF of this distribution at @var{f}) is
## returned in @var{pval}.
##
## If no output argument is given, the standard one-way ANOVA table is
## printed.
##
## BUG: DFE is incorrect for modeltypes != full
## @end deftypefn

## Author: Andy Adler <adler@ncf.ca>
## Based on code by: KH <Kurt.Hornik@ci.tuwien.ac.at>
## $Id: anovan.m 4893 2008-04-09 13:02:24Z adb014 $
##
## TESTING RESULTS:
##  1. ANOVA ACCURACY: www.itl.nist.gov/div898/strd/anova/anova.html
##     Passes 'easy' test. Comes close on 'Average'. Fails 'Higher'.
##     This could be fixed with higher precision arithmetic
##  2. Matlab anova2 test
##      www.mathworks.com/access/helpdesk/help/toolbox/stats/anova2.html
##     % From web site:
##      popcorn= [  5.5 4.5 3.5; 5.5 4.5 4.0; 6.0 4.0 3.0;
##                  6.5 5.0 4.0; 7.0 5.5 5.0; 7.0 5.0 4.5];
##     % Define groups so reps = 3
##      groups = [  1 1;1 2;1 3;1 1;1 2;1 3;1 1;1 2;1 3;
##                  2 1;2 2;2 3;2 1;2 2;2 3;2 1;2 2;2 3 ];
##      anovan( vec(popcorn'), groups, 'model', 'full')
##     % Results same as Matlab output
##  3. Matlab anovan test
##      www.mathworks.com/access/helpdesk/help/toolbox/stats/anovan.html
##    % From web site
##      y = [52.7 57.5 45.9 44.5 53.0 57.0 45.9 44.0]';
##      g1 = [1 2 1 2 1 2 1 2]; 
##      g2 = {'hi';'hi';'lo';'lo';'hi';'hi';'lo';'lo'}; 
##      g3 = {'may'; 'may'; 'may'; 'may'; 'june'; 'june'; 'june'; 'june'}; 
##      anovan( y', [g1',g2',g3'])
##    % Fails because we always do interactions
                                                                           

function [PVAL, FSTAT, DF_B, DFE] = anovan (data, grps, varargin)

    if nargin <= 1
        usage ("anovan (data, grps)");
    end

    # test supplied parameters
    modeltype= 'linear';
    for idx= 3:2:nargin
       param= varargin{idx-2};
       value= varargin{idx-1};

       if strcmp(param, 'model')
          modeltype= value;
#      elseif strcmp(param    # add other parameters here
       else 
          error(sprintf('parameter %s is not supported', param));
       end
    end

    if ~isvector (data)
          error ("anova: for `anova (data, grps)', data must be a vector");
    endif

    nd = size (grps,1); # number of data points
    nw = size (grps,2); # number of anova "ways"
    if (~ isvector (data) || (length(data) ~= nd))
      error ("anova: grps must be a matrix of the same number of rows as data");
    endif

    [g,grp_map]   = relabel_groups (grps);
    if strcmp(modeltype, 'linear')
       max_interact  = 1;
    elseif strcmp(modeltype,'interaction')
       max_interact  = 2;
    elseif strcmp(modeltype,'full')
       max_interact  = rows(grps);
    else
       error(sprintf('modeltype %s is not supported', modeltype));
    end
    ng = length(grp_map);
    int_tbl       = interact_tbl (nw, ng, max_interact );
    [gn, gs, gss] = raw_sums(data, g, ng, int_tbl);

    stats_tbl = int_tbl(2:size(int_tbl,1),:)>0;
    nstats= size(stats_tbl,1);
    stats= zeros( nstats+1, 5); # SS, DF, MS, F, p
    for i= 1:nstats
        [SS, DF, MS]= factor_sums( gn, gs, gss, stats_tbl(i,:), ng, nw);
        stats(i,1:3)= [SS, DF, MS];
    end

    # The Mean squared error is the data - avg for each possible measurement
    # This calculation doesn't work unless there is replication for all grps
#   SSE= sum( gss(sel) ) - sum( gs(sel).^2 ./ gn(sel) );  
    SST= gss(1) - gs(1)^2/gn(1);
    SSE= SST - sum(stats(:,1));
    sel = select_pat( ones(1,nw), ng, nw); %incorrect for modeltypes != full
    DFE= sum( (gn(sel)-1).*(gn(sel)>0) );
    MSE= SSE/DFE;
    stats(nstats+1,1:3)= [SSE, DFE, MSE];

    for i= 1:nstats
        MS= stats(i,3);
        DF= stats(i,2);
        F= MS/MSE;
        pval = 1 - fcdf (F, DF, DFE);
        stats(i,4:5)= [F, pval];
    end

    if nargout==0;
        printout( stats, stats_tbl );
    else
        PVAL= stats(1:nstats,5);
        FSTAT=stats(1:nstats,4);
        DF_B= stats(1:nstats,2);
        DF_E= DFE;
    end
endfunction


# relabel groups to a mapping from 1 to ng
# Input
#   grps    input grouping
# Output
#   g       relabelled grouping
#   grp_map map from output to input grouping
function [g,grp_map] = relabel_groups(grps)
    grp_vec= vec(grps);
    s= sort (grp_vec);
    uniq = 1+[0;find(diff(s))];
    # mapping from new grps to old groups
    grp_map = s(uniq);
    # create new group g
    ngroups= length(uniq);
    g= zeros(size(grp_vec));
    for i = 1:ngroups
        g( find( grp_vec== grp_map(i) ) ) = i;
    end
    g= reshape(g, size(grps));
endfunction

# Create interaction table
#
# Input: 
#    nw            number of "ways"
#    ng            number of ANOVA groups
#    max_interact  maximum number of interactions to consider
#                  default is nw
function int_tbl =interact_tbl(nw, ng, max_interact)
    combin= 2^nw;
    inter_tbl= zeros( combin, nw);
    idx= (0:combin-1)';
    for i=1:nw;
       inter_tbl(:,i) = ( rem(idx,2^i) >= 2^(i-1) ); 
    end

    # find elements with more than max_interact 1's
    idx = ( sum(inter_tbl',1) > max_interact );
    inter_tbl(idx,:) =[];
    combin= size(inter_tbl,1); # update value

    #scale inter_tbl 
    # use ng+1 to map combinations of groups to integers
    # this would be lots easier with a hash data structure
    int_tbl = inter_tbl .* (ones(combin,1) * (ng+1).^(0:nw-1) );
endfunction 

# Calculate sums for each combination
#
# Input: 
#    g             relabelled grouping matrix
#    ng            number of ANOVA groups
#    max_interact
#
# Output (virtual (ng+1)x(nw) matrices):
#    gn            number of data sums in each group
#    gs            sum of data in each group
#    gss           sumsqr of data in each group
function    [gn, gs, gss] = raw_sums(data, g, ng, int_tbl);
    nw=    size(g,2);
    ndata= size(g,1);
    gn= gs= gss=  zeros((ng+1)^nw, 1);
    for i=1:ndata
        # need offset by one for indexing
        datapt= data(i);
        idx = 1+ int_tbl*g(i,:)';
        gn(idx)  +=1;
        gs(idx)  +=datapt;
        gss(idx) +=datapt^2;
    end
endfunction

# Calcualte the various factor sums
# Input:  
#    gn            number of data sums in each group
#    gs            sum of data in each group
#    gss           sumsqr of data in each group
#    select        binary vector of factor for this "way"?
#    ng            number of ANOVA groups
#    nw            number of ways

function [SS,DF]= raw_factor_sums( gn, gs, gss, select, ng, nw);
   sel= select_pat( select, ng, nw);
   ss_raw=   gs(sel).^2 ./ gn(sel);
   SS= sum( ss_raw( ~isnan(ss_raw) ));
   if length(find(select>0))==1
       DF= sum(gn(sel)>0)-1;
   else
       DF= 1; #this isn't the real DF, but needed to multiply
   end
endfunction

function [SS, DF, MS]= factor_sums( gn, gs, gss, select, ng, nw);
   SS=0;
   DF=1;

   ff = find(select);
   lff= length(ff);
   # zero terms added, one term subtracted, two added, etc
   for i= 0:2^lff-1
       remove= find( rem( floor( i * 2.^(-lff+1:0) ), 2) );
       sel1= select;
       if ~isempty(remove)
           sel1( ff( remove ) )=0;
       end
       [raw_sum,raw_df]= raw_factor_sums(gn,gs,gss,sel1,ng,nw);
       
       add_sub= (-1)^length(remove);
       SS+= add_sub*raw_sum;
       DF*= raw_df;
   end

   MS=  SS/DF;
endfunction

# Calcualte the various factor sums
# Input:  
#    select        binary vector of factor for this "way"?
#    ng            number of ANOVA groups
#    nw            number of ways
function sel= select_pat( select, ng, nw);
   # if select(i) is zero, remove nonzeros
   # if select(i) is zero, remove zero terms for i
   field=[];

   if length(select) ~= nw;
       error("length of select must be = nw");
   end
   ng1= ng+1;

   if isempty(field)
       # expand 0:(ng+1)^nw in base ng+1
       field= (0:(ng1)^nw-1)'* ng1.^(-nw+1:0);
       field= rem( floor( field), ng1);
       # select zero or non-zero elements
       field= field>0;
   end
   sel= find( all( field == ones(ng1^nw,1)*select(:)', 2) );
endfunction


function printout( stats, stats_tbl );
  nw= size( stats_tbl,2);
  [jnk,order]= sort( sum(stats_tbl,2) );

  printf('\n%d-way ANOVA Table (Factors A%s):\n\n', nw, ...
         sprintf(',%c',toascii('A')+(1:nw-1)) );
  printf('Source of Variation        Sum Sqr   df      MeanSS    Fval   p-value\n');
  printf('*********************************************************************\n');
  printf('Error                  %10.2f  %4d %10.2f\n', stats( size(stats,1),1:3));
  
  for i= order(:)'
      str=  sprintf(' %c x',toascii('A')+find(stats_tbl(i,:)>0)-1 );
      str= str(1:length(str)-2); # remove x
      printf('Factor %15s %10.2f  %4d %10.2f  %7.3f  %7.6f\n', ...
         str, stats(i,:) );
  end
  printf('\n');
endfunction

# Test Data from http://maths.sci.shu.ac.uk/distance/stats/14.shtml
data=';
grp = [1,1; 1,1; 1,2; 1,2; 1,3; 1,3;
       2,1; 2,1; 2,2; 2,2; 2,3; 2,3;
       3,1; 3,1; 3,2; 3,2; 3,3; 3,3];
data=[7  9  9  8 12 10  9  8 ...
      9  8 10 11 13 13 10 11 ...
      9 10 10 12 10 12 10 12]';
grp = ;
# Test Data from http://maths.sci.shu.ac.uk/distance/stats/9.shtml
data=';
grp= [1:4,1:4,1:4]';
# Test Data from http://maths.sci.shu.ac.uk/distance/stats/13.shtml
data='; 
grp = [1,1;1,2;1,3;
       2,1;2,2;2,3;
       3,1;3,2;3,3;
       4,1;4,2;4,3;
       5,1;5,2;5,3];
# Test Data from www.mathworks.com/
#                access/helpdesk/help/toolbox/stats/linear10.shtml
data=[23  27  43  41  15  17   3   9  20  63  55  90];
grp= [ 1    1   1   1   2   2   2   2   3   3   3   3;
       1    1   2   2   1   1   2   2   1   1   2   2]';





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