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function [p, table] = anova_rm(X, displayopt)
% [p, table] = anova_rm(X, displayopt)
% Single factor, tepeated measures ANOVA.
%
% [p, table] = anova_rm(X, displayopt) performs a repeated measures ANOVA
% for comparing the means of two or more columns (time) in one or more
% samples(groups). Unbalanced samples (i.e. different number of subjects
% per group) is supported though the number of columns (followups)should
% be the same.
%
% DISPLAYOPT can be 'on' (the default) to display the ANOVA table, or
% 'off' to skip the display. For a design with only one group and two or
% more follow-ups, X should be a matrix with one row for each subject.
% In a design with multiple groups, X should be a cell array of matrixes.
%
% Example: Gait-Cycle-times of a group of 7 PD patients have been
% measured 3 times, in one baseline and two follow-ups:
%
% patients = [
% 1.1015 1.0675 1.1264
% 0.9850 1.0061 1.0230
% 1.2253 1.2021 1.1248
% 1.0231 1.0573 1.0529
% 1.0612 1.0055 1.0600
% 1.0389 1.0219 1.0793
% 1.0869 1.1619 1.0827 ];
%
% more over, a group of 8 controls has been measured in the same protocol:
%
% controls = [
% 0.9646 0.9821 0.9709
% 0.9768 0.9735 0.9576
% 1.0140 0.9689 0.9328
% 0.9391 0.9532 0.9237
% 1.0207 1.0306 0.9482
% 0.9684 0.9398 0.9501
% 1.0692 1.0601 1.0766
% 1.0187 1.0534 1.0802 ];
%
% We are interested to see if the performance of the patients for the
% followups were the same or not:
%
% p = anova_rm(patients);
%
% By considering the both groups, we can also check to see if the
% follow-ups were significantly different and also check two see that the
% two groups had a different performance:
%
% p = anova_rm({patients controls});
%
%
% ref: Statistical Methods for the Analysis of Repeated Measurements,
% C. S. Daivs, Springer, 2002
%
% Copyright 2008, Arash Salarian
% mailto://arash.salarian@ieee.org
%
if nargin < 2
displayopt = 'on';
end
if ~iscell(X)
X = {X};
end
%number of groups
s = size(X,2);
%subjects per group
n_h = zeros(s, 1);
for h=1:s
n_h(h) = size(X{h}, 1);
end
n = sum(n_h);
%number of follow-ups
t = size(X{1},2);
% overall mean
y = 0;
for h=1:s
y = y + sum(sum(X{h}));
end
y = y / (n * t);
% allocate means
y_h = zeros(s,1);
y_j = zeros(t,1);
y_hj = zeros(s,t);
y_hi = cell(s,1);
for h=1:s
y_hi{h} = zeros(n_h(h),1);
end
% group means
for h=1:s
y_h(h) = sum(sum(X{h})) / (n_h(h) * t);
end
% follow-up means
for j=1:t
y_j(j) = 0;
for h=1:s
y_j(j) = y_j(j) + sum(X{h}(:,j));
end
y_j(j) = y_j(j) / n;
end
% group h and time j mean
for h=1:s
for j=1:t
y_hj(h,j) = sum(X{h}(:,j) / n_h(h));
end
end
% subject i'th of group h mean
for h=1:s
for i=1:n_h(h)
y_hi{h}(i) = sum(X{h}(i,:)) / t;
end
end
% calculate the sum of squares
ssG = 0;
ssSG = 0;
ssT = 0;
ssGT = 0;
ssR = 0;
for h=1:s
for i=1:n_h(h)
for j=1:t
ssG = ssG + (y_h(h) - y)^2;
ssSG = ssSG + (y_hi{h}(i) - y_h(h))^2;
ssT = ssT + (y_j(j) - y)^2;
ssGT = ssGT + (y_hj(h,j) - y_h(h) - y_j(j) + y)^2;
ssR = ssR + (X{h}(i,j) - y_hj(h,j) - y_hi{h}(i) + y_h(h))^2;
end
end
end
% calculate means
if s > 1
msG = ssG / (s-1);
msGT = ssGT / ((s-1)*(t-1));
end
msSG = ssSG / (n-s);
msT = ssT / (t-1);
msR = ssR / ((n-s)*(t-1));
% calculate the F-statistics
if s > 1
FG = msG / msSG;
FGT = msGT / msR;
end
FT = msT / msR;
FSG = msSG / msR;
% single or multiple sample designs?
if s > 1
% case for multiple samples
pG = 1 - fcdf(FG, s-1, n-s);
pT = 1 - fcdf(FT, t-1, (n-s)*(t-1));
pGT = 1 - fcdf(FGT, (s-1)*(t-1), (n-s)*(t-1));
pSG = 1 - fcdf(FSG, n-s, (n-s)*(t-1));
p = [pT, pG, pSG, pGT];
table = { 'Source' 'SS' 'df' 'MS' 'F' 'Prob>F'
'Time' ssT t-1 msT FT pT
'Group' ssG s-1 msG FG pG
'Ineratcion' ssGT (s-1)*(t-1) msGT FGT pGT
'Subjects (matching)' ssSG n-s msSG FSG pSG
'Error' ssR (n-s)*(t-1) msR [] []
'Total' [] [] [] [] []
};
table{end, 2} = sum([table{2:end-1,2}]);
table{end, 3} = sum([table{2:end-1,3}]);
if (isequal(displayopt, 'on'))
digits = [-1 -1 0 -1 2 4];
statdisptable(table, 'multi-sample repeated measures ANOVA', 'ANOVA Table', '', digits);
end
else
% case for only one sample
pT = 1 - fcdf(FT, t-1, (n-s)*(t-1));
pSG = 1 - fcdf(FSG, n-s, (n-s)*(t-1));
p = [pT, pSG];
table = { 'Source' 'SS' 'df' 'MS' 'F' 'Prob>F'
'Time' ssT t-1 msT FT pT
'Subjects (matching)' ssSG n-s msSG FSG pSG
'Error' ssR (n-s)*(t-1) msR [] []
'Total' [] [] [] [] []
};
table{end, 2} = sum([table{2:end-1,2}]);
table{end, 3} = sum([table{2:end-1,3}]);
if (isequal(displayopt, 'on'))
digits = [-1 -1 0 -1 2 4];
statdisptable(table, 'repeated measures ANOVA', 'ANOVA Table', '', digits);
end
end |