function cufd22(D,alpha)
% CUFD22 Testing for Curvature of a 2^2 Factorial Design Analysis.
% This m-file is used in experiments where there is uncertain about the
% assumption of linearity over the region of exploration, and when the
% experimenter decides to conduct a 2^2 factorial design with a single
% or several replicates of each factorial run, augmented with some center
% points. A formal analysis of variance is running in order to see if
% there is evidence of curvature in the response over the region of
% exploration. The result can gives indication of some quadratic effects
% (second-order model) or not (first-order model).
%
% The formal curvature sum of squares in the analysis of variance is computed
% as follows,
%
% SSU = nf*nc*(mean of factorial points - mean of center points)^2/(nf+nc)
%
% where nf is the number of points in the factorial portion and nc the number
% of points at the center.
% For the mean square (MSU) SSU must divide by 1 degrees of freedom.
%
% The mean of squares error is calculated from the center points as follows,
%
% SSE = Sum_nc(y_center points - mean of center ponts)^2
%
% So,
%
% MSE = SSE/(nc-1)
%
% Next figure illustrates the four-factor level or treatment combinations
% and some center points on a 2^2 factorial design.
%
% b ab
% 2 .------------------.
% | |
% | |
% | |
% Level Factor B | . (nc) |
% | |
% | |
% | |
% 1 .------------------.
% (1) a
%
% 1 2
% Level Factor A
%
%
% Syntax: cufd22(D,alpha)
%
% Inputs:
% D - matrix data (=[X Y]) (last column must be the Y-dependent variable).
% (X-independent variable entry for the 2^2 factorial design).
% alpha - significance-value (default = 0.05).
%
% Outputs:
% A complete analysis of variance is summarized on a table.
%
% From the example 3.4 of Myers and Montgomery (2002, p.130), a chemical engineer is
% studing the yield of a process. There are two variables of interest as reaction
% time (A: 1=30, 2=40, 0=35 min) and reaction temperature (B: 1=150, 2=160, 0=155
% C degrees). Due that the engineer has uncertain about the assumption of linearity,
% decides to conduct a 2^2 design with a single replicate on each factorial run and
% with 5 center points. The data are as follow,
%
% -------------
% Factor-level combination Yield
% A B
% ----------------------------------------
% 1 1 39.3
% 1 2 40.9
% 2 1 40.0
% 2 2 41.5
% 0 0 40.3
% 0 0 40.5
% 0 0 40.7
% 0 0 40.2
% 0 0 40.6
% ----------------------------------------
%
% Data matrix must be:
% D=[1 1 39.3;1 2 40.9;2 1 40;2 2 41.5;0 0 40.3;0 0 40.5;0 0 40.7;
% 0 0 40.2;0 0 40.6];
%
% Calling on Matlab the function:
% cufd22(D)
%
% Answer is:
%
% Analysis of Variance of the 2^2 Factorial Design for the Curvature Test.
% ----------------------------------------------------------------------------------------------
% SOV SS df MS F P Conclusion
% ----------------------------------------------------------------------------------------------
% A 2.4025 1 2.4025 55.8721 0.0017 S
% B 0.4225 1 0.4225 9.8256 0.0350 S
% AB 0.0025 1 0.0025 0.0581 0.8213 NS
% Curvature 0.0027 1 0.0027 0.0633 0.8137 NS
% Error 0.1720 4 0.0430
% ----------------------------------------------------------------------------------------------
% Total 3.0022 8
% ----------------------------------------------------------------------------------------------
% Number of replicates on each factor-level are: 1
% With a given significance level of: 0.05
% The results are significant (S) and/or not significant (NS).
%
% Created by A. Trujillo-Ortiz, R. Hernandez-Walls and F.A. Trujillo-Perez
% Facultad de Ciencias Marinas
% Universidad Autonoma de Baja California
% Apdo. Postal 453
% Ensenada, Baja California
% Mexico.
% atrujo@uabc.mx
% Copyright (C) December 26, 2005.
%
% To cite this file, this would be an appropriate format:
% Trujillo-Ortiz, A., R. Hernandez-Walls, F.A. Trujillo-Perez. (2005). CUFD22:2^2 Testing for Curvature
% of a 2^2 Factorial Design Analysis. A MATLAB file. [WWW document]. URL http://www.mathworks.com/
% matlabcentral/fileexchange/loadFile.do?objectId=9461
%
% References:
% Myers, R. H. and Montgomery, D. C. (2002), Response Surface Methodology:Process and Product
% Optimization Using Designed Experiments. 2nd. Ed. NY: John Wiley & Sons, Inc.
%
if nargin < 2,
alpha = 0.05; %(default)
end;
if (alpha <= 0 | alpha >= 1),
disp(' ');
fprintf('Warning: significance level must be between 0 and 1\n');
return;
end;
if (length(find(D(:,1)==1))~=length(find(D(:,1)==2)))|(length(find(D(:,2)==1))~=length(find(D(:,2)==2))),
disp(' ');
disp('''Warning: Some of the factor-level combination(s) has(have) different number of replicates.''');
disp('Please, check it.');
return;
end;
if size(D,2)-1 ~= 2,
disp(' ');
disp('''Warning: The number of factors must be two.''');
disp('Please, check it.');
return;
end;
N = size(D,1); %total number of observations
n = length(find(D(:,1)==1))/2; %number of replicates
nc = length(find(D(:,1)==0)); %number of central points
nf = N-nc; %number of factorial points
Y = D(:,end);
O = sum(Y(1:n)); %factor-level combination 1 (low level A-low level B)
a = sum(Y(n+1:2*n)); %factor-level combination 2 (high level A-low level B)
b = sum(Y((2*n)+1:3*n)); %factor-level combination 3 (low level A-high level B)
ab = sum(Y((3*n)+1)); %factor-level combination 4 (high level A-high level B)
CA = ab+a-b-O; %contrast of the total effect of A
CB = ab+b-a-O; %contrast of the total effect of B
CAB = ab+O-a-b; %contrast of the total effect of AB
A = (1/(2*n))*CA; %average effect of A
B = (1/(2*n))*CB; %average effect of B
AB = (1/(2*n))*CAB; %average effect of AB
SSA = CA^2/(n*4); %sum of squares of A
SSB = CB^2/(n*4); %sum of squares of B
SSAB = CAB^2/(n*4); %sum of squares of AB
SSTo = sum(Y.^2)-(sum(Y)^2/N); %total sum of squares
SSU = (nf*nc)*(sum(Y(1:nf)/nf)-(sum(Y((4*n)+1:end))/nc))^2/N; %curvature sum of squares (formal)
SSE = SSTo-SSA-SSB-SSAB-SSU; %error sum of squares
%SSE = sum(((Y((4*n)+1:end))-repmat((sum(Y((4*n)+1:end))/nc),nc,1)).^2); %error sum of squares (formal)
%SSU = SSTo-SSA-SSB-SSAB-SSE; %curvature sum of squares
dfA = max(D(:,1)-1); %degrees of freedom of A
dfB = max(D(:,2)-1); %degrees of freedom of B
dfAB = dfA*dfB; %degrees of freedom of AB
dfTo = N-1; %total degrees of freedom
dfU = 1; %curvature degrees of freedom (formal)
dfE = dfTo-dfA-dfB-dfAB-dfU; %error degrees of freedom
%dfE = nc-1; %error degrees of freedom (formal)
%dfU = dfTo-dfA-dfB-dfAB-dfE; %curvature degrees of freedom
MSA = SSA/dfA; %mean square of A
MSB = SSB/dfB; %mean square of B
MSAB = SSAB/dfAB; %mean square of AB
MSE = SSE/dfE; %error mean square
MSU = SSU/dfU; %curvature mean square
FA = MSA/MSE;
PA = 1 - fcdf(FA,dfA,dfE);
FB = MSB/MSE;
PB = 1 - fcdf(FB,dfB,dfE);
FAB = MSAB/MSE;
PAB = 1 - fcdf(FAB,dfAB,dfE);
FU = MSU/MSE;
PU = 1 - fcdf(FU,dfU,dfE);
if PA >= alpha;
dsA ='NS';
else
dsA =' S';
end;
if PB >= alpha;
dsB ='NS';
else
dsB =' S';
end;
if PAB >= alpha;
dsAB ='NS';
else
dsAB =' S';
end;
if PU >= alpha;
dsU ='NS';
else
dsU =' S';
end;
disp(' ')
disp('Analysis of Variance of the 2^2 Factorial Design for the Curvature Test.')
fprintf('----------------------------------------------------------------------------------------------\n');
disp(' SOV SS df MS F P Conclusion');
fprintf('----------------------------------------------------------------------------------------------\n');
fprintf('A %11.4f%10i%18.4f%14.4f%14.4f%9s\n',SSA,dfA,MSA,FA,PA,dsA);
fprintf('B %11.4f%10i%18.4f%14.4f%14.4f%9s\n',SSB,dfB,MSB,FB,PB,dsB);
fprintf('AB %11.4f%10i%18.4f%14.4f%14.4f%9s\n',SSAB,dfAB,MSAB,FAB,PAB,dsAB);
fprintf('Curvature %11.4f%10i%18.4f%14.4f%14.4f%9s\n',SSU,dfU,MSU,FU,PU,dsU);
fprintf('Error %11.4f%10i%18.4f\n',SSE,dfE,MSE);
fprintf('----------------------------------------------------------------------------------------------\n');
fprintf('Total %11.4f%10i\n',SSTo,dfTo);
fprintf('----------------------------------------------------------------------------------------------\n');
fprintf('Number of replicates on each factor-level are: %i\n', n);
fprintf('With a given significance level of: %.2f\n', alpha);
disp('The results are significant (S) and/or not significant (NS).');
return;
