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What is the algorithm to estimate model coefficients in a Central Composite Rotatable Design?

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Chimezie Umezie
Chimezie Umezie 2012-9-24
The coded variables for the four factors are as follows: X1 =[-1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 0 0 -2 2 0 0 0 0 0 0]'; X2 =[-1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 0 0 0 0 -2 2 0 0 0 0]'; X3 =[-1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 0 0 0 0 0 0 -2 2 0 0]'; X4 =[-1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 -2 2]'; D=[X1 X2 X3 X4];
Y = [247, 250, 248, 253, 243, 243, 245, 245, 249, 254, 249, 258, 243, 244, 245, 246, 246, 246, 246, 267, 244, 248, 248, 242, 245, 247]';
The matrix D consist of the initial 16 runs and 2k(8) runs and 2 replicates at the center.It is required to determine the coefficients for the Main effects, Interaction effects and quadratic terms. In addition, I would appreciate if you can include the algorithm for the ANOVA table.

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Tom Lane
Tom Lane 2012-9-24
You could use backlash if you first form an array A of the constant, linear, interaction, and squared terms. Then compute A\Y.
If you have the Statistics Toolbox, you could use regstats:
regstats(Y,D,'quadratic')
If you have a recent release of the Statistics Toolbox you could use a LinearModel. It has an anova method (function):
m = LinearModel.fit(D,Y,'quadratic')
  2 个评论
Chimezie Umezie
Chimezie Umezie 2012-9-24
编辑：Chimezie Umezie 2012-9-24
I used the "D = x2fx(X,model)T" function to create the array and then used the backlash function to calculated the 1 coefficients. can I assume that that the coefficients. Can I are in this order: Constant, Linear, Interaction and quadratic?
The "regstats(Y,D,'quadratic')" returned only the Constant and Linear coefficients. Is there anyway the other 10 coefficients can be obtained using this function.
Thanks a million Tom
Tom Lane
Tom Lane 2012-9-25
You're right about the order of coefficients from x2fx.
When I use regstats, I get all the coefficients:
>> s = regstats(Y,D,'quadratic');
>> length(s.beta)
ans =
15

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