I have a data set arranged in 2^2 factorial design. There are 3 replicates for each combination of factors. While finding coefficients of a simple first order regression model (y= a1*x1+ a2*x2 +error), which y should I consider for a given x1 and x2 (as there are 3 repicates) ?
Use Y = Ab + e , where Y is 12 by 1, A is 12 by 3, e is 12 by 1 and b is 3 by 1.
Y is the response vector of all individual responses, block partition it such that first three y values correspond to replicates at (-1,-1), 2nd block corresponding to (-1,+1) replicate values, and the remaining two blocks be the replicate values corresponding to (+1,-1) and (+1,+1) factor settings respectively.
Construct the design matrix A compatible with the above factor setting arrangement as Y , the first four rows are ( +1 -1 -1 ), next four rows are ( +1 -1 +1 ), then subsequent 4 rows are ( +1 +1 -1 ) and last four rows are ( +1 +1 +1 ).
Estimate the model parameter vector b0 = (inv(A'*A))*A'*Y
In the above answer, the design matrix A is typed incorrect, it is actually as follows:
First 3 rows : ( +1 -1 -1 )
Next 3 rows: ( +1 -1 +1 )
Next 3 rows: ( +1 +1 -1 )
Last 3 rows: ( +1 +1 +1 )
Rest is as given in answer above.
Dr Ghosh did not give a reference but if you want one see:
https://www.amazon.com/Design-Analysis-Experiments-Douglas-Montgomery/dp/1118146921/ref=pd_rhf_se_p_img_2?_encoding=UTF8&psc=1&refRID=1SXAQFXE3C7DQ3GCH8G4
Best, David Booth
Thank you Debopam Ghosh and David Eugene Booth for your suggestions.
With regards
Tarini
ithis may depend on the definition of 'replicate.' if you mean 3 independent samples in each of the 4 categories, then
y(i)=aX1(i) + bX2(i) +e(i) is reasonable,
ASSUMING you know that there is no interaction between the effects of X1 and X2 (which you probably don't have power to detect in such a small sample).
if, however, you mean 3 replicates from the ''same' subjects (or beakers), the above is not correct. in the olden days, before GEE, we would have averaged the ys, leaving a sample size of 4. in truth, you study has somewhere between 4 and 12 effective 'subjects', depending on how correlated the replicates are within group (and if they were exactly equal, how would you tell whether that was nature or grouping?). these days GEE is used to take care of clustering within logical/natural subjects.
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