Multiple linear regression. Least squares fitting of response surface models

Multiple linear regression. Least squares fitting of response surface models 

Assume that we wish to test a model with linear terms and an interaction term to describe the variation in y in an experimental domain spanned by the scaled variables Xj andx, [-1 - ]  

If only one experiment is performed, e.g. with all variables at their high level, 

The observed response is a linear combination of the model parameters and the experimental error. Now, if an experiment is run with another setting of the variables (e.g Xi = -1, X2 = 1), which afforded the response yz, we obtain 

To be able to estimate values of each model parameter, we need to run at least as many experiments as there are unknown model parameters. The trouble is the error terms, Cj, which contain an unknown random experimental error which is not constant in the different experimental runs. We need to take these error terms into account and try to assign values to the model parameters in such a way that the error terms can be kept as small as possible over the whole set of experiments. 

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