Response surface design, degrees freedom

Note, however, there are two critical limitations to these "predicting" procedures. First, the mathematical models must adequately fit the data. Correlation coefficients (R ), adjusted for degrees of freedom, of 0.8 or better are considered necessary for reliable prediction when using factorial designs. Second, no predictions outside the design space can be made confidently, because no data are available to warn of unexpectedly abrupt changes in direction of the response surface. The areas covered by Figures 8 and 9 officially violate this latter limitation, but because more detailed... 

Replication is often included in central composite designs. If the response surface is thought to be reasonably homoscedastic, only one of the factor combinations (commonly the center point) need be replicated, usually three or four times to provide sufficient degrees of freedom for s. If the response surface is thought to be heteroscedastic, the replicates can be spread over the response surface to obtain an average purely experimental uncertainty. 

Assume that you have run experiments by a factorial design (with Np runs) with a view to assessing the significance of the experimental variables fi om estimates, hj, of the coefficients in a linear response surface model. Assume also that you have made Nq repeated runs of one experiment to obtain an estimate of the experimental error standard deviation. From the average response, J, in repeated runs, an estimate of the experimental error standard deviation, Sq, with (Nq - 1) degrees of freedom is obtained as... 

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