Heteroscedastic response surfaces

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. 

Is the response surface shown in Figure 12.6 homoscedastic or heteroscedastic ... 

These and most other equations developed by statisticians assume that the experimental error is the same over the entire response surface there is no satisfactory agreement for how to incorporate heteroscedastic errors. Note that there are several different equations in the literature according to the specific aims of the confidence interval calculations, but for brevity we introduce only two which can be generally applied to most situations. 

Finally it is often useful to be able estimate the experimental error (as discussed in Section 2.2.2), and one method is to perform extra replicates (typically five) in the centre. Obviously other approaches to replication are possible, but it is usual to replicate in the centre and assume that the error is the same throughout the response surface. If there are any overriding reasons to assume that heteroscedasticity of errors has an important role, replication could be performed at the star or factorial points. However, much of experimental design is based on classical statistics where there is no real detailed information about error distributions over an experimental domain, or at least obtaining such information would be unnecessarily laborious. 

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