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Hi all,

Suppose you're dealing with a Box-Behnken design (3 factors), and after running your RSM in R (second order polynomial with all two-way interactions), find find very high R and adjusted-R values (>0.98), and several significant terms,

but the lack of fit statistic is significant, the eigenvalues are of different signs, and the stationary point is way outside the region of experimentation.

How should you proceed or interpret these findings, if the eventual goal is optimizing the response based on those three factors?

Thanks in advance, any advice on the subject is welcome.

Jon

I'd suggest doing a few experimental runs along a path toward that distant stationary point, and see if that indeed leads you in an uphill direction. This is essentially a "ridge analysis" situation. May be wise to not explore too far away at first (maybe up to 2 coded units or so). In case this isn't fruitful, you don't want to break your budget on it. 

Jon,

A Box-Behnken design with 3 factors has 13 points for 10 quadratic polynomial coefficients. This is not an ideal situation. In particular, a gross error in one corner could have led to the wrong form for the surface and a gross error in the position of the stationary point. 

Secondly, the fact that eigenvalues are of different signs and the stationary point is outside the design region indicates that you were on the side of a ridge, which is not a situation in which it is natural to fit a quadratic. Wait until you have got up on the ridge.  

Thirdly, beware of replicates in the center point that are pseudo-replicates, i.e. do not represent all sources of experimental uncertainty. Such replicates would tend to indicate model lack of fit. 

Best wishes,

Rolf Sundberg, Stockholm University