If I have two parameters, A and B, is it always better to include A*A*B and A*B*B, or just A*B is enough. Generally, what is the basis for choice, does it depend on the number of factors, particularly in Minitab software. In response surface methodology, for example, software itself defines the terms, but in regression>fit model it is possible to include additional terms.
Hmmm first let me mention my fav resources in this area: https://www.amazon.com/Analysis-Experiments-Enhanced-Abridged-Companion/dp/1119593409/ref=sr_1_1?keywords=Douglas+montgomery+design+of+experiments&qid=1574824324&s=books&sr=1-1 and https://www.amazon.com/Response-Surface-Methodology-Optimization-Experiments/dp/1118916018/ref=sr_1_1?keywords=Douglas+montgomery+Response+Surface&qid=1574824473&s=books&sr=1-1. I would strongly suggest looking at these. Now you mention interaction. First, interaction appears in a factorial design structure. Following D of EXPTs above the model for a 2x2 complete factorial design is well-known to be y=a + bA + cB +dA*B + eps in regression form. Thus for 1 replication df(error)=0 It would make no sense to consider an A*A*B term in this context. Clearly we must replicate to get useful data. The basis of choice is the theory of 2x2 factorial designs and what makes sense there. This reasoning is applicable to all factorial designs where interaction can be studied. BTW software never defines the model. That is the purpose of the experimenter and the research question(s). I hope this and the Montgomery texts help clear this up. Best wishes, David Booth
A statistical modeling using RSM with two parameters A and B denote that only central composite design CCD can be used because it's the only design that can accommodate minimum of two factors. And also RSM model is based on the assumption of a polynomial model better still a quadratic model. And in a polynomial model, there is what we call linear terms A or B, interaction or cross-products term AB, and Finally the quadratic term A*A or B*B. Therefore, when analysis the model most software will look and the linear effect of the model, the interaction effect as well as the quadratic effect of the model. This will give the relationship between each term and the response you are considering. Kindly send a mail for more on DOE
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