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I'm trying to model 3 level DOE data using 4 factors. Each factor when compared to the other factors looks like:

multivariate scatterplot cell of 4 factors

Since I have a center point and 3 points in a row for each factor, I think I can use 2-factor interaction terms in my model. But could I also use squared terms if I see curvature in either the leverage plots or the predicted-vs-actual plot as long as the VIF's stay low? When I do add these terms, they often come out as significant and improve the R^2 adjusted of the model, but I am worried this is just overfitting. If adding squared terms to these models isn't valid, what diagnostics could I use to convince myself?

What about for data which looks like this?:

enter image description here

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The idea of adding centre points is to be able to assess if you need include quadratic terms. If you have enough degrees of freedom going spare you can estimate the coefficients - but it's more usual in this case to consider the experiment as exploratory & carry out further experiments (e.g. a central composite design) to estimate quadratic coefficients more accurately (& to check there's no need for cubic terms).

To address your question more directly, model selection techniques such as AIC or BIC apply just as much in a designed experiment as observational studies.

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