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I understand that when the factor is quantitative the $AB$ interaction can have a linear component and a quadratic component and so $AB^2$ makes sense there. But when we have qualitative elements how does this translate? The Design and Analysis of Experiments by Montgomery does say that components $AB$ and $AB^2$ have no actual meaning. But why would we use them then?
Can anyone provide me an intuitive answer to this or point to where I could find a better interpretation?
Well, for nominal factors the components have a meaning, but it's not "linear" or "quadratic", or generally of any particular interest, depending as it does on the arbitrary coding used for the qualitative variables. For ordinal factors, if you can consider the levels as more or less evenly spaced in some sense, then you can think of these components as representing more or less linear & quadratic trends in that same sense.
Why use them then?—I had a quick look, & M. says the only reason to bother decomposing interactions of qualitative variables like this is (sometimes) as a step in designing experiments. With quantitative factors, when you're building fractional factorials or blocking, you'd often want to alias higher-order interactions or blocks with a quadratic component of a second-order interaction. You can follow exactly the same approach with qualitative factors (& here you may as well alias the linear effect as the quadratic). These topics are covered a little later on in the same chapter.
