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I want to know how to interpret the quadratic-by-quadratic interaction (e.g., X^2*W^2). I have looked for textbooks or scholarly articles on this issue, but all I found is “Statistics methods and applications” a book written by Pawel Lewicki and Thomas Hill (2012), which just briefly noted that:

“… the interactions by the quadratic components can be interpreted as half the difference between the quadratic main effect of one factor at the respective settings of another; that is, either the high or low setting (quadratic by linear interaction), or the medium or high and low settings combined (quadratic by quadratic interaction).” (p.223)

I wonder if there would be an expert who can help me to understand how to interpret the quadratic-by-quadratic interaction, hopefully with a bit more detailed explanation than Lewicki and Hill (2012) and with an example.

Let’s say we would like to know the effects of providing financial incentives (=X) and interesting tasks (=W) on workers’ happiness (=Y).

Equation: Y = a + b1X + b2X^2 + b3W + b4W^2 +b5XW + b6X^2W + b7XW^2 + b8X^2W^2

In this case, for the quadratic-by-linear interaction terms, I think, depending on their signs, we could say, for example, for b6X^2W, that providing interesting tasks (W) to workers can flatten or steepen the financial incentives―happiness curve.

However, I am unsure how to interpret b8X^2W^2 in this case.

Thank you so much for your help in advance!

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    $\begingroup$ I would choose an interpretation suitable for the context: that is, it would depend on the other terms involving $X$ and $W$ that appear in the model, as well as on the objective, the data, and the audience. If you have a particular model in mind, could you please share that with us? $\endgroup$
    – whuber
    Dec 28, 2022 at 15:25
  • $\begingroup$ Thank you whuber for your suggestion! I've included an example and an equation :-) $\endgroup$
    – user376288
    Dec 28, 2022 at 23:58
  • $\begingroup$ Part of the understanding you seek might reside in a theoretical analysis of multivariate polynomial models, such as the one I gave at stats.stackexchange.com/a/408855/919. $\endgroup$
    – whuber
    Dec 29, 2022 at 2:11

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