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My question is conceptual rather than one based on data/findings. I am interested to know whether a situation where the main effects of two variables are linear whereas their interaction is nonlinear makes sense from a statistical standpoint and how it might be analyzed.

Three variables: x - a trait m - a situational variable y - behavior

I expect that the relationship between x and y is linear (positive relationship) and the relationship between m and y is linear (positive relationship). X and M are not causally related.

I further expect that an interaction between x and m on y is nonlinear. Specifically, y will be highest when x is high and m is at a moderate level. Y will be moderate when x is high and m is either high or low. And last, y will be smallest when x is low and m is either high or low.

Can someone help me sort through or suggest some resources, specifically how it might be tested? It seems that most quadratic approaches assume nonlinear main effects in order to test nonlinear interactions. I am sure this is a basic question for many of you but it's giving me fits

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In principle you could include a linear main effect and a non-linear interaction effect. In practice I would recommend against it. The problem is that if your main effect is (moderately) non-linear then that will influence the non-linearity you will find in the interaction effect. So I tend to allow the main effect to be at least as flexible as the interaction effect, even if the results show that it could just be linear. That way I show to the audience that the main effect is actually linear, rather than assume it. That way I can give them more confidence that the non-linearity in the interaction term is actually a non-linear interaction term and not some way regression tries to accomodate a non-linear main effect.

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