This isn't really a statistical problem, more like a curiosity. In any case, comments much appreciated.
I read a paper where they hypothesize the following:
Hypothesis 1. Z moderates the relationship between X and Y. When Z is high, the positive relationship between X and Y is stronger at intermediate levels of X than at lower or higher levels of X; when Z is low, X has a positive linear relationship with Y.
They tested this hypothesis by including in the regression equation of Y the interaction between X^3 and Z. My simple question is, according to this hypothesis, shouldn't they have included an X^2 x Z term instead - because the relationship that they are describing in the hypothesis seems more curvilinear/quadratic than cubic.
Thanks!