I am trying to moderate a curve and determine which equation I should follow.
A simple curvilinear relationship has the following equation:
(1) $Y = b_0 + b_1X + b_2X^2$ (i.e. a linear term and a squared term).
We have a moderator ($M$) that we hypothesize changes the strength of the curve. From my reading of several references on moderation (Aiken and West, 1991; Miller et al. 2013) this then leads to the following equation:
(2) $Y = b_0 + b_1X + b_2X^2 + b_3M + b_4XM$.
Thus, the interaction term is between the moderator and the linear term only.
I also see other studies using the following equation:
(3) $Y = b_0 + b_1X + b_2X^2 + b_3M + b_4XM + b_5X^2M$.
So, then both the linear term and the squared term interact with the moderator.
I am curious whether there are reasons to use one or the other. Is there something wrong with following equation (2), or would anybody know when using (2) is warranted?
Aiken LS, West SG. 1991. Multiple Regression: Testing and Interpreting Interactions. Sage: Thousand Oaks.
Miller JW, Stromeyer WR, Schwieterman MA. 2013. Extensions of the Johnson-Neyman technique to linear models with curvilinear effects: Derivations and analytical tools. Multivariate Behavioral Research 48: 267-300.