# Curvilinear relationship with moderator

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?

### References

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.

Look at the slopes of the curves:

1. $\partial Y/\partial X=b_1+2b_2X$
2. $\partial Y/\partial X=b_1+b_4M+2b_2X$
3. $\partial Y/\partial X=b_1+b_4M+2(b_2+b_5M)X$

and the convexities:

1. $\partial^2 Y/\partial X^2=2b_2$
2. $\partial^2 Y/\partial X^2=2b_2$
3. $\partial^2 Y/\partial X^2=2(b_2+b_5M)$

So, which model to choose depends on whether you believe that the slopes and convexities of your model curves should follow 1),2) or 3). For instance, only in model 3 the convexity depends on the moderator variable.

The main reason to choose one or another form is understanding the underlying phenomenon. The second reason is fit. If you data fits better model 2, then you should consider it over others as long as your theory supports the relationship.

• I don't think it is a good idea to call the second derivatives the "curvature", since that word is usually used for a quite different concept (the reciprocal of the radius of the osculating circle). – Silverfish Dec 19 '14 at 0:34
• I think that numbers 2 and 3 have a typo unless I am misinterpreting what you mean by "curvature"? – Silverfish Dec 19 '14 at 0:36
• @Silverfish, I'll use a word convexity, and you're right about typos. Fixed. – Aksakal Dec 19 '14 at 3:47