As a teacher, with some frequency, I hear the idea from students that when statistically significant interaction terms are present, one does not need to interpret main effects at all, a contention apparently heard by others (Grace-Martin, n.d.), yet which to me appears mathematically and statistically unfounded.
I’m thankful to a StataList post by Kolev (2022) for providing a mathematical foundation to this thinking.
Consider the following equation with an interaction term:
\[y = \beta_0 + \beta_1 x + \beta_2 m + \beta_3 xm + e_i \tag{1}\]
Kolev (2022) uses derivatives to make a point. I’m going to use partial derivatives, because I think they are more accurate.
\[\frac{\partial y}{\partial x} = \beta_1 + \beta_3 m \tag{2}\]
So, using the partial derivative of \(y\) with respect to \(x\), we see that this partial derivative depends both upon the main effect of \(x\), \(\beta_1\), and the effect of \(m\), \(\beta_3 m\).
That main effect, \(\beta_1\), might be 0, but even then, it needs to be interpreted.
Interactions and main effects need to be interpreted together.