# 2-way interaction with only one main effect in the model - is it a "real" interaction?

I'm estimating a GLM with a continuous variable $X_1$, a dummy variable $X_2$ and the interaction term $X_1*X_2$.

After my understanding, when interpreting the coefficient of $X_1$ you should assume the value of zero for $X_2$, i.e. also for the interaction term.

Now a colleague suggested that if I skipped one of the main effects, then the interaction would not be a real interaction, i.e. I could simply interpret the coefficients of both - the remaining main effect and interaction term as if there was no interaction term in the model. I could not find any literature that supported this view. Is it correct?

• So, the model is $g(\mu) = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_1 X_2$? I think the reasonable thing to do would be to interpret the effect of $X_1$ conditional on $X_2$, i.e. look at $\beta_1$ for $X_2 = 0$ and $\beta_1 + \beta_3$ for $X_2 = 1$. Essentially, interpret the regression separately for $X_2 = 0$ and $1$.
– guy
Aug 13 '12 at 4:59
• @guy, that's basically the answer here. Would you care to re-post you comment as an answer (possibly w/ a little elaboration)? Aug 13 '12 at 16:12

So, the model is $g(\mu) = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_1 X_2$. I think the natural thing to do would be to interpret the effect of $X_1$ conditional on $X_2$, i.e. look at $\beta_1$ for $X_2 = 0$ and $\beta_1 + \beta_3$ for $X_2 = 1$. This is tantamount to fitting a separate regression for $X_2 = 0$ and $X_2 = 1$; we have $g(\mu) = \beta_0 + \beta_1 X_1$ and $g(\mu) = (\beta_0 + \beta_2) + (\beta_1 + \beta_3) X_1$ respectively.
I would say $X_1 X_2$ is as pure an interaction term as there is, given that you are using a function linear in $X_1$. If $\beta_3 = 0$, this says that the effect of a unit increase in $X_1$ is the same regardless of whether $X_2 = 1$ or $X_2 = 0$. Things get hairier as you start adding more predictors since the number of potential interaction terms grows exponentially; without some simplifying assumptions (e.g. a limit on the order and nature of the interactions) things can get unmanageable from a meaningful interpretation standpoint pretty fast.
• That would be an odd thing to do I think. Usually one wouldn't remove a main effect while retaining an interaction term without having a good reason. Given that you want to do it that way, though, you can still interpret things by stratifying on $X_2 = 0$ and $X_2 = 1$. The coefficient on $X_1 X_2$ still represents the interaction between $X_1X_2$, but you are imposing the restriction that $g(\mu) = \beta_0$ when $X_1 = 0$, regardless of the value of $X_2$; that is, you are restricting the intercept of the two regressions to be the same. I guess one might call this a "shared intercept" model.
• If $X_2$ "had no relevant effect" then it ought not be in the model at all. $X_2$ does have a relevant effect if it is changing the influence of $X_1$. I'm really not sure why you want to do this. Even if $\beta_2$ is insignificant, if the interaction is significant it is standard practice to leave $\beta_2$ in. It only makes sense to me to drop $\beta_2$ if you have subject-matter knowledge that $X_2$ should not influence $g(\mu)$ at the value $X_1 = 0$, or maybe that $X_2$ only influences the outcome by magnifying/shrinking the effect of $X_1$, but even then I think you only stand to lose.