# Interpretation of the magnitude of the effect of interaction terms

I have a model with interaction terms: $Y = a + b X_1 + c X_2 + d Z + f X_1 Z + g X_2 Z + error$, where $X_1$ and $X_2$ are (0,1) indicator variables and $Z$ is a continuous variable (with a distribution whose standard error is 0.25).

After an OLS regression, I get that $a$, $c$ and $f$ are the only significant coefficients. Specifically, $f$ is equal to 0.05.

Can I interpret the effect of $X_1$ on $Y$ using the standard deviation of $Z$? Specifically, is it correct to state that, if $X_1 = 1$, a one standard increase in $Z$ (0.25) at the mean increases the value of $Y$ in 0.05 * 0.25 = 0.0125?

At the same time, if $c$ is equal to 2, can I simply conclude (given the significant coefficients) that, for $X_2 = 1$, $Y$ increases in 2?

Interpreting interaction terms is actually quite controversial and the literature is loaded with contradictions. My comments are based on a close reading of Aiken and West's book Multiple Regression which has one of the best methodological breakdowns for dealing with interaction terms that I've ever read. More recent papers (e.g., Understanding Interaction Models: Improving Empirical Analyses by Brambor, Clark and Golder, Political Analysis, 2006) aren't anywhere near as rigorous, contradict solidly documented A&W findings and add to the existing confusion.

First off, introducing an interaction shifts the interpretation from a main effects only model interpreted at the means of the Xs to a model with interactions that are interpreted at zero. This suggests mean centering your variables before taking an interaction term. The coefficients and std devs don't change, but the interpretation does.

Next, the main effects for the X's serve only to adjust the location of the intercept on the y-axis. It doesn't matter for the interaction term whether the X's are 0 or 1.

This is all a long way of saying that your suggested calculations need more information. In other words, it's not enough to assume that Y changes by '2' if the X2 coefficient, 'c,' is 2 since the effect of X2 is a conditional adjustment to the intercept. So, the impact on Y of X2 is not '2' when X2=1, it's 'a + 2'. Similarly for the calculation of the interaction between X1 and Z, it's not about shifting Z by 0.25 SDs. The full equation for estimating Y when X1=1 and Z=0.25 would be:

Ytilda = a + b + (0.5 * 0.25)


which would not equal 0.0125.

• Thanks, Mike. The continuous variable included in the interaction term are mean-centered. I was not explicit on this. Given mean-centering, that is why I made the interpretation in terms of average values. Since b is not significant, the key point of your answer, if I have understood you correctly, is that the intercept must be taken into account to compute the value of Y. Nevertheless, I still have a doubt. What about if we speak about CHANGES in Z? That is, can I say that the effect of a change in Z equal to one stand. dev. (0.25) results in a change of Y equal to 0.5 * 0.25.? – madu May 28 '15 at 15:59
• Madu, given mean centering, Z would be conditionally evaluated at zero. So, plugging in a value of 0.25 would "shift" Z from 0 to 0.25 and the Ytilda equation above would remain the same. – Mike Hunter May 28 '15 at 16:20
• Thanks again, Mike. You have helped me a lot. Just one more question. The derivative of Y in X1 informs on how Y changes in X1. This derivative is equal to b + f. Since b is not significant, it would be equal to f. Therefore, and bearing in mind that Z is mean-centered and f = 0.05, what can be said is that being of type X1 = 1 implies, at the average value of Z, that Y is 0.5 higher than under any other type. Am I right? – madu May 28 '15 at 16:48
• "Since b is not significant..." This is a grey area not covered in the textbooks that I'm aware of. Theoretically, you are correct but bear in mind that your statement concerns the distinction between statistical significance (b is not significant and therefore defined to be zero) and the applied fact that that the parameter b for X1 is included in the model, as is the also not significant parameter for the interaction between X2 and Z. All of these coefficients, significant or not, will adjust the Y predictions in real terms. – Mike Hunter May 28 '15 at 18:45