Solved – Interpretation of the magnitude of the effect of interaction terms

interactioninterpretation

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?

Thanks in advance.

Best Answer

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.

Related Solutions

Solved – Sign of Estimate (Coefficient) of Interaction Terms

When you discuss the interaction you don't discuss it in terms of what happens to y but what happens to a1 and a2. If it's positive then as a1 increases a2 increases. If it's negative then as a1 increases a2 decreases. Interactions are about differences in effects. The slopes in regression are the effects and so it's most efficiently discussed in those terms.

Once you consider the foregoing you can understand why many say that main effects don't mean anything when there is an interaction. I don't agree with that as a generalization but certainly there are many cases where it's true.

Solved – Interpreting the direction of an interaction effect in Binary Logistic Regression

It looks like you're confused as to what an interaction is in general. In particular, from the bit "suggesting a positive interaction - as HR increases so does PD", it sounds like you believe that the interaction term for a pair of independent variables (IVs) tells you how the IVs are associated with each other. It doesn't. Rather, it tells you how the dependent variable (DV) is associated with the product of the two IVs.

To interpret an interaction, it's often less helpful to look at the coefficients themselves than to look at the effect of interaction in combination with the associated main effects. For example, we can plot the effects of HP, BD, and HP × BD like this (change HP and BP values as appropriate for your actual data):

d = expand.grid(hp = c(-1, 0, 1), bd = c(-1, 1))
d = transform(d, effect = -.0324*hp + 1.4095*bd + 1.1849*hp*bd)
library(ggplot2)
qplot(bd, color = ordered(hp), effect, data = d)

A plot

We see that as HP increases, the effect of BD on the DV also increases.

Best Answer

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Solved – Sign of Estimate (Coefficient) of Interaction Terms

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