Linear Regression – Interpreting Lower Order Coefficients When the Model Contains an Interaction

A Linear Regression Model with an interaction between two predictors (X1 and X2) has the form:

Y = B0 + B1X1 + B2X2 + B3X1*X2.

It doesn’t really matter if X1 and X2 are categorical or continuous, but let’s assume they are continuous for simplicity. One important concept is that B1 and B2 are not main effects, the way they would be if there were no interaction term. Rather, they are conditional effects. A main effect means that B1 is the effect of X1–the difference in the mean of Y for each one unit change in X1. But B1 is the effect of X1 conditional on X2 = 0. For all values of X2 other than zero, the effect of X1 is B1 + B3X2.

The biggest practical implication is that when you add an interaction term to a model, B1 and B2 change drastically by definition (even if B3 is not significant) because B1 and B2 are measuring a different effect than they were in a model without the interaction term. So don’t be upset if B1 suddenly isn’t signficant. It’s measuring something else altogether.

So B1, in the presence of an interaction, is the effect of X1 only when X2 = 0.

If X2 never equals 0 in the data set, then B1 has no meaning. None.

This is a good reason to center X2. If X2 is centered at its mean, then B1 is the effect of X1 when X2 is at its mean. Much more interpretable. Even better is to center X2 at some meaningful value, even if it’s not its mean. For example, if X2 is Age of children, perhaps the sample mean is 6.2 years. But 5 is the age when most children begin school, so centering Age at 5 might be more meaningful, depending on the topic being studied.

If X2 is categorical, the same approach applies, but with a different implication. If X2 is dummy coded 0/1, B1 is the effect of X1 only for the reference group. The effect of X1 for the comparison group is B1 + B3. To see why, plug in 0 for X2 for the reference group and write out the regression equation. Then plug in 1 for X2 for the comparison group. Do the algebra.

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