# Interpreting multiple regression coefficients with 2 continuous variables interacting and 2 categorical variables interacting

I find it challenging to interpret interaction effects in OLS multiple regression where the interactions are between two categorical variables and between two continuous variables.

Say for instance that the interaction between sexfemale: medmobility is -10.1, where sexfemale is a categorical variable with male as the reference group and medmobility is a categorical variable where low mobility is the reference group. What is the meaning of that coefficient, -10.1?

Likewise, what if you have an interaction between two continuous variables, say for instance between weight and IQ (weight:IQ) If the coefficient is, for example, 1.3 what does that mean? How does the interpretation change if both have been centered?

What about the main effects of these variables?

• The basic answer is that you don't interpret main effects or interaction effects when the interaction is 'significant', you look at / interpret simple effects instead. We could possibly be more specific if you could state what your study, your response variable & the full regression equation is. Dec 9, 2012 at 17:37

## 1 Answer

As @gung said, it would help if you gave your full equation and DV, but, here, if the interaction between sex-female and mobility is -10.1, it means that the effect of high mobility on the dependent variable is 10.1 units less for women then men. Similarly, the effect of being female on the DV is 10.1 units less for high mobility people than for low.

For continuous variables, it is much the same, except that it is per unit of the other IV. So, 1.3 for the interaction between weight and IQ means that the effect of IQ on the DV is 1.3 units higher for each increase of one unit (pound? kilogram?) of weight, and the effect of weight is 1.3 units higher for each increase of 1 point in IQ. In other words, the effect is more positive for people who are both smart and heavy than for people who are one or the other.

• Thanks this was helpful. The scenario is just hypothetical. For the categorical one you mean the effect of medium mobility (not high mobility right?) Also, what if a continuous variable such as IQ is also in a different interaction with another continuous variable such as weight centered at average. Does this factor in to what 1.3 means? (ie does it only hold for people of average weight? or does it not matter?) Dec 9, 2012 at 19:01
• I don't know how you coded mobility. It's for wherever that variable = 1. A second interaction does not affect the meaning of the first. Dec 9, 2012 at 19:25