# Interaction term usage

In which case should I use an interaction term in regression analysis? What does it depend on that I should be using it?

When you interpret coefficients in regression analysis, you look at the effect of one unit increase in a predictor that is associated with the coefficient on the response variable, while holding other variables constant. However, this ignores the scenario in which predictors have an effect on each other.

For instance, in a multiple linear regression setting, we have a model:

$$\text{Pollution Level in a city} = \beta_0 + \beta_1 \text{Population} + \beta_2 \text{Number of Cars}$$

Normally, if the population increases by 1 (million or thousand), pollution level will be increased by $\beta_1$, while we hold the number of cars as a constant. However,this is inappropriate because with more people, we would expect the increasing purchase of vehicles. Therefore, if we add a interaction term, we will have:

$$\text{Pollution Level in a city} = \beta_0 + \beta_1 \text{Population} + \beta_2\text{Number of Cars} +$$$$\beta_3 \text{Number of Cars} \times \text{Population}$$

$$=\beta_0 + \beta_1 \text{Population} + (\beta_2 + \beta_3 Population) \text{Number of Cars}$$

Thus, the increased population will also have an impact on the coefficient associated with the number of cars, which allows us to model the pollution level non-linearly. So if you see a non-linear pattern in the residual plot and you suspect that changes in one predictor would have effect on another, you probably should try to include an interaction term.

• Thank you Jack Shi for the answer. It is quite helpful. As you have mentioned here that if I see a non-linear pattern in the residual plot then I should consider adding an interaction term. Is there any other case scenario where I should consider adding an interaction term? – Rahul Wadhwani Oct 21 '15 at 8:32
• The main reason is that one is substantively interested in that interaction effect, e.g. has the effect of parental background on offspring's educational attainment increased or decreased over time. – Maarten Buis Oct 21 '15 at 8:50
• @RahulWadhwani Non-linearality in multiple linear regression can be caused by many factors. A transformation of a predictor or several predictors may also be a simple, valid solution. Or the relationship between predictors and response variable is not linear to begin with, in which case you might want to try polynomial regression, etc. If you feel like there might be a relation between two predictors in your model, an interaction term may be worth a try. Also, you should notice that including an interaction term may also introduce collinearity in the model. – Jack Shi Oct 21 '15 at 16:05