So far in the definition of the multiple regression model, it is declared (implicitly) that a change in  results in a constant change in , while keeping fixed the values for the rest of the predictor variables. But of course, this is not necessarily true. It could happen that changes in  affects , which is modulated by changes in _. A classic example of this behavior is the interaction between drugs. For example, increasing the dose of drug A results in a positive effect on a patient. This is true in the absence of drug B (or for a low dose of B) while the effect of A is negative (even lethal) for increasing doses of B.

In all of the examples we have seen so far, the dependent variables contribute additively to the predicted variable. We just add variables (each one multiplied by a coefficient). If we wish to capture effects, like in the drug example, we need to include terms in our model that are not additive. One common option is to multiply variables, for example:

Notice that the  coefficient is multiplying a third variable that is the product of  and . This non-additive term is an example of what is know in statistics as interaction. There are other ways to introduce interactions, but we are going to restrict the discussion to the multiplicative case, as it is the most common expression for interactions.

Interpreting linear models with interactions is not as easy as interpreting linear models without them. Let's rewrite the expression 3.21:

This shows us the following:

• The interaction term can be understood as a linear model. Thus, the expression for the mean, , is a linear model with a linear model inside of it!
• The interaction is symmetric; we can think of it as the slope of  as a function of  and at the same time as the slope of  as a function of .
• In a multiple regression model without interactions, we get a hyperplane, that is, a flat hypersurface. An interaction term introduces a curvature in such a hypersurface. This is because slopes are not constant anymore but functions of another variable.
• The coefficient  describes the influence of predictor  only at . This is true because for that value , and then the slope of  reduces to . By symmetry, the same reasoning can be applied to .