9.2 Interactions

We touched on the concept of an interaction in the gas consumption model given in Example 6.4.6. In this example, the relationship between gas consumption and outside temperature was altered by the installation of cavity wall insulation. This is an interaction between a factor (insulated or not) and a covariate (temperature).

Suppose that the response variable is $Y_i$ and there are two explanatory variables $x_{i,1}$ and $x_{i,2}$. We could either

1. 1

   Model the main effects only,

   $𝔼[Y_i]={\beta }_1+{\beta }_2{x}_{i,1}+{\beta }_3{x}_{i,2}$

2. 2

   Or include an interaction as well

   $𝔼[Y_i]={\beta }_1+{\beta }_2{x}_{i,1}+{\beta }_3{x}_{i,2}+{\beta }_4{x}_{i,1}{x}_{i,2}$

Note the interaction term is sometimes written as $x_{i,1}\times x_{i,2}$.