• Know what an interaction term is and use a scatterplot to understand the interaction effects
• Write appropriate multiple linear regression model equations for categorical predictors and interaction terms

Interaction Effects on Equations

A linear regression model with two or more explanatory variables is called a multiple linear regression model. Since there is more than one explanatory variable, the model is no longer a line. In fact, we can include [latex]p[/latex] explanatory variables in our model. The equation for the estimated model that uses [latex]p[/latex] variables is

[latex]\hat{y} = a + b_1 \cdot x_1 + b_2 \cdot x_2 + ... + b_p \cdot x_p[/latex]

where [latex]b_1, b_2, ... ,b_p[/latex] are the regression coefficients for explanatory variables [latex]x_1, x_2, ... ,x_p[/latex], respectively. In multiple linear regression, [latex]b_1, b_2, ... , b_p[/latex] are called partial slopes.

An interaction occurs when an explanatory variable has a different effect on the response variable, depending on the values of another explanatory variable.

Let’s assume that there are 2 explanatory variables in the model. The full equation would be:

[latex]\hat{y} = a + b_1 \cdot x_1 + b_2 \cdot x_2[/latex]

And here is the same regression equation with an interaction:

[latex]\hat{y} = a + b_1 \cdot x_1 + b_2 \cdot x_2 + b_{12} \cdot x_1x_2[/latex]

where [latex]b_3[/latex] is the regression coefficient and [latex]x_1x_2[/latex] is a two-way interaction.

Note: We are not going into details on how to calculate the regression equation, instead, we focused on the interpretation of the equation.