Assessing an interaction term

Question

Suppose we have a variable $x$ with 3 levels (e.g. $x_1, x_2$ and $x_3$). We want to see if there is an interaction between $x$ and $y$ where $y$ is a continuous variable. To test this would we include the following terms: $x_{1}y, x_{2}y, x_{3}y$ in a regression model?

Answer 1

First, I want to be clear that $y$ is not your response variable, right? I'll call your response variable $z$. Now, you have a continuous covairate, $y$, and a factor, $x$, with three levels. How you want to do this depends, in part, on the coding scheme you use to indicate the $k$ levels of your factor. I will review methods based on reference cell coding (also called 'dummy coding'). In this scheme, you pick a default level of your factor (I'll arbitrarily pick $x_1$). Then you form $k-1$ new, categorical variables to represent the remaining levels of your factor. For each of these new variables, each observation gets a 1, if it is associated with that level, and a 0 otherwise. Now, to form interaction terms, you will create $k-1$ new variables by computing the products of those dummies with your continuous covariate $y$. Thus, in your case the first few rows of the data might look like:

 z     y     x2     x3     x2y     x3y  
6.7   3.4    0      0       0       0  
7.3   2.7    1      0     2.7       0  
5.8   4.4    0      1       0     4.4

And your model would be:

$$ z=\beta_0+\beta_1y+\beta_2x_2+\beta_3x_3+\beta_4x_2y+\beta_5x_3y $$

Within this scheme:

• $\beta_0$ is the level of $z$ for those observations in the $x_1$ level of your factor when $y=0$
• $\beta_1$ is the slope of the relationship between $y$ & $z$ for observations in the $x_1$ level of your factor
• $\beta_2$ is the intercept for level $x_2$
• $\beta_3$ is the intercept for level $x_3$
• $\beta_4$ is the slope for observations in the $x_2$ level
• $\beta_5$ is the slope for observations in the $x_3$ level

To test these effects for 'significance', you first enter all of the factor level dummies into the model together and perform a simultaneous test that all effects are equal to zero, then repeat this procedure by entering all of the interaction terms together.