how does adding a interaction of categorical variable influence the number of betas

Question

I have a multiple linear regression model:

$$y= \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 + \beta_4 x_4 + \beta_5 x_5$$

where $x_1$ is a continuous variable and $x_2$, $x_3$, $x_4$ and $x_5$ are all dummy variables come from a 5 levels categorical variable. Now I found out the continuous variable and the categorical variable are dependent, so I have to add the interaction term into my model.

In this case, how many terms do I have to add? In other words, how many betas do I have to add to my model?

I don't think it is simply 4 or 5 betas, will it be something like 15 by combination?

Answer 1

A way to think about this is suppose you had a quadratic effect of a continuous predictor, requiring 2 coefficients. Interacting that predictor with the 5-level categorical variable means that you have 5 quadratic curves so this is 10 parameters. Rephrasing the model in the usual way you'd have 2 main effect parameters for the quadratic relationship for the reference category and 8 interaction terms.

Answer 2

Think about what the interaction would mean: it says that the slope of the continuous variable ($x_1$) depends on the level of the categorical variable (although you can interpret the model also as the mean-differences between the levels of the categorical variable changing according to $x_1$). As you have 5 "groups", you need only 4 interaction terms. Your regression then would look like

$y = b_0 + b_1x_1 + b_2x_2 + b_3x_3 + b_4x_4+b_5x_5 + b_{12}x_1x_2 + b_{13}x_1x_3 + b_{14}x_1x_4 + b_{15}x_1x_5 + \epsilon.$

Now, for "group" 1 (all dummies are equal to zero) you have

$E[y|x_1, x_2=1,x_3=0,x_4=0,x_5=0] = b_0 + b_1 x_1$

for "group" 2 (all dummies are equal to zero except $x_2$), you have

$E[y|x_1, x_2=0,x_3=1,x_4=0,x_5=0]= (b_0+b_2) + (b_1+b_{12})x_1$,

for "group" 3 (all dummies are equal to zero except $x_3$), you have

$E[y|x_1, x_2=0,x_3=1,x_4=0,x_5=0]= (b_0+b_3) + (b_1+b_{13})x_1$,

and so on...