# Interaction (in linear regression) ignores one of the discrete variable's levels

I am fitting a dataframe with the linear regression model that includes interaction:

lm0 <- lm(Mean_RT ~ POS*Length, data = df)


The output of summary() function is:

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)   582.040     16.701  34.851   <2e-16 ***
POSVB         -55.523     33.810  -1.642    0.101
Length         25.064      1.958  12.803   <2e-16 ***
POSVB:Length    4.147      4.131   1.004    0.316


I don't understand why it just uses POSVB and not POSNN (and POSNN:Length) in the coefficients?

The dataframe:

It is accounting for POSNN. Consider the equation that you (i.e. R) constructed:

$$y_i = \beta_0 + \text{POS}_i\times \beta_1 + \text{Length}_i\times \beta_2 + \text{POS}_i\times \text{Length}_i\times \beta_3$$

Where $$\text{POS}_i$$ may take values 0 (NN) and 1 (VB) by the way R converts factors into dummy-coded binary predictors. Length, presumably, can take some positive real-valued number.

Should you have a NN item, then the equation simplifies to $$y_i = \beta_0 + \text{Length}\times \beta_2$$. Therefore, we have the effect of NN (just the intercept) in interaction with $$\text{Length}$$. You need estimated coefficients $$\beta_0$$ and $$\beta_2$$. There's no $$\text{POS}_i\times \text{Length}_i$$ for $$\text{POS}_i=0$$ because this would just be 0.

Should you have an VB item, then you need estimated coefficients $$\beta_0$$, $$\beta_1$$, $$\beta_2$$, and $$\beta_3$$ (since $$\text{POS}_i$$ will be 1).

• Thanks! Makes sense. – user3639557 Feb 21 at 22:34
• Great! I'm pretty sure there are many similar questions on dummy-coded variables in regressions on this SE, consider searching if you'd like other answers. – Alex L Feb 21 at 22:39
• what would happen if the factor could take three values? I assume it is no longer binary coded? Basically imagine $POS_i$ could take another value like DD. What would be the corresponding $y_i$ for DD? Assuming NN is still 0, and VB is 1. – user3639557 Feb 21 at 23:03
• @user3639557 good question, simple answer. There are a lot of resources on regression, and almost any stats textbook will answer your questions. Here's one arbitrary (but seemingly decent) resource I found. – Alex L Feb 22 at 0:58