Interpretation of coefficients in a linear regression model with two binary predictors and their interaction

Question

This is the output from a linear regression model that is trying to predict math scores based on the student's gender and whether or not they took a test preparation course. It is just a toy example for me to understand how to interpret the coefficients.

> str(StudentsPerformance)
'data.frame':   1000 obs. of  8 variables:
 $ gender                     : chr  "female" "female" "female" "male" ...
 $ test.preparation.course    : chr  "none" "completed" "none" "none" ...
 $ math.score                 : int  72 69 90 47 76 71 88 40 64 38 ...

The model includes an interaction term between gender and test preparation course.

lm(formula = math.score ~ 1 + gender * test.preparation.course, data = StudentsPerformance)

Residuals:
    Min      1Q  Median      3Q     Max 
-61.671  -9.671   0.329   9.804  38.329 

Coefficients:
                                       Estimate Std. Error t value Pr(>|t|)    
(Intercept)                             67.1957     1.0857  61.889  < 2e-16 ***
gendermale                               5.1434     1.5574   3.303 0.000992 ***
test.preparation.coursenone             -5.5250     1.3521  -4.086 4.74e-05 ***
gendermale:test.preparation.coursenone   3.1258     1.9440  -0.065 0.050426 . 

I made the following statements :

• the intercept is the average math score for female students who completed the test preparation course
• males have a math score that is 5.1434 points higher than females between those who completed the course
• females who didn't take a test preparation course have a math score that is 5.5250 points lower than females who took the course
• the effect of a test preparation course is different for male and females. Male students who did not take the test preparation course have a math score 3.1258 points higher than female students who took the test preparation course

is that a correct interpretation?

Answer 1

Personally, I like to draw a table for these kinds of situations (see this post). Your model is:

$$ \operatorname{E}(\mathrm{mathscore}|\mathrm{gender}, \mathrm{course}) = \beta_{0} + \beta_{1}\mathrm{gender_{m}} + \beta_{2}\mathrm{course_{none}} + \beta_{3}\underbrace{\mathrm{gender_{m}}\times\mathrm{course_{none}}}_{\text{= interaction term}} $$

Tabulating all possible combinations with their corresponding coefficients, we have:

$$ \begin{array}{l|l|l} & \text{Gender = m} & \text{Gender= f} & \text{Difference} \\ \hline \text{Course = completed} & \beta_{0} + \beta_{1} & \beta_{0} & \beta_{1} \\ \text{Course = none} & \beta_{0} + \beta_{1} + \beta_{2} + \beta_{3} & \beta_{0} + \beta_{2} & \beta_{1} + \beta_{3} \\ \hline \text{Difference} & \beta_{2} + \beta_{3} & \beta_{2} & \beta_{3} \end{array} $$

Let's summarize the interpretation:

• $\beta_{0}$ is the estimated average math score for a female with a completed course.
• $\beta_{1}$ is the difference between males and females that completed the course.
• $\beta_{2}$ is the difference between females who completed the course and those who didn't.
• $\beta_{3}$ is the additional difference between the average math scores between males and females when we change from course completed to none. To make it specific: The difference between males and females who did complete the course is $\beta_{1} = 5.14$ but the difference between males and females who did not complete the course is $\beta_{1} + \beta_{3} = 5.14 + 3.13 = 8.27$.

Here is a schematic plot that hopefully aids the interpretation (note that the depicted effects do not correspond to the values of your coefficient):