Interpreting -Regress- Coefficients with Interaction Terms

[Elliot Smith]

Dear all,

I posted this questions in a different thread, however as I got not answer and I still find myself with troubles interpreting this, I wanted to create my own post, to see if I find some clarification. So I saw in somebodys work a model which had the structure:

y = b1*x1 + b2*x2 + b3*x3 + b4*(x1*x2) + b5*(x1*x3)

In particular, it explored gender differences (x1=gender) in performance results in a exam (y = performance), conditional on number of days of preparation to the exam (x2= number of days) and subject type (x3 = stem/non-stem). Hence, the model was something like:

Model a: performance = b1*female + b2*STEM + b3* days + b4(female*STEM) + b5(female*days)

As I understand from manuals, and also following other posts in this forum, then each coefficient can be interpreted as (I am bulleting my questions for easiness of answering):

• b1 = Female gap in performance in non-STEM subjects and 0 days preparation (or average days preparation when centered)?
• b2 = STEM gap for in performance for MALE students (ceteris parbus days of preparation) ?
• b3 = Effect of days preparation in performance for MALE students (ceteris parbus STEM vs non-STEM subjects) ?
• b4 = Difference in the gap in STEM for female students vs male (hence, FEMALE gap in STEM would be b4 + b2) ceteris parbus preparation days ?
• b5 = Difference in the effect of days for female vs male (hence, FEMALE slope would be b3 + b5) ceteris parbus subject type ?

Are these interpretations correct?

Additionally, an additional doubt emerges that is causing me a lot of confusion when looking at this model. So imagine I would only have this model instead:

Model b: performance = b6*female + b7*STEM + b8(female*STEM)

• As I understand,in model b, b8 could be interpreted not only as the "Difference in the gap in STEM for female students vs male" (as I interpreted for the model above), but it could also be interpreted as the difference in the gender gap in STEM vs non-STEM, so that the gender gap in STEM subjects would be b6 + b8. Is this correct?
• If this is so, going back to model A, how does this apply to the interpretation I made of b3? Would also reflect the gender gap of STEM vs non-STEM ceteris parbus days preparation? Except, if I am correct, it could not be estimated by adding it to b1, as b1 would describe the gap when there are 0 days of preparation, so they are not really comparable.

Thanks in advance for anybodys time spent responding to my questions. Best regards,

Elliot

[Clyde Schechter]

The foolproof way to interpret interaction models is to do the algebra yourself. The model is, as you note:

performance = b0 + b1*female + b2*STEM + b3* days + b4(female*STEM) + b5(female*days)-------------(you left out the constant term)

Consider the gender-gap in non-STEM subjects with no preparation. So for males, female = 0, STEM = 0 and days = 0.
performance = b0 + b1*0 + b2*0 + b3*0 + b4*0*0 + b5*0*0 = b0.
And for females, female = 1, STEM = 0 and days = 0. So
performance = b0 + b1*1 + b2*0 + b3*0 + b4*1*0 + b5*1*0 = b0 + b1.
So the gender gap is the difference: b0 + b1 - b0 = b1.

If you want the gender gap for STEM with 0 days preparation:
male performance = b0 + b1*0 + b2*1 + b3*0 + b4*0*1 + b5*0*0 = b0 + b2
female performance = b0 + b1*1 + b2*1 + b3*0 + b4*1*1 + b5*1*0 = b0 + b1 + b2 +b4
So the gender gap is the difference: b0 + b1 + b2 + b4 - (b0 + b2) = b1 + b4

And so on.

It is, frankly, much harder, at least for me, and I think for most people, to start from the perspective of assigning word interpretations to coefficients: words can be slippery, and particularly when a model has multiple interaction terms it gets very hard to come up with phrases that are simple enough to use and still accurately describe what the term does. So just do the algebra: it's much easier that way.

Added: After you have done this enough times, it becomes second nature and you can start doing the algebra quickly in your head.

[Carlo Lazzaro]

Elliot:
welcome to this forum.
As an aside to Clyde's enlightening advice, I usually try to calculate the fitted values myself using coefficients and variables and then compare my results with the ones from -predict,xb-.
I'm still learning a lot from this exercise, especially when interactions are included in the right-hand side of the regression equation.