interpretation of interaction terms

[Michael Lee]

When we have interaction terms in a model, how we can interpret the coefficient of the main variables' coefficient:

Y=a0+a1*x1+a2*x2+a3*x1*x2

what is the interpretation of a1 and a2 and also a0?
is this interpretation right? : a1 is the effect of x1 on y when x2=0.
and if yes, what is the source for citing this fact?

Best Regards,
Michael

[Michael Lee]

And also when the model is similar to this:

Y=a0+a1*x1+a2*x2+a3*x3+a4*x1*x2+a5*x1*x3

what are interpretation of a1, a2, a3,a4 and a5.

[Richard Williams]

This handout might be helpful:

https://www3.nd.edu/~rwilliam/stats2/l53.pdf

The critical thing to understand is that the interpretation of main effects and significance tests changes once you add interactions.

[Michael Lee]

Thank your Richard.
What is your idea about post #2.

[Michael Lee]

And also when the model is similar to this:

Y=a0+a1*x1+a2*x2+a3*x3+a4*x1*x2+a5*x1*x3

what are interpretation of a1, a2, a3,a4 and a5.

Any idea?

[Maxence Morlet]

You're allowing x1 to have a different relationship with the dependent variable as x2 changes (and x3).

[Maarten Buis]

Rich's Handout seems clear enough. Maybe just read it again more carefully. If you still don't understand, then you need to tell us what it is about the handout that you did not understand. Otherwise, our explanation will be very similar to Rich's, so is unlikely to help.

[Joro Kolev]

These questions appear with some regularity here, so OP can also search a bit Statalist.

The way how I interpret parameters in linear in the parameters, nonlinear in the variables models is by taking derivatives.

E.g., in the model E(Y|x)=a0+a1*x1+a2*x2+a3*x3+a4*x1*x2+a5*x1*x3

d(EY)/d(x1) = a1 + a4*x2 + a5*x3, and this is your interpretation; you can evaluate the derivative at values of interest of the x2 and x3.

Or alternatively you can reparametrise the model by demeaning x2 and x3, and in this reparametrised model the interpretation of a1 would be d(EY)/d(x1) evaluated at the means of x2 and x3.