Interpreting Interaction Coefficients within Multiple Linear Regression Model - r

Question

I am struggling with the interpretation of the coefficients within interaction models.
I am looking at the outcome of an interaction model of 2 binary (dummy variables). I was just wondering how I interpret the:
- Intercept (is everything at 0)?
- The slope coefficients?
- The interaction coefficients?
In standard multiple linear regression, we talk about the change in y when we have a 1-unit change in x, holding everything else constnat. How do we interpret this in interactions? Especially since both my variables are dummy?
Hope this makes sense and thanks very much in advance.

Answer 1

How do we interpret this in interactions?
The meaning of the regression coefficients in models having interaction do not remain the same as in the case of simple linear regression without interaction simply because of the added interaction term/terms.
The regression coefficients no longer indicate the change in the mean response with a unit increase of the predictor variable, with the other predictor variable held constant at any given level. This interpretation is only valid after accounting for the dependence on the level of the other predictor variable.
Ex:
A SLRM with interaction terms:
E(Y) = B0 + B1X1 + B2X2 + B3X1X2
Interpretation:
It can be shown that the change in the mean response with a unit increase in X1 when X2 is held constant is:
B1 + B3X2
And, the change in the mean response with a unit increase in X2 when X1 is held
constant is:
B2 + B3X1
I was just wondering how I interpret the: - Intercept (is everything at 0)?
The intercept is the prediction from the regression model when all the predictors are at level zero.
The slope coefficients?
In case of no interaction coefficients.
E(Y) = B0 + B1X1 + B2X2
The coefficients B1, B2 indicate, respectively, how much higher (lower) the response functions for dummies X1, X2 are than the one for, both dummies zero.
Thus, B1 and B2 measure the differential effects of the dummy variables on the height of the response function i.e. E(Y).
You can inspect that only the slope changes:
When X1 = 1 and X2 = 0.
E(Y) = B0 + B1
and, when X1 = 0 and X2 + 1.
E(Y) = B0 + B2
The interaction coefficients?
By interaction coefficients, I understand the regression coefficients for model with interaction.
The model:
E(Y) = B0 + B1X1 + B2X2 + B3X1X2
When both X1 and X2 are 1, then the model becomes:
E(Y) = B0 + B1 + B2 + B3.
Which translates to an increase or decrease in the height of the response function.
You can create a more interesting example with a third continuous predictor and explore the interaction relationship of the continuous variable with the dummies, in which case the slope of the regression would also change instead of only the intercept. And, hence the interpretation that how much higher (lower) one response function is than the other for any given level of X1 and X2 would not be valid as the slope also would have changed and thus the effect of the dummy predictor also would have been more evident.
When interaction effects are present, the effect of the qualitative predictor (dummy variable) can be studied by comparing the regression functions within the scope of the model for the different classes of the dummy variable.
Reference: Kutner et. al. Applied Linear Statistical Models