# Interpretation of two dummy variables in interaction with addition of new variables

I have a mixed effects model with an interaction between two dummy variables. Y variable is continuous. My questions:

1. Does the interpretation of the variables included in the interaction (and the interaction itself) change with the addition of variables to the model which are NOT included in the interaction? For example: in model 1 below:

• Model 1: Y= intercept + dummy3 + dummy1 + dummy2 + dummy1*dummy2

I interpret the variables as follows:

• dummy1 = the change in Y moving from 0 to 1 on dummy1 when dummy2= 0

• dummy2 = the change in Y moving from 0 to 1 on dummy2 when dummy1= 0

• dummy1*dummy2 = difference between dummy1=0 and dummy1=1 in change in Y when move from dummy2=0 to dummy2=1

In model 2 below, new variables have been added (which are not part of the interaction)-specifically, a continuous variable, its squared term, and its cubed term. is the general interpretation of dummy1, dummy2, and dummy1*dummy2 the same in Models 1 and 2?

Model 2: Y= intercept + cont1 + cont1^2 + cont1^3 + dummy3 + dummy1 + dummy2 + dummy1*dummy2

1. For dummy3 in model 2 (not part of an interaction), is its interpretation influenced by the interaction? Or is it just interpreted as a standard dummy? (i.e., as the change in the Y variable that occurs when moving from dummy3=0 to dummy3=1).

Thank you!