Vartanian: Data Analysis 541
Interaction vs. Separate Models
If you were to run separate models for two groups or one model
with an interaction term, you will be getting essentially the
same information for your parameter estimates.
For example, examine the following three regression models:
Full model -- All Observations
Variable
B
Intercept
Black
X
XBlack
4.333333
1.49359
-0.33333
0.006410
Where XBlack is the interaction between the variables Black and
X.
We could also examine the separate models, which divides the
observations into White and Black observations. Since there is
no race variable in these models, there will be no interaction
variable.
White Model -- Only White Observations
Variable
B
Intercept
X
4.333333
-0.33333
Black Model -- Only Black Observations
Intercept
X
5.826923
-0.326923
If we go back and examine the full, interactive model (with the
full set of observations) we would determine the following
estimates for the intercept and the slope for each of the groups:
Whites:
Y=4.33333 + 1.49359 (0) - 0.33333X1 + 0.006410 (0) or
Y=4.33333 - 0.3333X1
Blacks:
Y= 4.33333 + 1.49359 (1) - 0.3333X1
+ 0.006410 (1) or
Y= (4.33333 + 1.49359) + (-0.33333 + 0.006410)X1 or
Y = 5.826923 - 0.326923 X1
These are the same values we get when we run the regressions
separately. However, running the models separately will help you
interpret coefficient estimates, especially when you have three
or four way interaction terms. The Chow Test will also tell you
whether the entire models are significantly different from one
another for the different groups. The interaction terms within a
single model will tell you whether or not particular variables
within the models are significantly different from one another
for the different groups.