I have a question regarding interactions in GLM. I run a Poisson regression with the purpose of predicting claims in insurance.
I have the following problem :
model 1 : CLAIMS ~ X1 + X2 model 2 : CLAIMS ~ X1 + X2 + X1:X2
X1 has 10 levels, X2 has 4 levels
When I do a deviance test (Chi Square), my result is that model 2 is significantly better compared to model 1 (at the 5% level).
On the other side the AIC of model 2 is worse. (higher)
When I look at the individual parameter estimates and their corresponding standard error, I see that only 3 out of 27 levels from X1:X2 differ significantly from the base level.
I am not sure how to interpret these observations. Can someone help me and say something meaningful about it? And more important, how to deal with it?
My attempt would be: "There are interactions in the model that are meaningful, however most of them are not. This explains the increasing AIC, although the deviance test says that model 2 is the better one." Should I try to construct a few dummy variables for these three interactions that seem significant? (However, I am not sure this is a correct approach, because it is only significant relative to the base level...)