This question/topic came up in a discussion with a colleague and I was looking for some opinions on this:
I am modeling some data using a random effects logistic regression, more precisely a random intercept logistic regression. For the fixed effects I have 9 variables that are of interest and come into consideration. I would like to do some sort of model selection to find the variables that are significant and give the “best” model (main effects only).
My first idea was to use the AIC to compare different models but with 9 variables I was not too exciting to compare 2^9=512 different models (keyword: data dredging).
I discussed this with a colleague and he told me that he remembered reading about using stepwise (or forward) model selection with GLMMs. But instead of using a p-value (e.g. based on a likelihood ratio test for GLMMs), one should use the AIC as entry/exit criterion.
I found this idea very interesting, but I did not find any references that further discussed this and my colleague did not remember where he read it. Many books suggest using the AIC to compare models but I did not find any discussion about using this together with a stepwise or forward model selection procedure.
So I have basically two questions:
Is there anything wrong with using the AIC in a stepwise model selection procedure as entry/exit criterion? If yes, what would be the alternative?
Do you have some references that discuss the above procedure that (also as reference for a final report?