# AIC Model Fitting - Multiple Models

I have many predictor variables (100+) and many outcome variables (10+). I am attempting to optimise which predictor variables to use when modelling my outcome variables. I am using R for the analysis and the outcome variables must be modeled separately using the glm function.

My issue is that the GLMs should all use the same predictor variables, and so when I am choosing which variables to use, I have to take all models into consideration and not just one. Since there are so many predictor variables, I am using backward selection for simplicity. If I were optimising a single model, I would minimise the AIC. However, since I have more than one model, I can't minimise the AIC for just one of them as the AIC for the others might be really high.

My question is this: When optimising more than one model, what value should I minimise at each iteration?

I believe I read somewhere that you are to take the sum of the AICs, but I can't find where I read that (sources would be incredibly helpful). I have a niggling feeling however that it might be beneficial to minimise the product of the AICs as that would reduce influence of any outlying values (this boils down to minimising the arithmetic vs geometric mean)

P.S. If anyone can suggest a different information criterion (such as BIC, etc.), how to use it in this case and why it is superior I would be more than happy to have the feedback.

• For independent data sets adding would be the correct way of getting the overall AIC of the collection of models. – Glen_b -Reinstate Monica Aug 10 '16 at 10:17
• @Glen_b In my opinion the output variables should be independent. I think this requirement is not satisfied by definition. – Cagdas Ozgenc Aug 10 '16 at 11:12
• @Cagdas I mentioned that independence requirement already; in the absence of independence the sum would generally not be suitable. – Glen_b -Reinstate Monica Aug 10 '16 at 11:41
• @Glen_b You said independent data sets. I am not sure what that even means. If the output variables are independent then the multivariate distribution will factor and then the log likelihood can be written as sum however summing AICs will mean 2k component will be summed several times as well which I think is not correct. We just need to add 2k only once on top of the sum of negative log likelihoods from the marginals. – Cagdas Ozgenc Aug 10 '16 at 12:31
• Another way to do it, would be to perform some kind reduction technique on the predictors (e.g. factor analysis), or perhaps use an PLS regression. You'd get to control how much of the original variation you want to keep, and it'd solve your uniformity issue, as you can simply keep the same factors across your models. The modeling choice depends somewhat on your substantive setting, as with inferential analysis factors may pose problems with interpretation. Alternatively, you could use the mean change in adjusted R² across all models for each variable as a criterion. – Maxim.K Aug 10 '16 at 12:36