Multiple imputation is fairly straightforward when you have an a priori linear model that you want to estimate. However, things seem to be a bit trickier when you actually want to do some model selection (e.g. find the "best" set of predictor variables from a larger set of candidate variables - I am thinking specifically of LASSO and fractional polynomials using R).
One idea would be to fit the model in the original data with missing values, and then re-estimate this model in MI datasets and combine estimates as you normally would. However, this seems problematic since you are expecting bias (or else why do the MI in the first place?), which could lead to selecting a "wrong" model from the start.
Another idea would be to go through whatever model selection process you are using in each MI dataset - but how would you then combine results if they include different sets of variables?
One thought I had was to stack a set of MI datasets and analyze them as one large dataset that you would then use to fit a single, "best" model, and include a random effect to account for the fact you are using repeated measures for each observation.
Does this sound reasonable? Or perhaps incredibly naive? Any pointers on this issue (model selection with multiple imputation) would be greatly appreciated.