Variable selection when data is nested

[BISTGP]

I have a large set of candidate predictors that could be selected for a logistic regession model. However the data has a nested structure with individuals nested within institutions. In a regression analsis GEE methods or a random intercept could be used to deal with the nesting. However I am not sure how best to do variable selection here. Ignoring the nesting is the simple solution, but is there a better idea?

[PGStats]

An interesting exercise would be to do variable selection within each institution... if you have enough data... and try to reach some concensus among the variety of models that come out.

[BISTGP]

Thank you for the suggestion.  This occurred to me too but while there are nearly 500 facilities, the median sample size per facility is 150 and an average prevalence of about 10%.  My intuition is that except for a handful of the largest facilities the sample size isn't really large enough to support variable selection methods, but might be worth a try.