Imagine you were interested in student dropout rates across schools over time. You had data on the number of dropouts, but not the number of students enrolled at each school-month.
A single dropout for a school with 10 students is a higher relative rate than 2 dropouts in a school of a 100.
An increased number of observed dropouts might mean either something is causing dropouts or that the dropout rate is remaining constant and there are just more kids enrolling.
Is there anything meaningful that can be said about how independent variables influence the dropout rate, given the unmeasured enrollment level?
Current ideas include treating enrollment as an unmeasured confounding variable and including unit-time fixed effects (for say each school quarter). Another idea is to apply different priors about enrollment and generate some kind of bounds for coefficient estimates.
Would either of these work, and are there alternatives? Any help would be greatly appreciated.