I am working on a field experiment where assignment to treatment vs. comparison was random, but participation uptake was not. The design is pre-post, and attrition is certainly not MCAR. This is a clustered design (randomization at level 1). Outcomes are continuous or binary.
One possibility for the uptake issue is propensity matching or computing (trimmed) inverse probability of treatment weights. An IV approach using assignment as an instrument for uptake to get an ATT estimate is also possible.
The problem arises where some outcomes are only defined for selected participants and selection may be endogenous to treatment. For example, in an intervention to reduce pregnancy, an outcome might be receipt of adequate prenatal care, which is only defined if the participant is pregnant, and whether they are pregnant reflects an endogenous selection process.
I'm skeptical of an instrumental variable approach--not sure what I would have that would cause meaningful differences in pregnancy rates but not also in prenatal care. A seeming answer is a Heckman correction, but I am trying to avoid having to run selection models for each of several such conditionals.
Are there any suggestions that get around the problem of multiple selection processes for different outcomes in the same data? Perhaps creating matched pairs instead of IPTW? But I'm not sure what the next step to that would be (though block deletion would then become an option for attrition).
Thank you.
