I am not sure I understand your question.
Just putting the idea out for comment in case there's some way this fails to deliver what I want it to deliver. Excerpting out all the comparisons and writing up the mixture model in JAGS would be a lot of work; just reading the papers takes long enough as it is.
If I got such data I would (a) be very happy, (b) use the RCT to inform policy, and (c) use the pair to point out how correct causal inference methods can recover the RCT result if assumptions hold (hopefully they hold in the observational study)
Indeed. You can imagine that when I stumbled across Deeks and the rest of then in Google Scholar (my notes), I was overjoyed by their obvious utility (and because it meant I didn't have to do it myself, as I was musing about doing using FDA trials) but also completely baffled: why had I never heard of these papers before?
I am not following your mixture model idea. For every data point you know if it comes from the RCT or observational study. You don't need uncertainty about treatment assignment. What you need is figuring out how to massage observational data to get causal conclusions (e.g. what I think about all day long).
If you have specific observational data you want to look at, email me if you want to chat more.
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.
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