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I am pretty sure it is not going to let you take an effect size and a standard error from a correlation study and get out a accurate posterior distribution of the causal effect without doing something similar to what I'm proposing.

Ok, and

how do we model them? I am proposing a multilevel mixture model to compare them.Which is not going to work since in most, if not all, of these studies, the original patient-level data is not going to be available and you're not even going to get a correlation matrix out of the published paper, and I haven't seen any intervention-style algorithms which work with just the effect sizes which is what is on offer.

To work with the sparse data that is available, you are going to have to do something in between a meta-analysis and an interventionist analysis.

Ok. You can use whatever statistical model you want, as long as we are clear what the underlying object is you are dealing with. The difficulty here isn't the statistical modeling, but being clear about what it is that is being estimated (in other words the interpretation of the parameters of the model). This is why I don't talk about statistical modeling at first.

If all you have is reported effect sizes you won't get anything good out. You need the data they used.