I would suggest at the very least getting an idea, maybe from other papers, of the lowest and highest plausible correlation. This is a bit of a slippery idea. Ideally, you would switch to a Bayesian calculation, or as health economists put it so as not to scare the children, “probabilistic sensitivity analysis”, but I understand that could be daunting or not acceptable to collaborators sometimes. In which case you then have a minimum and maximum correlation scenario and see what difference, if any, they make on the total meta-analysis. If not much, then you could just go with the midrange correlation, but if it tips the final result either side of significant, for example, then you’d have to say there is uncertainty arising from the unreported stats and give both cases.
I’m hoping to run a course on Bayesian meta-analysis later this year, if you’re interested send me an email at robert at bayescamp.com or just keep an eye on the website. It’ll be London-based.

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