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I have a question regarding the use of random-effects GLM meta-analytic models and the minimum number of studies needed and its relation to the estimation of $ \tau^2 $. (I understand that only two studies are needed for a fixed effect estimate.)

My hunch is that one needs at least $k=4$ studies, since 1 degree of freedom will be used for the average effect estimate and the remaining 3 will then go towards estimation of $\tau$. However, I have see published reports that will report random-effects estimates with $k=2$ studies, and indeed any popular meta-analytic packages will fit such models because it is mathematically possible. With so few studies, will $\hat{\tau}^2$ be meaningful? or (badly) biased?

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The main problem is that it is likely to be imprecisely estimated. The preferred way of showing this is to present it with a confidence interval but this is not often done. An article by Ioannidis and colleagues in the British Medical Journal here entitled "Uncertainty in heterogeneity estimates in meta-analyses" argues for the routine presentation of confidence intervals. Any meta-analysis based on a few studies would need to be treated with caution anyway but perhaps some extra caution is needed if a random effects model is proposed.

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    $\begingroup$ I agree with you on the imprecision point. Certainly a model with few studies and heterogeneity could yield very wide confidence intervals. $\endgroup$ Feb 20, 2018 at 19:38

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