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I have $2n$ paired populations, on which I measure a rate of success, let's say $(r_{group}^i)_{group \in (0,1)}^{i \in[1,n]}$ where $r_0^i$ and $r_1^i$ are the rates of success in the $i$-th pair of populations.

I want to test whether there is a significant difference of rate between the $(r_0^i)$ and the $(r_1^i)$ across all pairs i, in other words whether an individual in a population of group 1 will have a higher rate of success than one in the paired population in group 2, irrespective of the specific pair $i$.

I did a 2-sample proportion test on individual pairs of populations ($n=1$), but I am not sure which test to apply with an arbitrary number of pairs. Since I am looking for an effect of the group membership, it seems wrong to just repeat the test on all pairs of populations.

Is there a way to test this group effect directly?

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Okay, so after some investigation, it appears that Fisher's method for meta-analysis can do this type of job, since the hypotheses $\mathcal{H}_0$ (identical rates for both populations in pair $i$) and $\mathcal{H}_1$ are the same for all pair of populations.

Once the $n$ p-values corresponding to the $n$ 2-sample proportion tests are calculated, they can be meta-analysed using Fisher's method. Since the sample sizes are different and the rate difference is signed, a better alternative might be to use Stouffer's z-score method.

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