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I am preparing to use the Paule-Mandel method for determining the weighting factors in a meta-analysis. The paper I'm reading is the original one available here.

I believe only those who are familiar with the paper can answer my question, so I won't summarize the paper here. I would be happy to if need be, though.

My question is how the final equation on page 379 (the third page of the linked PDF) follows from the equations given so far. Specifically:

The authors establish that $\text{Var}(\sqrt{\omega_{i}}\bar{Y}_{i}) = 1$ and then go on to say

for any given set of $\omega_{i}$ this variance can be estimated from the sample by the formula

$$s^{2}(\omega_{i}\bar{Y}_{i}) = \frac{\sum_{i=1}^{m}\omega_{i}(\bar{Y}_{i}-\tilde{Y})^{2}}{m-1}$$

Equating this estimate to its expected value (unity, see above) we obtain

$$s^{2}(\omega_{i}\bar{Y}_{i}) = \frac{\sum_{i=1}^{m}\omega_{i}(\bar{Y}_{i}-\tilde{Y})^{2}}{m-1} = 1$$

I'm not following these statements.

The authors defined $\text{Var}(\bar{Y}_{i})$ as having a within-set component and between-set component. The $s^{2}(\omega_{i}\bar{Y}_{i})$ expression seems only to be an estimate of the between-set component of variance: you're summing the squared difference between set averages and the consensus value.

So $s^{2}(\omega_{i}\bar{Y}_{i})$ should not equal 1. This number plus the within-set component of variance should be 1.

But that's not what the paper says. What am I missing? Thanks!

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