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In the traditional fixed effects model (FEM) of meta analysis, given the estimated effects from K studies θ1,…, θK , with θi ~N(θ, σi2) , the combined effect θ is estimated as the weighted mean θest =(wi θ1+…+ wK θK)/W ~N(θ ,1/ W ) , where wi=σi-2 and W= (w1+…+ wK ).
If the homogeneity of the effects is rejected, the random effects model can be used: θi ~N(θ, σi-2 +τ2). (Sutton et al, 2000).
When the variance stabilizing transformation (vst) is applied to the estimated effects, we deal instead with the transformed standardised effects K(δi). They are estimated by κi =ni-1/2h(Si) ~N(K(δ),1/ni) and can be added with known weights ni in meta-analysis (Kulinskaya, Morgenthaler and Staudte, 2008)
Given variance stabilized statistics from K studies T1,…,TK , with T1 ~N(ni1/2 κ ,1)
the combined effect κest=(n1 κ1+…+nK κK))/ N ~N(K(δ),1/ N ) where N= n1 +…+nK.
The back-transformation is used to obtain the inference on the standardised effects δ.
If the homogeneity of the transformed effects is rejected, the random transformed effects model can be used: κ i ~N(κ , ni-1 +τ2).
When there are no nuisance parameters (as in the 1-sample Binomial or Poisson case) these two approaches to meta analysis are equivalent. In the general case, the variance stabilization approach can be used even when the inference on the original, non-standardised effects is of primary interest. In this case the optimal weights depend on the nuisance parameters. An example is the variance stabilizing arcsine transformation for the difference in absolute risks, with the average risk as the nuisance parameter.
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