[R-meta] Metafor: Reliability Generalization and Bonett-Transformation

Dear Daniel,

One cannot back-transform the value of tau^2 or tau in that manner. One possibility would be to apply the delta method (https://en.wikipedia.org/wiki/Delta_method). For the back-transformation of the ABT measure, this would be:

dat <- dat.bonett2010
dat <- escalc(measure="ABT", ai=ai, mi=mi, ni=ni, data=dat)
dat

res <- rma(yi, vi, data=dat)
res

c(res$tau2 * (exp(-res$beta))^2) # estimated tau^2 in the original units

That should be somewhat close to just analyzing the raw alpha values directly:

dat <- escalc(measure="ARAW", ai=ai, mi=mi, ni=ni, data=dat)
rma(yi, vi, data=dat)$tau2

But I wouldn't recommend doing this (either analyzing the raw alpha values or back-transforming tau^2 in this manner). Instead, one can just compute a prediction/credibility interval and back-transform its bounds:

dat <- escalc(measure="ABT", ai=ai, mi=mi, ni=ni, data=dat)
res <- rma(yi, vi, data=dat)
predict(res, transf=transf.iabt)

The cr.lb/ci.ub values reflect the amount of heterogeneity in a much more understandable way than reporting tau^2.

Best,
Wolfgang

-----Original Message-----
From: R-sig-meta-analysis [mailto:r-sig-meta-analysis-bounces at r-project.org] On Behalf Of Rupp, Daniel
Sent: Tuesday, 22 October, 2019 12:22
To: r-sig-meta-analysis at r-project.org
Subject: [R-meta] Metafor: Reliability Generalization and Bonett-Transformation

I am currently working on a Meta-Analysis of Cronbachs Alphas in Knowledge Tests. I am using the escalc function with the ABT Measure to calculate the alphas and the respective sampling variances. Everything is working out, but I am wondering about the inverse Transformation of tau2 after the rma. Just transforming ist shouldn't be right, since the variance is on another scale than alpha. So i thought about calculating the square root of tau2, conducting the inverse transformation and square it to get a backtransformed tau2. Is this the right way to do it or is there another way?

Best,
Daniel