# [R-meta] R-sig-meta-analysis Digest, Vol 67, Issue 2

James Pustejovsky jepu@to @end|ng |rom gm@||@com
Thu Dec 15 16:53:39 CET 2022

```Hi Yuhang,

For the sort of probability calculations you are trying to do, it is best
to use the transformed scale (i.e., the Fisher z scale). These sorts of
calculations rely on ALL of the model's distributional assumptions, so it
is critical to use the scale that best conforms to those assumptions. That
said, you can still interpret the results of these calculations in the
original metric.

Say that zi = transf.rtoz(ri) and that zi is an approximately unbiased
estimate of the parameter zeta-i. We can interpret zeta-i as the r-to-z
transformation of the correlation parameter rho-i. Assume that zeta-i
follows a normal distribution (possibly with a hierarchical structure, like
in your example, but I'll ignore that for now). Then this assumption
implies that rho-i also follows some distribution, but a non-normal one.
However, you can still do percentile calculations on the zeta-i scale. For
instance, based on the model, we can calculate
Pr(zeta-i > 0)
Since transf.rtoz(0) = 0, then
Pr(zeta-i > 0) = Pr(rho-i > 0)
More generally, suppose you want to find Pr(rho-i > c_r). Then you can take
c_z = transf.rtoz(c_r) and find
Pr(zeta-i > c_z),
which will correspond to Pr(rho-i > c_r). For small values of c_r, the
cutoffs are very similar, e.g.
c_r = 0.2 implies c_z = 0.2027
c_r = 0.3 implies c_z = 0.3095
c_z = 0.4 implies c_z = 0.4236
etc.

All of these calculations work more generally with the sort of multi-level
random effects model that you described in your example.

Cheers,
James

On Wed, Dec 14, 2022 at 7:33 PM Yuhang Hu <yh342 using nau.edu> wrote:

> Dear Wolfgang,
>
> Many thanks for your response. To make sure, as a general principle, do you
> recommend always computing those probabilities based off of the
> r2z-transformed effect size estimates?
>
> My intuition regarding transforming the variance components back to their
> original scale came from the fact that the emmeans package does a similar
> thing (for standard errors) via the delta method (
>
> https://cran.r-project.org/web/packages/emmeans/vignettes/transformations.html#after
> ).
>
> And my desire to accurately back transform the variance components was
> solely because I thought it could improve interpretability given the ease
> of working with correlations than r2z transformed correlations.
>
> Thanks again,
> Yuhang
>
> On Wed, Dec 14, 2022 at 4:01 AM <r-sig-meta-analysis-request using r-project.org
> >
> wrote:
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> >    1. Using variance components with effect size transformation
> >       (Yuhang Hu)
> >
> > ----------------------------------------------------------------------
> >
> > Message: 1
> > Date: Tue, 13 Dec 2022 20:28:55 -0700
> > From: Yuhang Hu <yh342 using nau.edu>
> > To: R meta <r-sig-meta-analysis using r-project.org>
> > Subject: [R-meta] Using variance components with effect size
> >         transformation
> > Message-ID:
> >         <
> > CA+dzWjp7MVtvytKgzK8eAQmGUzRsVex2zdvsi86UGP2QwHcgTA using mail.gmail.com>
> > Content-Type: text/plain; charset="utf-8"
> >
> > Dear Meta Community,
> >
> > I have a 3-level meta-regression model with a categorical moderator (3
> > levels) plus some covariates fit as:
> >
> > m6 = rma.mv(r2z_transformed ~ 0 + cat_mod + covariates, random = ~ 1 |
> > study/effect)
> >
> > I wonder to calculate probabilities from the distribution of effects as
> > explained in this post (
> >
> https://stat.ethz.ch/pipermail/r-sig-meta-analysis/2022-August/004136.html
> > ),
> > whether I should use the transformed variance components (A) or
> > back-transformed variance components (B)?
> >
> > A: confint.rma.mv(m6)   # transformed i.e., 'r2z'
> > B: confint.rma.mv(m6, transf = transf.ztor) # back-transformed i.e.,
> 'z2r'
> >
> > Currently, A and B give the same result, and I wonder why?
> >
> > Thanks,
> > Yuhang
> >
> >         [[alternative HTML version deleted]]
> >
> >
> >
> >
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> > End of R-sig-meta-analysis Digest, Vol 67, Issue 2
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