[R-meta] Can z-transformed R-squared be used as an effect size?

Viechtbauer, Wolfgang (SP) wo||g@ng@v|echtb@uer @end|ng |rom m@@@tr|chtun|ver@|ty@n|
Sat Mar 19 12:09:41 CET 2022

I sent a response yesterday along the same lines, but it never made it through to the list. Very strange. Maybe the spam catcher was trying to tell me to stop posting so much ...

In any case, just for the record, let me try to post my response one more time:


Hi Matthew,

And for once, there is also a pretty quick answer: Doing this is nonsense.

The r-to-z transformation is a variance-stabilizing transformation for Pearson product-moment correlation coefficients. It does not work the same way for squared correlations or R^2 values. Also, the variance of an r-to-z transformed R^2 value is not just 1/(n-3) as for r-to-z transformed Pearson product-moment correlation coefficients.

So yes, your intuition that this is problematic is absolutely right.


Let's see if this one goes through.


>-----Original Message-----
>From: R-sig-meta-analysis [mailto:r-sig-meta-analysis-bounces using r-project.org] On
>Behalf Of Michael Dewey
>Sent: Friday, 18 March, 2022 17:06
>To: Matthew Yates; r-sig-meta-analysis using r-project.org
>Subject: Re: [R-meta] Can z-transformed R-squared be used as an effect size?
>Dear Matthew
>I share your concerns and doubts about this. I would hesitate to go out
>on a limb over it but if you are reviewing it for a journal I would
>suggest politely telling the authors that you find it hard to justify
>and ask them to provide a reference or other wise justify it.
>I do not make confidential comments to the editors on principle but if I
>did I would tell them that unless the authors can justify it the paper
>demands instant rejection.
>On 17/03/2022 17:47, Matthew Yates wrote:
>> Hello SIG-meta folks,
>> I have (what I think is) a pretty quick question. I'm currently conducting a
>peer-review of a meta-analysis in my field.
>> The authors of this manuscript elected to use fisher-Z-transformed R-squared
>values (note: NOT Pearson correlation coefficients, but their squared-values) as
>their 'effect size' statistic, and then calculated variance for the z-transformed
>R-squared values as for a typical Pearson correlation coefficient (r).
>> Is this a valid statistical effect size? The z-transformation, as I understand
>it, was developed specifically for Pearson correlation coefficients, so this
>strikes me as potentially problematic - the z-transformation, itself, is meant
>for variables that can span -1 to 1, so the underlying distribution of the
>transformed variables (0 to 1 for r-squared values) are inherently different.
>Similarly, estimating the variances based on the sample size (n) of the z-
>transformed R-squared values again strikes me as potentially problematic as well.
>> As far as I can tell, I think pretty much all of the studies being analysed
>were bivariate linear regressions, so there isn't an issue with non-linear
>relationships, covariates, etc (I saw another post on here asking that
>question....). I've just never seen this done before in a meta-analysis, or read
>of it in any literature, guides, etc. on how to conduct a meta-analysis. Most
>people typically just use the z-transformed Pearson correlation coefficients,
>rather than the R-squared values!
>> I've done a few meta-analyses myself, so am familiar with general techniques
>but would not consider myself an expert/specialist (most of mine were pretty
>basic). However, this strikes me as potentially problematic, and I was wondering
>what others with more statistical expertise in meta-analytic techniques might
>think of this issue.
>> Any input would be appreciated.
>> Thanks,
>> Dr. Matthew C. Yates
>> Post-doctoral Researcher
>> Great Lakes Institute for Environmental Research (GLIER)
>> University of Windsor
>> 2990 Riverside Dr W,
>> Windsor, ON N9C 1A2
>> (514) 919 5613
>> Website: https://matthewyates6.wixsite.com/ecologist

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