# [R-meta] Meta-Analysis using different correlation coefficients

Lukasz Stasielowicz |uk@@z@@t@@|e|ow|cz @end|ng |rom un|-o@n@brueck@de
Tue Feb 15 08:03:19 CET 2022

```Dear Lena,

since you haven't received a response yet, I will address some points:

It is important to define a target effect size for the meta-analysis,
e.g. Pearson correlation r. If other effect sizes are reported in the
primary studies then one could transform them or use the raw data to
compute the target effect size.

If Pearson correlation is the target effect size and you have
point-biserial correlations then no transformation is necessary because
it is mathematically equivalent to Pearson correlation.

Spearman correlation and Pearson correlation will not always lead to
similar values, as the latter is less likely to detect nonlinear
(monotonic) relationships. Therefore, transforming Spearman correlations
to Pearson correlations could ensure, that the effect size values can be
interpreted the same way. In this particular case one could use arcsin
formulas:
de Winter, J. C. F., Gosling, S. D., & Potter, J. (2016). Comparing the
Pearson and Spearman correlation coefficients across distributions and
sample sizes: A tutorial using simulations and empirical data.
Psychological Methods, 21(3), 273–290. doi.org/10.1037/met0000079
You'll find formulas for other transformations in textbooks and other
journal articles.

While many meta-analysts tend to use all available information and apply
all possible transformations (d --> r, Spearman --> Pearson, OR --> r),
there are some people, who would point out that under certain
circumstances not all transformations are valid (e.g., d --> r  can the
distinction between control group and experimental group be thought as a
part of an underlying continuous distribution or is it a natural
dichotomy?).
However, some people would argue that every effect size is useful (in
particular, if the meta-analysis is small).
Sometimes meta-analysts conduct a moderator analysis and compare
converted effect sizes (e.g., d-->r) with effect sizes that didn't
require converting (e.g. r), in order to check the influence of pooling
different designs/effects.

Multiple effect sizes: If there are multiple studies with multiple
effect sizes then a three-level meta-analysis could be useful.
Alternatively, one could use cluster-robust variance estimation, e.g.
cran.r-project.org/web/packages/clubSandwich/vignettes/meta-analysis-with-CRVE.html
If there is only one sample with multiple effects then choosing one
effect size (e.g., the typical operationalization of the constructs, the
most valid operationalization) could be an option. Alternatively, one
could compute a mean effect size, as you have mentioned in your message.

project!

Best,
Lukasz
--
Lukasz Stasielowicz
Osnabrück University
Institute for Psychology
Research methods, psychological assessment, and evaluation
Seminarstraße 20
49074 Osnabrück (Germany)

Am 09.02.2022 um 07:54 schrieb r-sig-meta-analysis-request using r-project.org:
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> Today's Topics:
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>     1. Meta-Analysis using different correlation coefficients
>        (Lena Pollerhoff)
>     2. Meta-Analysis using different correlation coefficients
>        (Lena Pollerhoff)
>     3. Comparison to baseline- remove intercept or keep? (Danielle Hiam)
>     4. questions on some functions in metafor and clubsandwich
>        (Brendan Hutchinson)
>
> ----------------------------------------------------------------------
>
> Message: 1
> Date: Tue, 8 Feb 2022 13:39:52 +0100
> From: Lena Pollerhoff <lena using pollerhoff.de>
> To: r-sig-meta-analysis using r-project.org
> Subject: [R-meta] Meta-Analysis using different correlation
> 	coefficients
> Message-ID: <AA0B9D07-6B89-4BF7-91F7-D32B5FDE031A using pollerhoff.de>
> Content-Type: text/plain; charset="utf-8"
>
> Hello,
>
> We are currently conducting a meta-analysis based on correlation coefficients.
> We have received a huge amount of raw datasets, so that we are able to calculate effect sizes/correlations coefficients on our own for many datasets, and  we have other correlations extracted from the original pubs. Therefore I have a couple of questions:
>
> 1. If one variable is dichotomous and the other variable is continuous but not normally distributed, what kind of coefficient should be calculated? We’d go for point-biserial if the variable is naturally dichotomous (not artificially dichotomized), and for biserial correlation if the dichotomous variable was artificially dichotomized, but are worried that both require normal distribution of the continuous variable?
>
> 2. We are wondering how to best integrate person’s product moment correlation coefficients (both continuous, normally distributed variables), (point-) biserial correlation coefficients (for 1 (artificial) dichotomous and 1 continuous variable) and spearman rang correlation coefficients (for non-parametric, both continuous variables) in one meta-analysis? Just use the raw values? Or is it better to transform them in a homogenous way (I’ve read Fisher’s z makes less sense for anything else than Pearson’s r as a variance-stabilizing procedure?)? Can spearman rho be converted using fisher’s z transformation? I’ve also read that it is not advisable to include product-moment correlation and point-biserial correlation in one meta-analysis, is there a way to convert the point-biserial correlation to something that can be integrated with Pearson’s r and Spearman’s rho?
>
> 3. I have multiple effect sizes within one sample and I want to aggregate them, how do I define rho in the aggregate function from the metafor package? Is it possible to calculate rho based on the raw datasets? Or would it better to think in a conservative way and assume perfect redundancy (i.e., rho = 0.9)?
>