[R-meta] Calculating variances and z transformation for tetrachoric, biserial correlations?

[Mark White]

Hello,

I have converted a number of summary statistics (contingency tables, *t-* and
*F*-statistics,* M*s and *SD*s) to tetrachoric and biserial correlations.
The other effect sizes that I directly observed were raw correlations. I
have my model all set up to run, but I am unsure as to what to do about
these effect sizes. I see two options:

1. Submit raw, tetrachoric, and biserial correlations and their variances
to analyses directly (what I have now).

2. Do Fisher's r-to-z transformation and *then *submit those to analyses.
The problem here is: How do I convert tetrachoric and biserial correlations
to Fisher's z? And if I do that, can I just use N to calculate the
variance? Or, do I have to also convert the variances of tetrachoric and
biserial correlations?

In either case, I am not sure how `metafor::escalc` calculates variances
for tetrachoric (`RTET`) and biserial (`RBIS`) correlations. I tried
looking through the code for `metafor::escalc` on GitHub, but could not
figure out the calculations.

I have included a table describing my effect sizes and how I calculated
them/their variances below.

What do you all think would be the best way to handle these data?

*Effect size*

*k*

*Effect size calculation*

*Variance calculation*

Raw correlation

217

Directly observed

Typical large-samples estimation (see Hedges, 1989, Equation 5), using
`metafor::escalc`

Tetrachoric correlation

12

From 2 x 2 contingency tables, using `metafor::escalc`

From 2 x 2 contingency tables, using `metafor::escalc`


*Unsure what the formula is*

Biserial correlation (from *t*- or *F*-statistic)

8

From *t*- or *F*-statistic to point-biserial correlation (using
`compute.es::tes`
and `compute.es::fes`) to biserial correlation (self-written function based
on Jacobs & Viechtbauer, 2016, assuming *n*s equal across conditions)

From *n*, using self-written function based on Soper's method (Jacobs &
Viechtbauer, 2016, Equation 13, assuming *n*s equal across conditions)

Biserial correlation (from *M *and *SD*)

2

From means and standard deviations directly, using `metafor::escalc`

From means and standard deviations directly, using `metafor::escalc`


*Unsure what the formula is*

Square-root of eta-squared

1

F-statistic to Cohen’s *f*  (Cohen, 1988) to eta-squared to square-root of
eta-squared as an approximation of a raw correlation coefficient (Lakens,
2013), using self-written function



*This was a one-way ANOVA with three means (low, medium, high prejudice).*

Typical large-samples estimation (see Hedges, 1989, Equation 5), using
`metafor::escalc`

Best,
Mark

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