Solved – How to convert $eta^2$ to Pearson’s r

I am doing a meta-analysis of associations between sets of variables that are essentially all continuous. However, many studies operationalised them at 'lower' levels, and I have (converted to) most effect sizes you can think of (Cohen's $d$, Pearson's $r$, Cramer's $V$, $eta^2$, $omega^2$, and $OR$'s). Converting between most of these is no problem (that is, technically – of course there remain the methodological considerations).

However, now that I'm looking for a method to convert $eta^2$ I can't find any sources.

With Cohen's Statistical Power Analyses for the Behavioral Sciences (or the internet) it's easy to find how to convert $eta^2$ to Cohen's $f$:

$$text{Cohen's }f = sqrt{frac{eta^2}{1-eta^2}}$$

But, I can't find a formula for computing Pearson's $r$ from Cohen's $f$. What I did find is the formula for computing Cohens's $f^2$ from the squared multiple correlation, $R^2$, which is equal to the squared Pearson correlation for a bivariate association, in which case the following is therefore true:

$$text{Cohen's }f^2 = frac{R^2}{1-R^2} = frac{r^2}{1-r^2}$$

Now, assuming Cohen's $f^2$ is the square of Cohen's $f$ (doesn't seem too unreasonable, does it?), this would mean that you could compute Cohen's $f$ from Pearson's $r$ using:

$$text{Cohen's }f = sqrt{frac{r^2}{1-r^2}}$$

Which looks suspiciously like the formula for computing Cohen's $f$ from $eta^2$. It would in fact mean that a formula for computing Pearson's $r$ from $eta^2$ would just be:

$$text{Pearson's } r = sqrt{eta^2}$$

(of course, you have to take into account that $eta^2$ doesn't have a sign, but in my case, i.e. for a meta-analysis, I can add that manually)

Which makes sense – after all, $eta^2$ is an estimate of the proportion of explained variance, just like Pearson's $r^2$. In his discussion of his threshold for a 'large' value of $f$, Cohen (1988, p. 287-288) happily concludes:

In terms of correlation and proportion of variance accounted for, $f$ = .40 implies a correlation ratio ($eta$) of .371 and a PV (here $eta^2$) of .1379, somewhat more than twice the PV for a medium effect ($eta^2$ = .0588).

[…]

However, it is smaller than the criterion of a large ES in hypotheses concerning the Pearson $r$, where large $r$ is defined as .50, $r^2$ = PV = .25 (Section 3.2).

But that's where he stops. He doesn't go on to explain whether, if you want to convert $f$ to $r$, you have to correct for this inconsistency in any manner.

Does anybody have experience with this, or insight into the matter that can help determine how to proceed?

(Note: I know that converting everything to the same effect size metric is dubious. However, as Borenstein, Hedges, Higgins and Rothstein (2009) argue quite convincingly, so is omitting studies, especially if this could quite well lead to a systematic bias.)

[NOTE based on accepted answer: Laken's spreadsheet converts $eta^2$ to $r$ by taking its square root – so this would mean that indeed, $r^2$ is considered equivalent to $eta^2$.]

Check out Laken's (2013) article on effect sizes. One of the supplemental materials provided with his article is a spreadsheet titled, "From_R2D2.xlsx" (check out his Open Science Framework profile–he keeps the files updated). It is very helpful for converting between effect sizes (I've been using it for a meta-analysis of my own), and includes a conversion between $eta^2$ and $r$.

References

Lakens, D. (2013). Calculating and reporting effect sizes to facilitate cumulative science: A practical primer for t-tests and ANOVAs. Frontiers in Psychology, 4, 863. http://dx.doi.org/10.3389/fpsyg.2013.00863

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