# How do I 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 $\eta^2$.]

• Good question. I don't have an answer, but I would like to say that if you are concerned about the impact of converting, you could test the effect of converting as a moderator effect in your analysis. Simply adding a categorical variable for each observation stating the original measure of ES. – Emilie Aug 12 '15 at 2:37
• That's a great idea, thank you!!! I'll definitely do this :-) – Matherion Aug 14 '15 at 8:43
• You could probably use R^2 (equivalent to eta squared) as your relevant statistic. However, take note of the difference between eta squared and partial eta squared. – jona Aug 30 '15 at 15:31
• The suggestion by @jona jives with the Laken's spreadsheet I suggested below, as the spreadsheet just takes the square root of $\eta^2$ to convert to $r$. Still, I would stick with meta-analyzing $r$ vs. $R^2$ for a number of reasons (e.g., convention, retaining information about direction of effects, standard error/variance calculations, etc.,) – jsakaluk Aug 30 '15 at 16:06
• @jsakaluk you'd then of course have to somehow deal with the fact that R^2 is always positive. – jona Aug 31 '15 at 7:07

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

• Lakens does note, however, that the spreadsheet itself was not peer-reviewed. – jsakaluk Aug 30 '15 at 14:51
• It is not clear which effectsize you are working on for a meta analysis. – Subhash C. Davar Aug 30 '15 at 15:59
• I too am meta-analyzing Pearson correlation coefficients, but I don't see how that is important seeing as I didn't ask the question. – jsakaluk Aug 30 '15 at 16:01
• shall appreciate if you make a reading of Hedges and Olkin(1985) p.101-102 and that of Hunter , Schmidt and Jackson(1982) pp. 105-110. I simply got scared - it was d or r statistic. – Subhash C. Davar Aug 31 '15 at 15:00

Pearson's 𝑟 is the relationship between continuous variables, but eta squared is a measure of ANOVA, it means y is a continuous variable, and the x is a nominal variable. Pearson's 𝑟 is not identical to eta.

• R squared is commonly reported for anova, and is reported by software like SAS and R for anova, particularly if the analysis is done in the context of a general linear model (lm in R, glm in SAS). – Sal Mangiafico Oct 5 '19 at 9:53