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

correlationeffect-sizemeta-analysis

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 $\eta^2$.]

Best Answer

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

Related Solutions

Solved – Interpretation when converting correlation of continuous data to Cohen’s d

You've hit on a personal pet peeve of mine. I don't think that the interpretation given in the book (of an r-to-d transformed value of a correlation coefficient that is based on two continuous variables) makes any sense. There is no explicit or implicit dichotomization happening here (and of which variable, the first or the second? and dichotomized at what point?) and I've never seen any proper analytic demonstration that this interpretation is justified.

In the other direction, the conversion of d to r makes perfect sense, as long as we realize that the conversion yields a point-biserial correlation coefficient (not a Pearson product-moment correlation coefficient of a bivariate normal distribution or some other bivariate distribution of two continuous variables). And to be precise, the correct equation for the conversion is $$r_{pb} = \frac{d}{\sqrt{d^2 + h}},$$ where $$h = \frac{m}{n_1} + \frac{m}{n_2}$$ and $m = n_1 + n_2 - 2$. Equation 7.7 in the book is not quite right, although the difference will usually be small. An example:

grp <- c(0,0,0,0,1,1,1,1,1,1)
out <- c(2,4,3,4,2,3,5,4,5,5)
cor(grp, out) ### point-biserial correlation

This yields:

[1] 0.3340213

First, we compute the standardized mean difference:

m <- c(by(out, grp, mean))
v <- c(by(out, grp, var))
n <- c(by(out, grp, length))
vp <- ((n[1] - 1)*v[1] + (n[2] - 1)*v[2]) / (n[1] + n[2] - 2)
d <- (m[2] - m[1]) / sqrt(vp)

Exact conversion:

m <- n[1] + n[2] - 2
h <- m/n[1] + m/n[2]
d / sqrt(d^2 + h)

This yields:

[1] 0.3340213

as it should. Now try the equation from Borenstein et al.:

a <- (n[1] + n[2])^2 / (n[1]*n[2])
d / sqrt(d^2 + a)

This yields:

0.3021478

Not quite right.

Also, this conversion does not assume that "a continuous variable was dichotomized to create the treatment and control groups." (p. 49). No such assumption is necessary. The assumption (that a continuous variable was dichotomized) comes into play when we want to convert a d value into a biserial correlation coefficient. See:

Jacobs, P., & Viechtbauer, W. (2017). Estimation of the biserial correlation and its sampling variance for use in meta-analysis. Research Synthesis Methods, 8(2), 161-180. doi:10.1002/jrsm.1218

Quite problematic is also the last paragraph on page 49. It suggests that we can just transform measures in any direction. But the interpretation of the converted measures is more than dubious (as in the case of going from a correlation of two continuous variables to d). Moreover, the sampling variances of such converted measures is often much more complicated than this paragraph suggests. For example, if you do d-to-r-to-z (so, going from a standardized mean difference to a point-biserial correlation and then applying Fisher's r-to-z transformation), then the sampling variance of the resulting value is not $1/(n-3)$. The conversion of r-to-z applies when r is a correlation between two continuous variables (that are bivariate normal), which isn't the case here. Another relevant article in this regard is:

Pustejovsky, J. E. (2014). Converting from d to r to z when the design uses extreme groups, dichotomization, or experimental control. Psychological Methods, 19(1), 92-112.

Solved – How to compare concordance correlation coefficient to Pearson’s r

Pearson's r measures linearity, while CCC measures agreement. Imagine a scatterplot between the two measures. High agreement implies that the scatterplot points are close to the 45 degrees line of perfect concordance which runs diagonally to the scatterplot, whereas a high Pearson's r implies that the scatterplot points are close to any straight line.

In practice,

CCC = r * C_b

where r is Pearson's r, and C_b is a bias correction factor.

Therefore, CCC cannot be compared to Pearson's r. To calculate Pearson's r from CCC for a direct comparison you will need to divide CCC by C_b.

Best Answer

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Solved – Interpretation when converting correlation of continuous data to Cohen’s d

Solved – How to compare concordance correlation coefficient to Pearson’s r

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