## 15.6 Hedges’ g from a correlation

For equally sized groups ($$n_1=n_2$$), we can use the following formula to derive the SMD from the pointbiserial correlation (Rosenthal 1984).

$r_{pb} = \frac{d}{\sqrt{d^2+4}}$ And this formula for unequally sized groups (Aaron, Kromrey, and Ferron 1998):

$r_{pb} = \frac{d}{\sqrt{d^2+ \frac{(N^2-2 \times N)}{n_1 n_2} }}$ To convert $$r_{pb}$$ to Hedges’ g, we can use the esc_rpb function with the following parameters:

• r: The r-value. Either r or its p-value must be given.
• p: The p-value of the correlation. Either r or its p-value must be given.
• grp1n: The sample size of group 1.
• grp2n: The sample size of group 2.
• totaln: Total sample size, if grp1n and grp2n are not given.
• es.type: the effect measure we want to calculate. In our case this is "g". But we could also calculate Cohen’s d using "d".
esc_rpb(r = 0.25, grp1n = 99, grp2n = 120, es.type = "g")
##
## Effect Size Calculation for Meta Analysis
##
##      Conversion: point-biserial r to effect size Hedges' g
##     Effect Size:   0.5170
##  Standard Error:   0.1380
##        Variance:   0.0190
##        Lower CI:   0.2465
##        Upper CI:   0.7875
##          Weight:  52.4967

### References

Rosenthal, R. 1984. “Meta-Analysis Procedure for Social Research.” Sage Publications.

Aaron, Bruce, Jeffrey D Kromrey, and John Ferron. 1998. Equating R-Based and d-Based Effect Size Indices: Problems with a Commonly Recommended Formula. ERIC Clearinghouse. 