Several sources (here here here) claim that there is a relation between Cohen's d and Pearson's r if the data is paired (bivariate). This strikes me as odd since, for example, evaluating a "before and after" scenario, one could end up with "after" values being the same as "before". This would yield a Cohen's d = 0 (the mean did not change) but Pearson's r = 1 (correlation of x = y). The result in this case is completely turned around.
I'd appreciate if someone would give an explanation why this relation makes sense. Worked-out examples would also be welcome.
Best Answer
The relationship described for Cohen's d and Pearson's r isn't for paired data. It's for unpaired data. For r, one variable is the two groups and the other is measurement variable. I've attached a plot to illustrate this, and some R code that works through an example. As is, I think it only works if the two samples have equal numbers of observations.
Note that for the simplest statement of this relationship,
d = 2*r / sqrt(1 - r^2)
, that the formula for Cohen's d needs to use n in the denominator for the pooled standard deviation and not n - 2, as is common. Also note that I think the formulas presented work only with equal sample sizes. The webpages provided don't seem to address these assumptions.