Solved – the relation between the effect size and correlation

Question

Correlation refers to the degree to which a pair of variables is linearly related. The effect size quantifies some difference between two groups (e.g. the difference between the means of two datasets). For example, there's the Cohen's effect size.

It seems to me that these concepts are related, but how exactly are they related?

Answer 1

You are right, Cohen's $d$ and the correlation coefficient $r$ are conceptually related, in at least two ways:

• Both are effect sizes, because both quantify the size of an effect (yes, it's that litteral!). Cohen's $d$ quantifies the difference between the means of two populations, and $r$ quantifies how robust is the relationship between two variables.
• $r^2$ is the proportion of variance shared by the two variables. The greater the $r^2$, the more closely the variables vary together. Cohen's $d$ is not a proportion of variance strictly speaking but it's conceptually related. Take a group of teenagers, all of the same sex. You could probably not tell 15 from 16-year-olds based on their height alone: most of the variability across the two sub-populations is not due to age because the effect size $d$ is very small (around 0.2). However, if you mix preteens and young adults, a substantial amount of variability in heigts can be ascribed to differences in age, and the effect size is indeed much larger (around 1).

However, it is important to remember that $r$ and $d$ are very different and bear no direct relationship. They quantify entirely different types of phenomena. Two series of values can have exactly the same mean ($d=0$) and be totally uncorrelated ($r=0$) or highly correlated ($r=1$); every combination is possible.