by Adam
Last Updated April 21, 2018 18:19 PM

All sources I looked at say that correlation measures effect size. Hence, I presume, if $Y$ is a dependent variable of $X$ and $cor(X,Y)=1$ then the effect size of $X$ on $Y$ is considered huge? Perhaps even infinitely large? (since $1$ is the largest possible value of correlation). This doesn't make sense to me as the dependence between $X$ and $Y$ may be 100% certain but very small.

For example $X$ ranging between $0$ and $1$ and $Y=100+0.001X$ have correlation $1$.

Let's separate the magnitude of the correlation from one's level of certainty about it. If X = the two values {0,5} and Y = {10,15}, the observed correlation will be 1.00, indicating complete dependence or the maximum possible effect size that can appear using this metric. But who would feel certain about this based on just 2 observations?

Conversely, with ten million observations, a correlation of 0.05 would seem nearly certain to reflect a population correlation that's greater than zero, and its 95% or 99% confidence interval would be tiny. But .05 would indicate only a slight association between X and Y, and thus would constitute (in most contexts) a very small effect size.

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