Suppose X is a cause of Y that explains r^2 of the variance in Y. If you have someone who is z standard deviations about average in Y, then in expectation r^2 z of that is due to being above average in X. That is, the variance explained tells you how much of the Y is explained by the X.

The trouble is that people are mixing up two different questions: explanation and intervention. If you want to intervene on Y with some variable X, then it doesn't matter how well X explains the preexisting variance in Y. It just matters how big the effect of X is. But conversely, if someone wants to explain Y, then it doesn't matter how big the interventions are that it supports, it only matters how much variance it explains. It just so happens that there is a simple quadratic relationship between the effect size and the explanation validity (in the linear-Gaussian case).

Maybe this is a bit simplistic, but the way I think about it is that you use r^2 when comparing two indepedent variables to each other and you use r when comparing one independent variable to the depedent variable.

“The ballpark is ten miles away, but a friend gives you a ride for the first five miles. You’re halfway there, right? Nope, you’re actually only one quarter of the way there.”

One notion of variance explained I like:

Suppose X is a cause of Y that explains r^2 of the variance in Y. If you have someone who is z standard deviations about average in Y, then in expectation r^2 z of that is due to being above average in X. That is, the variance explained tells you how much of the Y is explained by the X.

The trouble is that people are mixing up two different questions: explanation and intervention. If you want to intervene on Y with some variable X, then it doesn't matter how well X explains the preexisting variance in Y. It just matters how big the effect of X is. But conversely, if someone wants to explain Y, then it doesn't matter how big the interventions are that it supports, it only matters how much variance it explains. It just so happens that there is a simple quadratic relationship between the effect size and the explanation validity (in the linear-Gaussian case).

See also:

https://kirkegaard.substack.com/p/variance-explained-is-mostly-bad

https://openpsych.net/paper/61/

Maybe this is a bit simplistic, but the way I think about it is that you use r^2 when comparing two indepedent variables to each other and you use r when comparing one independent variable to the depedent variable.

“The ballpark is ten miles away, but a friend gives you a ride for the first five miles. You’re halfway there, right? Nope, you’re actually only one quarter of the way there.”

Classic.

Taleb is coping on Twitter about this: https://x.com/nntaleb/status/1653072081154715651?s=20