I stumbled on a discussion regarding the usefulness of $R^2$ as a metric. Where $R^2$ is defined as:

$$ \frac{\sum (\hat{y} - \bar{\hat{y}})^2 } {\sum(y - \bar{y})^2 }.$$

The criticism is backed by Carnegie Mellon's Cosma Shalizi and focused on these 4 points:

  • $R^2$ does not measure goodness of fit.
  • $R^2$ does not measure predictive error.
  • $R^2$ does not allow you to compare models using transformed responses.
  • $R^2$ does not measure how one variable explains another.

He even suggested that "I have never found a situation where it helped at all.".

The article I read can be found here.

What do you think. Have you encountered any situation where $R^2$ would be useful?

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    $\begingroup$ I calculate R-squared (R2) as "R2 = 1.0 - (absolute_error_variance / dependent_variable_variance)" and I use it to tell me what fraction of the variance in the dependent varable is explained by the model. For example, an R-squared of 0.95 tells me my model explains 95% of the variance in the dependent variable. This is exact for a straight line and approximate for other models. It has the advantage of telling you the same thing about the model whether the units are in milliliters or light years, which is one of the reasons it is so widely used. $\endgroup$ – James Phillips Sep 15 '18 at 22:40
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    $\begingroup$ Well, that's Cosma Shalizi, who has his opinions. Yes, I do find $R^2$ useful on occasion, and have for many years. Like any statistical tool, you have to understand its strengths and weaknesses, understand how what you have been doing to build a model plays into them, understand how the objectives of your model-building exercise relate to them, and understand how the specifics of your problem relate to them. $\endgroup$ – jbowman Sep 16 '18 at 0:44
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    $\begingroup$ To describe it as a distraction and a nuisance is consistent with my experience. It’s bad enough in the context where it belongs. Stray outside of linear regression, and be prepared to answer either the question “but what’s $R^2$?” or “but why is $R^2$ so low (less than 90%)?” $\endgroup$ – The Laconic Sep 16 '18 at 3:33

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