My Math Forum R Squared Not Ranging From 0 to 1

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 August 3rd, 2018, 12:20 PM #1 Senior Member   Joined: Oct 2013 From: New York, USA Posts: 639 Thanks: 85 R Squared Not Ranging From 0 to 1 https://en.wikipedia.org/wiki/Coeffi...Interpretation says: "Values of R^2 outside the range 0 to 1 can occur when the model fits the data worse than a horizontal hyperplane. This would occur when the wrong model was chosen, or nonsensical constraints were applied by mistake. If equation 1 of Kvalseth[8] is used (this is the equation used most often), R^2 can be less than zero. If equation 2 of Kvalseth is used, R^2 can be greater than one." How can R^2 be less than 0 if the data are real numbers?
August 3rd, 2018, 01:03 PM   #2
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Quote:
 Originally Posted by EvanJ https://en.wikipedia.org/wiki/Coeffi...Interpretation says: "Values of R^2 outside the range 0 to 1 can occur when the model fits the data worse than a horizontal hyperplane. This would occur when the wrong model was chosen, or nonsensical constraints were applied by mistake. If equation 1 of Kvalseth[8] is used (this is the equation used most often), R^2 can be less than zero. If equation 2 of Kvalseth is used, R^2 can be greater than one." How can R^2 be less than 0 if the data are real numbers?
Sounds like the data sets I measured in my grad lab class. There's a reason why I'm a theorist.

-Dan

August 3rd, 2018, 01:58 PM   #3
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Quote:
 Originally Posted by EvanJ https://en.wikipedia.org/wiki/Coeffi...Interpretation says: "Values of R^2 outside the range 0 to 1 can occur when the model fits the data worse than a horizontal hyperplane. This would occur when the wrong model was chosen, or nonsensical constraints were applied by mistake. If equation 1 of Kvalseth[8] is used (this is the equation used most often), R^2 can be less than zero. If equation 2 of Kvalseth is used, R^2 can be greater than one." How can R^2 be less than 0 if the data are real numbers?
The same article you linked has an explcit description of the formula for $R^2$. Note that it is not computed by just squaring something so negative numbers are possible. From inspection of the formula, you can get negative values if $SS_{res} > SS_{tot}$.

Intuitively, this happens if the regression model "often" takes values which do not lie between the mean and the observed values. This can be seen directly by applying the triangle inequality to the formulas specified in your article.

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