# Interpretation of $R^2$ (coefficient of determination)

I found two different and contradictory definitions / interpretations of $$R2$$. They cannot be true at the same time and I don't know how to tell which one is true.

1. $$R^2$$ is just a square of Pearson's correlation. Hence its value ranges between $$[0, 1]$$
2. $$R^2$$ is equal to $$1 - \dfrac{SSE}{SST}$$

and can be negative if the model is worse than random. For example - in Python's sklearn function r2_score can return a negative number.

Are those two different $$R^2$$s? Why are there two contradictory definitions? Can anyone explain it comprehensively?

• Just because two formulas are not algebraically equivalent does not mean they are contradictory: one can be a special case of the other, for instance. For 14 additional formulas for $R^2$ see stats.stackexchange.com/questions/70969. (Strictly speaking, you need to square those formulas for $\rho.$)
– whuber
Commented Dec 27, 2022 at 22:31
• By contradiction I mean that one definition assumes $R2$ can be negative and the other that it ranges between 0 and 1 Commented Dec 28, 2022 at 8:14
• Neither definition makes such an assumption. It merely turns out that, in some circumstances (where definition (1) is inapplicable), definition (2) can yield negative values.
– whuber
Commented Dec 28, 2022 at 14:55

In a simple setting where you fit a linear model, using ordinary least squares, and include an intercept, multiple definitions of $$R^2$$ give equal values.

1. In simple linear regression (just one slope and the intercept), Pearson correlation between the $$x$$ and $$y$$ values

2. Squared Pearson correlation between predicted and true $$y$$

3. Proportion of variance explained by the regression

4. Your second definition: a comparison of the square loss of your model to the square loss of a naïve model that always predicts the overall (pooled, marginal, unconditional) mean of $$y$$

(There are others, as discussed in this link given in a comment by whuber.)

In simple settings, these give the same answer, so people use them interchangeably as the definition of $$R^2$$. However, in more complex settings, there are differences. I get into some of those differences here and explain my support for the last of the four.

If you fit both slope and intercept with linear regression, both your definitions will be true (and both give the same value of $$R^2$$).

If you only fit one of those (slope or intercept) and constrain the other to a fixed value (often the intercept is forced to equal 0), then the second definition is correct but the first definition will be wrong. And in this case, yes $$R^2$$ can be negative: What does negative R-squared mean?