Coefficient of Determination

The coefficient of determination, denoted $R^2$, measures the proportion of the total variance in the response variable that is explained by a regression model, computed as $R^2=1-\frac{S{S}_{res}}{S{S}_{tot}}$, one minus the ratio of the residual (unexplained) sum of squares to the total sum of squares. Bounded between 0 and 1 for ordinary least squares regression (0 indicating the model explains none of the variance beyond the response's own mean, 1 indicating a perfect fit), $R^2$ is one of the most widely reported and widely misused statistics in applied regression, a high R-squared indicates the model closely tracks the observed data's variation but says nothing directly about whether the model's coefficients are unbiased, whether important variables have been omitted, or whether the model would generalize to new data, which is why R-squared should always be interpreted alongside, not instead of, coefficient significance, residual diagnostics, and out-of-sample validation.

• R-squared always increases (or stays exactly the same) as more predictors are added to a model, regardless of whether those predictors have any real relationship with the response, this mechanical property is why adjusted R-squared, which penalizes model complexity, is the standard tool for comparing models with different numbers of predictors.
• A high R-squared does not imply that a regression model's coefficients are unbiased or causally meaningful, a model can fit the observed data extremely well while still suffering from omitted variable bias, reverse causality, or other specification problems that R-squared alone cannot detect.
• In simple linear regression with one predictor, R-squared is exactly equal to the square of the Pearson correlation coefficient between the predictor and the response, this direct relationship breaks down in multiple regression, where R-squared instead reflects the combined explanatory power of all predictors together.