R-Squared

Study guide: Exam MAS-I

R-Squared (R^2), also called the coefficient of determination, measures the proportion of total variance in the response variable Y that is explained by the model.

R^2 = 1 - \frac{\text{SS}_\text{Res}}{\text{SS}_\text{Tot}}

= 1 - \frac{\sum(y_i - \hat{y}_i)^2}{\sum(y_i - \bar{y})^2}

• For a least-squares fit that includes an intercept, R^2 \in [0, 1]; a value of 1 means the model explains all variability; 0 means it explains none. Fitted without an intercept (Regression Through the Origin), the formula above can be negative
• R^2 always increases (or stays the same) when predictors are added, even if they are irrelevant
• Adjusted R^2 corrects for this by penalizing for the number of predictors p: R^2_{\text{adj}} = 1 - \frac{\text{SS}_\text{Res}/(n-p-1)}{\text{SS}_\text{Tot}/(n-1)}
• In Generalized Linear Models, the analogue is the pseudo-R^2 based on Deviance: R^2_D = 1 - D/D_0 where D_0 is the null deviance
• R^2 measures in-sample fit only and does not directly assess predictive performance on new data

Computing R-Squared for a Linear Regression A simple regression of claim severity on vehicle age gives \text{SS}_\text{Tot} = 10{,}000 and \text{SS}_\text{Res} = 3{,}500. Calculate R^2.

Answer R^2 = 1 - \frac{3{,}500}{10{,}000} = 1 - 0.35 = 0.65 Vehicle age explains 65% of the variance in claim severity.

Adjusted R-Squared Penalizing for Additional Predictors With n = 50 observations, R^2 = 0.65 for a model with p = 5 predictors. Compute R^2_{\text{adj}}.

Answer R^2_{\text{adj}} = 1 - \frac{(1-0.65)(50-1)}{50-5-1} = 1 - \frac{0.35 \times 49}{44} = 1 - \frac{17.15}{44} = 1 - 0.390 = 0.610