13.5 R-squared, and Adjusted R-squared

• $R^2$ is the Coeficient of Determination. It represents the proportion of variation in $y$ (about its mean) explained by the multiple linear regression model with predictors, $x_1,x_2, \ldots$

$$R^2=\frac{SSR}{SSTO}=1-\frac{SSE}{SSTO}$$ - $R^2$ always increases (or stays the same) as more predictors are added to a multiple linear regression model, even if the predictors added are unrelated to the response variable. Thus, by itself, $R^2$ cannot be used to help us identify which predictors should be included in a model and which should be excluded.

• $$\text{Adjusted-} R^2 = 1-\left(\frac{n-1}{n-(k+1)}\right)(1-R^2)$$

• $\text{Adjusted-} R^2$, does not necessarily increase as more predictors are added, and can be used to help us identify which predictors should be included in a model and which should be excluded. Simply stated, when comparing two models used to predict the same response variable, we generally prefer the model with the higher value of $\text{Adjusted-}R^2.$