R-squared vs. adjusted R-squared

Two common measures of how well a model fits to data are $R^2$ (the coefficient of determination) and the adjusted $R^2$. The former measures the percentage of the variability in the response variable that is explained by the model. To compute this, we define $$ R^2 = 1 - \frac{SSE}{SST} \,, $$ where $SSE$ and $SST$ are the sum of the squared residuals, and the total sum of the squares, respectively. One issue with this measure is that the $SSE$ can only decrease as new variable are added to the model, while the $SST$ depends only on the response variable and therefore is not affected by changes to the model. This means that you can increase $R^2$ by adding any additional variable to your model—even random noise.

The adjusted $R^2$ includes a term that penalizes a model for each additional explanatory variable (where $p$ is the number of explanatory variables). $$ R^2_{adj} = 1 - \frac{SSE}{SST} \cdot \frac{n-1}{n-p-1} \,, $$

We can see both measures in the output of the summary() function on our model object.

Use summary() to compute $R^2$ and adjusted $R^2$ on the model object called mod.
Use mutate() and rnorm() to add a new variable called noise to the mario_kart data set that consists of random noise. Save the new dataframe as mario_kart_noisy.
Use lm() to fit a model that includes wheels, cond, and the random noise term.
Use summary() to compute $R^2$ and adjusted $R^2$ on the new model object. Did the value of $R^2$ increase? What about adjusted $R^2$?

Take Hint (-30 XP)

Exercise

R-squared vs. adjusted R-squared

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