Unlike $r^2$, adjusted $r^2$ explicitly accounts for which of the following in regression analysis?

• A. The scale of the dependent variable
• B. The number of predictors in the model
• C. The correlation between independent variables
• D. The presence of outliers in the data

Answer: (B)

Verified step by step guidance

1. Understand that \(r^2\) (coefficient of determination) measures the proportion of variance in the dependent variable explained by the independent variables in a regression model.
2. Recognize that \(r^2\) tends to increase or at least not decrease when more predictors are added to the model, regardless of whether those predictors are meaningful.
3. Learn that adjusted \(r^2\) modifies the \(r^2\) value by incorporating a penalty for adding more predictors, thus adjusting for the number of predictors in the model.
4. Recall the formula for adjusted \(r^2\): \[\text{Adjusted } r^2 = 1 - \left(1 - r^2\right) \times \frac{n - 1}{n - p - 1}\] where \(n\) is the sample size and \(p\) is the number of predictors.
5. Conclude that adjusted \(r^2\) explicitly accounts for the number of predictors in the model, helping to prevent overfitting by penalizing unnecessary variables.