Question

Which of the following statements describe(s) the relationship between \( R^2 \) and adjusted \( R^2 \) (denoted by \( \bar{R}^2 \))?

• A. If \( R^2 = 1 \), then \( \bar{R}^2 = 1 \)
• B. If \( R^2 = 0 \), then \( \bar{R}^2 \) can be negative
• C. If \( R^2 = 1 \), then \( \bar{R}^2 = 0 \)
• D. If \( R^2 = 0 \), then \( \bar{R}^2 = 1 \)

Hint:

Adjusted \( R^2 \) accounts for the number of predictors in the model and can be negative if the model does not improve the fit.

Answer 1

The Correct Option is B

Step 1: Understand the relationship between \( R^2 \) and adjusted \( R^2 \).
- \( R^2 \) measures the proportion of the variance in the dependent variable that is explained by the independent variables in the regression model. However, \( R^2 \) always increases when more independent variables are added, even if those variables are not meaningful.
- Adjusted \( R^2 \) (denoted \( \bar{R}^2 \)) adjusts for the number of explanatory variables in the model. It can decrease if the new variable does not improve the model significantly.
Step 2: Analyze the options.
- Option (A) is incorrect because while \( R^2 = 1 \) indicates a perfect fit, the adjusted \( R^2 \) can also be 1 if the model is perfectly fit, but adding extra variables that don't contribute might decrease it.
- Option (B) is correct because when \( R^2 = 0 \), the model explains no variance, and the adjusted \( R^2 \) can become negative, which reflects that the model is worse than a simple mean model.
- Option (C) is incorrect because if \( R^2 = 1 \), then \( \bar{R}^2 \) will also be 1, as no adjustments are needed.
- Option (D) is incorrect because \( R^2 = 0 \) implies no explanatory power, and \( \bar{R}^2 \) would likely be negative, not 1.
Final Answer: \[ \boxed{\text{If } R^2 = 0, \text{ then } \bar{R}^2 \text{ can be negative.}} \]