Answer: Will never be greater than the regression R2

## Explanation **Correct Answer: B** - Adjusted R-squared ($\bar{R}^2$) will never be greater than the regression R-squared ($R^2$). ### Key Concepts: 1. **R-squared ($R^2$)**: Measures the proportion of variance in the dependent variable explained by the independent variables. It always increases when adding more variables, even if they're irrelevant. 2. **Adjusted R-squared ($\bar{R}^2$)**: Adjusts for the number of predictors in the model. It penalizes the addition of irrelevant variables. ### Why Option B is Correct: - $\bar{R}^2 = 1 - \frac{(1-R^2)(n-1)}{n-p-1}$ where n = sample size, p = number of predictors - Since $\frac{n-1}{n-p-1} \geq 1$ when $p \geq 0$, $\bar{R}^2 \leq R^2$ - $\bar{R}^2$ can be equal to $R^2$ only when $p=0$ (no predictors) ### Why Other Options are Incorrect: **A: Is always positive** - False. $\bar{R}^2$ can be negative when the model fits worse than a horizontal line (when $R^2$ is very small relative to the number of predictors). **C: Cannot increase when an additional independent variable is incorporated** - False. $\bar{R}^2$ can increase if the new variable significantly improves the model beyond what would be expected by chance. **D: Is always negative** - False. $\bar{R}^2$ is typically positive when the model has explanatory power. ### Practical Implication: Researchers use $\bar{R}^2$ for model selection because it accounts for model complexity, preventing overfitting by penalizing unnecessary variables.

Author: Nikitesh Somanthe