Answer: The adjusted R² is always less than the R²

## Explanation ### Understanding R² and Adjusted R² **R² (Coefficient of Determination):** - Measures the proportion of variance in the dependent variable explained by the independent variables - Always increases when adding more independent variables, even if they're not statistically significant - Range: 0 to 1 (or 0% to 100%) **Adjusted R²:** - Modified version of R² that accounts for the number of predictors in the model - Penalizes the addition of irrelevant variables - Can be negative when the model performs worse than a simple mean model - Formula: \(\bar{R}^2 = 1 - \left(\frac{n - 1}{n - k - 1}\right)(1 - R^2)\) where n = number of observations, k = number of independent variables ### Why Option C is Correct 1. **Adjusted R² is always less than or equal to R²:** - From the formula: \(\bar{R}^2 = 1 - \left(\frac{n - 1}{n - k - 1}\right)(1 - R^2)\) - Since \(\frac{n - 1}{n - k - 1} > 1\) when k > 0 (more than zero predictors) - Therefore, \(\bar{R}^2 < R^2\) when k > 0 - Only when k = 0 (no predictors), they are equal 2. **Why other options are incorrect:** - **A:** False - Adjusted R² is always less than or equal to R², never greater - **B:** False - Adjusted R² can be negative when the model is very poor - **D:** False - Adjusted R² decreases when adding irrelevant variables that don't improve the model ### Practical Implications - **Model Selection:** Use adjusted R² when comparing models with different numbers of predictors - **Overfitting Prevention:** Adjusted R² helps avoid adding unnecessary variables - **Interpretation:** A model with higher adjusted R² is generally better, even if its R² is slightly lower ### Example If you have: - R² = 0.85 with 5 predictors - R² = 0.86 with 10 predictors The model with 10 predictors might have a lower adjusted R² if the additional 5 predictors don't contribute meaningfully to explaining the variance.

Author: Nikitesh Somanthe