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.