Question

In the multiple regression model, the adjusted $R^{2}$:

• A. Cannot be negative.
• B. Will never be greater than $R^{2}$.
• C. Equals the squares of the correlation coefficient, r.
• D. Cannot decrease when an additional explanatory variable is added.

Hint:

- $R^{2}$ can only increase or remain constant as variables are added.
- Adjusted $R^{2}$ can decrease if added variables are statistically insignificant.
- Adjusted $R^{2}$ can be negative, whereas regular $R^{2}$ is bounded between 0 and 1.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The coefficient of determination, $R^{2}$, measures the proportion of variance in the dependent variable explained by the explanatory variables.
However, $R^{2}$ always increases or stays constant when new explanatory variables are added, even if they are irrelevant.
To address this, the adjusted $R^{2}$ (denoted as $\bar{R}^{2}$) incorporates a penalty for adding variables that do not contribute significantly to the model's explanatory power.
Key Formula or Approach:
The relationship between Adjusted $R^{2}$ ($\bar{R}^{2}$) and $R^{2}$ is given by:
\[ \bar{R}^{2} = 1 - (1 - R^{2}) \frac{n - 1}{n - p - 1} \]
where:
- $n$ is the number of observations (sample size).
- $p$ is the number of explanatory variables (predictors).

Step 2: Detailed Explanation:
Let us analyze the properties of the adjusted $R^{2}$ based on this formula:
1. Comparison with $R^{2$:}
Since $p \ge 1$, we have $n - p - 1 < n - 1$.
This implies that the ratio $\frac{n - 1}{n - p - 1}$ is strictly greater than 1.
Consequently, the term $(1 - R^{2}) \frac{n - 1}{n - p - 1}$ is greater than $(1 - R^{2})$.
Therefore, $\bar{R}^{2}$ will always be less than or equal to $R^{2}$ ($\bar{R}^{2} \le R^{2}$).
Thus, Adjusted $R^{2}$ will never be greater than $R^{2}$ (Option B is correct).
2. Possibility of being negative:
If $R^{2}$ is very low and the model contains many irrelevant variables, the term $(1 - R^{2}) \frac{n - 1}{n - p - 1}$ can exceed 1, making $\bar{R}^{2}$ negative. Thus, Option A is incorrect.
3. Relationship with Additional Variables:
Adjusted $R^{2}$ can decrease if the marginal improvement in $R^{2}$ from adding a new variable is too small to compensate for the reduction in degrees of freedom (increase in $p$). Thus, Option D is incorrect.

Step 3: Final Answer:
Therefore, adjusted $R^{2}$ will never be greater than $R^{2}$.