Question

Which of the following is NOT correct regarding R-squared ($R^2$) and Adjusted R-squared ($\overline{R^2}$)?

• A. $R^2$ is a scale invariant statistic.
• B. $\overline{R^2}$ is always positive.
• C. $R^2$ tends to increase if we add an additional explanatory variable.
• D. $\overline{R^2} = 1-(1-R^2) (\frac{n-1}{n-k})$ where k is the number of parameters and n is the number of observations.

Answer 1

The Correct Option is B

To determine which statement is NOT correct regarding R-squared (\(R^2\)) and Adjusted R-squared (\(\overline{R^2}\)), let's evaluate each option:

1. \(R^2\) is a scale invariant statistic.
   - \(R^2\) measures the proportion of variance in the dependent variable that is predictable from the independent variable(s). It is scale invariant, meaning it does not change with different scales of measurement. This statement is correct.
2. \(\overline{R^2}\) is always positive.
   - Adjusted R-squared, represented as \(\overline{R^2}\), adjusts the \(R^2\) value by the number of predictors in the model. It can be negative if the model does not explain the data well. Therefore, this statement is incorrect.
3. \(R^2\) tends to increase if we add an additional explanatory variable.
   - \(R^2\) typically increases or remains the same when additional explanatory variables are added regardless of the quality of the model, due to increased fit to the existing data. This statement is correct.
4. \(\overline{R^2} = 1-(1-R^2) \left(\frac{n-1}{n-k}\right)\) where \(k\) is the number of parameters and \(n\) is the number of observations.
   - This is the correct formula for calculating Adjusted R-squared. The formula accounts for the number of predictors in the model, making it more reliable than \(R^2\) when comparing models with different numbers of predictors. This statement is correct.

After evaluating all the options, we conclude that the statement "\(\overline{R^2}\) is always positive." is NOT correct. Thus, the correct answer is the statement regarding Adjusted R-squared being always positive.