Explanation

Adjusted R² is a modified version of R² that penalizes for adding unnecessary variables to the model.

Key properties:

• Can be negative: When the model performs worse than using the mean as predictor, adjusted R² can be negative (option A is correct)
• Penalizes overfitting: It adjusts for the number of predictors, unlike regular R² which always increases with more variables
• Not for bias detection: It doesn't indicate whether predictions are biased (option B is incorrect)
• Not a significance test: F-test or t-tests are used for significance testing, not adjusted R² (option C is incorrect)

Formula: $\text{Adjusted } R^2 = 1 - \left[\frac{(1-R^2)(n-1)}{n-k-1}\right]$ Where n = sample size, k = number of predictors

When the model is very poor, this formula can yield negative values.