How does Adjusted R^2 differ from R^2 in model evaluation?

How does Adjusted R^2 differ from R^2 in model evaluation?

• A. Adjusted R^2 always increases with more predictors
• B. Adjusted R^2 is the same as AIC
• C. Adjusted R^2 ignores model fit
• D. Adjusted R^2 penalizes model complexity, potentially decreasing when adding irrelevant predictors

Answer: (D)

Adjusted R-squared builds in a penalty for adding more predictors, unlike R-squared which tends to rise merely by including additional variables. R-squared measures how much of the variation in the outcome is explained by the model and will not decrease as you add predictors, even if those predictors don’t actually help. Adjusted R-squared, however, weighs improvements in fit against the cost of model complexity (the number of predictors relative to the sample size). If new predictors genuinely improve the model, adjusted R-squared increases; if they don’t add real explanatory power, the penalty can cause it to decrease. That’s why it can decline when irrelevant predictors are included. It’s also different from AIC and does not ignore model fit—it purposely blends fit with parsimony.

Adjusted R-squared builds in a penalty for adding more predictors, unlike R-squared which tends to rise merely by including additional variables. R-squared measures how much of the variation in the outcome is explained by the model and will not decrease as you add predictors, even if those predictors don’t actually help. Adjusted R-squared, however, weighs improvements in fit against the cost of model complexity (the number of predictors relative to the sample size). If new predictors genuinely improve the model, adjusted R-squared increases; if they don’t add real explanatory power, the penalty can cause it to decrease. That’s why it can decline when irrelevant predictors are included. It’s also different from AIC and does not ignore model fit—it purposely blends fit with parsimony.