What is the relationship of Adjusted R-squared to R-squared?

Adjusted R-squared provides a more accurate measure of goodness of fit when comparing statistical models, especially when the number of predictors varies. While R-squared can only increase or stay the same when additional predictors are included in a model, Adjusted R-squared adjusts for the number of predictors. This means that if adding a predictor does not improve the model significantly, the Adjusted R-squared can actually decrease.

The inclusion of the adjustment for the number of predictors is crucial because it accounts for the complexity of the model, helping to prevent overfitting. Consequently, it's possible for Adjusted R-squared to be lower than R-squared, reflecting that the additional predictors may not significantly contribute to the explanatory power of the model.

Understanding this distinction is fundamental for evaluating model performance and ensuring that models are both adequately complex and appropriately simple, making Adjusted R-squared a valuable tool for model selection in regression analysis.