R-squared corrected for the number of predictors: 1 − (1 − R²)(n − 1)/(n − p − 1), with n rows and p predictors. Unlike R², it only rises when a new predictor improves the fit by more than chance would, so it can fall when you add a useless one.
Card 1 of 259. Answer: R-squared corrected for the number of predictors: 1 − (1 − R²)(n − 1)/(n − p − 1), with n rows and p predictors. Unlike R², it only rises when a new predictor improves the fit by more than chance would, so it can fall when you add a useless one.
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In least squares, adding a predictor never lowers the training R², even if the predictor is pure noise, because the model can always set its coefficient to zero. Adjusted R² charges for each predictor through the degrees of freedom, n − p − 1, so it measures fit per parameter spent. It can be negative when the model explains less than the penalty costs.
Example: a model with 5 predictors on 50 rows has R² = 0.60, so adjusted R² = 1 − 0.40 × 49/44 ≈ 0.555. Add 5 noise columns and R² might creep up to 0.62, but adjusted R² falls to 1 − 0.38 × 49/39 ≈ 0.523, flagging that the new columns didn't earn their place.
It is still an in-sample number, so for choosing between models, cross-validated error or information criteria such as AIC and BIC are better guides. The gap between R² and adjusted R² is a quick warning sign of too many predictors for the amount of data, and it shrinks as n grows relative to p.
