What is adjusted R-squared?

Adjusted R-squared (adjusted R²) is a regression fit metric that modifies R² to account for the number of predictors in the model.

Unlike ordinary R², which cannot decrease when additional predictors are added to an ordinary least-squares model on the same training data, adjusted R² can decrease if a new predictor contributes too little improvement.

Formula

Adjusted R² = 1 - (1 - R²) × (n - 1) / (n - p - 1)

where:

• R² is the ordinary coefficient of determination
• n is the number of observations
• p is the number of predictors, excluding the intercept

This standard formula assumes an intercept and requires n > p + 1.

Why is the adjustment needed?

Adding more predictors may slightly reduce training error even when those predictors provide little real value.

Adjusted R² imposes a penalty for model complexity, helping compare regression models with different numbers of predictors.

Simple example

Suppose a model has:

• R² = 0.80
• n = 100 observations
• p = 5 predictors

Then:

Adjusted R² = 1 - 0.20 × 99 / 94 ≈ 0.789

The adjusted value is slightly lower because the model uses five predictors.

How should it be interpreted?

• Adjusted R² increases: a new predictor improves fit enough to justify the extra complexity under this metric.
• Adjusted R² decreases: the improvement is too small relative to the complexity penalty.
• Adjusted R² can be negative: the fitted model performs poorly relative to the mean baseline after adjustment.

Important limitation

Adjusted R² is mainly useful for comparing linear regression models fitted to the same target and observations. A higher adjusted R² does not guarantee better predictions on unseen data; use validation or test-set metrics such as MAE and RMSE to check generalization.

In short, adjusted R² modifies R² to discourage unnecessary predictors, making it more informative for comparing regression models of different complexity.