Why is the adjusted R2 more informative than the R2?

Question

BlackTom AI · Accepted Answer

To evaluate why adjusted R2 is more informative than R2, let's examine what each metric does with new predictors.
Option A: 'Because the adjusted R Squared adjusts for potential outliers.' This is not accurate. Adjusted R2 does not specifically penalize for outliers; it adjusts for the number of predictors to account for overfitting. Outliers are not the central factor here, so this statement misattributes the purpose of adjusted R2.
Option B: 'Because the adjusted R Squared can never go down.' This is misleading about adjusted R2. In fact, adjusted R2 can decrease when adding a useless or irrelevant variable because the penalty for extra parameters may outweigh any slight improvement in fit. Therefore, this claim is incorrect.
Option C: 'Because the R Squared can't go down even when a useless variable is added.' This is a true property of R2: it does not decrease with the addition of new predictors, regardless of usefulness. However, this statement is describing R2, not adjusted R2, and it highlights why adjusted R2 provides more information: R2 can stay the same or increase, masking overfitting, while adjusted R2 can decrease to reflect unnecessary variables.
Option D: 'Because there is a penalty when a new, non-useful variable is added.' This captures the essence of adjusted R2: it introduces a penalty for adding extra predictors that do not improve the model fit, helping to guard against overfitting by potentially lowering the statistic when a useless variable is included.
In sum, A misidentifies the rationale (outliers aren't the central issue), B and C describe properties of R2 rather than adjusted R2, and D accurately describes the key mechanism that makes adjusted R2 more informative than R2 when comparing models with different numbers of predictors.

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