Why does adjusted R-squared matter more than R-squared when comparing models?

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AcadiFi · Accepted Answer

R-squared measures the fraction of total variation in the dependent variable explained by the regressors. Its formula is 1 - SSE/SST. Because adding any regressor can only reduce SSE (or leave it unchanged), R-squared is monotonically non-decreasing in the number of predictors. This creates an incentive to overfit. Adjusted R-squared corrects this by penalizing additional parameters. The formula is 1 - [(1-R^2)(n-1)/(n-k-1)], where n is the sample size and k is the number of slope coefficients. When you add a variable, k increases and the penalty term grows. The adjusted measure rises only if the new variable improves fit more than the penalty costs. Imagine analyst Priya Ramanujan is modeling Fendleton Industries stock returns. With three predictors her R-squared is 0.42 and adjusted R-squared is 0.40. She adds two dummy variables for calendar effects. R-squared climbs to 0.44 but adjusted R-squared falls to 0.38. The signal is clear: the dummies are not pulling their weight. ```mermaid flowchart LR A[Add variable] --> B{SSE
drops
enough?} B -->|Yes| C[Adjusted R^2
rises] B -->|No| D[Adjusted R^2
falls] ``` Use adjusted R-squared, information criteria like AIC and BIC, and out-of-sample testing together. No single metric should drive model selection.

Why does adjusted R-squared matter more than R-squared when comparing models?

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