Read an R-squared honestly with adjusted R-squared

Question

You fit a model with k = 15 features on n = 120 observations and get R^2 = 0.35. Compute the adjusted R^2 and the R^2 you would expect from pure noise. What do they tell you about the fit?

Accepted Answer

Adjusted R^2 penalizes each regressor by inflating the residual term: R^2{	ext{adj}} = 1 - (1 - R^2)\frac{n-1}{n-k-1} = 1 - 0.65 \cdot \frac{119}{104} = 1 - 0.65(1.144) \approx 0.256. Noise floor. Regressing pure noise on k predictors gives E[R^2] = \frac{k}{n-1} = \frac{15}{119} \approx 0.126. So of your raw R^2 = 0.35, about 0.13 is what any 15 random columns would deliver. Reading it. The adjusted figure 0.256 sits above the noise floor 0.126, so there is some real explanatory power, but the honest number is well below the headline 0.35. Roughly a third of the raw fit is the free lift fro…

Read an R-squared honestly with adjusted R-squared

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