In simple linear regression, R^2 equals the square of the sample correlation coefficient r. Which option is true?

In simple linear regression, R^2 equals the square of the sample correlation coefficient r. Which option is true?

• A. True
• B. False
• C. Only when standardized
• D. Only when x is centered

Answer: (A)

In simple linear regression, the fraction of the outcome’s variance explained by the predictor (R^2) matches the strength of the linear relationship between X and Y measured by the Pearson correlation r, squared. The algebra links these two: R^2 = SSR/SST, where SSR is the regression sum of squares and SST is the total sum of squares. With b1 = SXY/SXX, SSR = b1^2 SXX = SXY^2 / SXX, and SST = SYY. The squared correlation is r^2 = SXY^2 / (SXX SYY). Dividing SSR by SST gives SSR/SST = (SXY^2 / SXX) / SYY = SXY^2 / (SXX SYY) = r^2. Hence R^2 = r^2 in simple linear regression. This equivalence relies on there being only one predictor (and an intercept) and does not necessarily extend in the same way to multiple regression.

In simple linear regression, the fraction of the outcome’s variance explained by the predictor (R^2) matches the strength of the linear relationship between X and Y measured by the Pearson correlation r, squared. The algebra links these two: R^2 = SSR/SST, where SSR is the regression sum of squares and SST is the total sum of squares. With b1 = SXY/SXX, SSR = b1^2 SXX = SXY^2 / SXX, and SST = SYY. The squared correlation is r^2 = SXY^2 / (SXX SYY). Dividing SSR by SST gives SSR/SST = (SXY^2 / SXX) / SYY = SXY^2 / (SXX SYY) = r^2. Hence R^2 = r^2 in simple linear regression. This equivalence relies on there being only one predictor (and an intercept) and does not necessarily extend in the same way to multiple regression.