The correlation test is the slope test

Topic: Linear regression · more questions on this topic
‹ Where is the prediction interval narrowest?

Show that \widehat\beta_1/S_1=R\sqrt{n-2}/\sqrt{1-R^2}, where R is the sample correlation. Use R^2=\widehat\beta_1^2S_{xx}/S_{YY} and (n-2)S^2=S_{YY}(1-R^2).

Worked answer (click to show)

Start from \widehat\beta_1/S_1=\widehat\beta_1\sqrt{S_{xx}}/S. Square it and use the two identities:

\frac{\widehat\beta_1^2S_{xx}}{S^2}=\frac{R^2S_{YY}}{S_{YY}(1-R^2)/(n-2)}=\frac{(n-2)R^2}{1-R^2}.

\widehat\beta_1 and R have the same sign, both being S_{xY} divided by something positive. Taking square roots, \widehat\beta_1/S_1=R\sqrt{n-2}/\sqrt{1-R^2}. Testing \beta_1=0 in the fixed-x model and testing \rho=0 in the bivariate normal model use the same statistic and the same t_{n-2} reference distribution.