e-Statistics

R Square

The residual sum of squares

$\displaystyle S_{\mbox{error}} = \displaystyle\sum_{i=1}^{n} (Y_i - \hat{\beta}_0 - \hat{\beta}_1 x_i)^2$

and the total sum of squares

$\displaystyle S_{\mbox{total}} = \displaystyle\sum_{i=1}^{n} (Y_i - \bar{Y})^2$

are introduced. They are used to calculate

and the error variance $\hat{\sigma}^2$ .

The data set consists of

The coefficient of determination

$R^2 = 1 - \displaystyle\frac{S_{\mbox{error}}}{S_{\mbox{total}}} =$

takes a value between 0 and 1, and represents the proportion which can be explained by the linear regression. The value

indicates how close the data points are to the regression line as

gets larger, and it is simply the square of the sample correlation coefficient $\hat{\rho}$ .

$\hat{\sigma}^2 = \displaystyle\frac{S_{\mbox{error}}}{n-2} =$

can be obtained as the point estimate of the variance $\sigma^2$ of error terms.