R2

R-squared (R2) is a statistic that explains the amount of variance accounted for in the relationship between two (or more) variables. Sometime R2 is called the coefficient of determination, and it is given as the square of a correlation coefficient.

[Page 1340]Given paired variables $(X_i,Y_i)$, a linear model that explains the relationship between the variables is given by

$$Y={\beta }_0+{\beta }_{1}X+e,$$

where e is a mean zero error. The parameters of the linear model can be estimated using the least squares method and denoted by ${\widehat{\beta }}_0$ and ${\widehat{\beta }}_1$, respectively. The parameters are estimated by minimizing the sum of squared residuals between variable $Y_i$ and the model ${\beta }_0+{\beta }_1{X}_i$, that is,

$$({\widehat{\beta }}_0,{\widehat{\beta }}_1)=\underset{{\beta }_{\text{0}},{\beta }_1}{\text{argmin}}{(Y_i-{\beta }_0+{\beta }_1{X}_i)}^2.$$

It can be shown that the least squares estimations are

$${\widehat{\beta }}_0=\overline{Y}-\overline{X}\frac{S_{xy}}{S_{xx}}\text{and}{\widehat{\beta }}_1=\frac{S_{xy}}{S_{xx}},$$

where the sample cross-covariance $S_{xy}$ is defined as

$$S_{xy}=\frac{1}{n}{\displaystyle \sum _{i=1}^n(X_i-\overline{X})(Y_i-\overline{Y})}=\overline{XY}-\overline{X}\overline{Y}.$$

Statistical packages such as SAS, SPLUS, ...