2.10 The coefficient of determination
Definition 2.10.1. The coefficient of determination is \[ R^2 = \frac {SS_R}{SS_T} = 1 - \frac {SS_{\mathrm {Res}}}{SS_T} , \] the proportion of the variation in \(y\) that is explained by the regression on \(x\).
By Theorem 2.9.1, \(SS_R\) and \(SS_{\mathrm {Res}}\) are both non-negative and add up to \(SS_T\), so \(0 \le R^2 \le 1\). \(R^2 = 1\) when every residual is zero, so the points lie exactly on a line, and \(R^2 = 0\) when \(\hat \beta _1 = 0\), so the fitted line is flat.
For the advertising data, \(R^2 = 4.9/6 = 0.8167\): about \(82\%\) of the variation in sales is explained by advertising. For the yield data of Example 2.2.7, \(R^2 = 571.4286/573.3333 = 0.9967\).
Remark. \(R^2\) must be read with care.
- 1.
- A large \(R^2\) does not mean a straight line is the right model. Points on a gentle curve can give \(R^2\) above \(0.9\); the residual plots of a later section reveal the curvature.
- 2.
- \(R^2\) depends on the spread of the \(x_i\): since \(E(SS_R) = \sigma ^2 + \beta _1^2 S_{xx}\), spreading the \(x\) values out increases \(R^2\) even when the model and \(\sigma ^2\) are unchanged.
- 3.
- \(R^2\) says nothing about predictions outside the range of the data.
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