Proportionate reduction of error

Proportionate reduction of error (PRE) is the gain in precision of predicting a dependent variable $y$ from knowing the independent variable $x$ and forms the mathematical basis for several correlation coefficients.

Example

If both $x$ and $y$ vectors have cardinal (interval or rational) scale, then without knowing $x$, the best predictor for an unknown $y$ would be ${\bar {y}}$, the arithmetic mean of the $y$-data. The total prediction error would be $E_{1}=\sum _{i=1}^{n}{(y_{i}-{\bar {y}})^{2}}$ .

If, however, $x$ and a function relating $y$ to $x$ are known, for example a straight line ${\hat {y}}_{i}=a+bx_{i}$, then the prediction error becomes $E_{2}=\sum _{i=1}^{n}{(y_{i}-{\hat {y}})^{2}}$. The coefficient of determination then becomes $r^{2}={\frac {E_{1}-E_{2}}{E_{1}}}=1-{\frac {E_{2}}{E_{1}}}$ and is the fraction of variance of $y$ that is explained by $x$. It's square root is Pearson's product-moment correlation $r$.

There are several other correlation coefficients that have PRE interpretation and are used for variables of different scales:

predict	from	coefficient	symmetric
nominal, binary	nominal, binary	Guttman's λ	yes
ordinal	nominal	Freeman's θ	yes
cardinal	nominal	η$^{2}$	no
ordinal	binary, ordinal	Wilson's e	yes
cardinal	binary	point biserial correlation	yes