Proportionate reduction of error
Proportionate reduction of error is the gain in precision of predicting a dependent variable from knowing the independent variable and forms the mathematical basis for several correlation coefficients.
Example
If both and vectors have cardinal scale, then without knowing, the best predictor for an unknown would be, the arithmetic mean of the -data. The total prediction error would be .If, however, and a function relating to are known, for example a straight line, then the prediction error becomes. The coefficient of determination then becomes and is the fraction of variance of that is explained by. It's square root is Pearson's product-moment correlation.
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 | no |
| ordinal | binary, ordinal | Wilson's e | yes |
| cardinal | binary | point biserial correlation | yes |