Proportionate reduction of error

Proportionate reduction of error (PRE) is the gain in precision of predicting dependent variable from knowing the independent variable (or a collection of multiple variables). It is a goodness of fit measure of statistical models, and forms the mathematical basis for several correlation coefficients.[1] The summary statistics is particularly useful and popular when used to evaluate models where the dependent variable is binary, taking on values {0,1}.

Example

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If both   and   vectors have cardinal (interval or rational) 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  . Its 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[2] yes
ordinal nominal Freeman's θ[3] yes
cardinal nominal η [4] no
ordinal binary, ordinal Wilson's e [5] yes
cardinal binary point biserial correlation yes

References

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  1. ^ Freeman, L.C.: Elementary applied statistics, New, York, London, Sidney (John Wiley and Sons) 1965
  2. ^ Guttman, L. The quantification of a class of attributes: A theory and method of scale construction. In: The prediction of personal adjustment. Horst, P.; Wallin, P.; Guttman, L. et al. (eds.) New York (Social Science Research Council) 1941, pp. 319–348.
  3. ^ Freeman, L.C.: Elementary applied statistics, New, York, London, Sidney (John Wiley and Sons) 1965
  4. ^ anonymous. Fehlerreduktionsmaße [web-site, accessed 2017-07-29]. 2016. Available from: https://de.wikipedia.org/wiki/Fehlerreduktionsma%C3%9Fe#.CE.B72.
  5. ^ Freeman, L.C.: Order-based statistics and monotonicity: A family of ordinal measures of association. J. Math. Sociol. 1986, vol. 12, no. 1, pp. 49–69. Available from: http://moreno.ss.uci.edu/41.pdf.