Proportionate reduction of error (PRE) is the gain in precision of predicting dependent variable

y

from knowing the independent variable

x

(or a collection of multiple variables). It is a

goodness of fit The goodness of fit of a statistical model describes how well it fits a set of observations. Measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under the model in question. Such measure ...

measure of statistical models, and forms the mathematical basis for several correlation coefficients. The summary statistics is particularly useful and popular when used to evaluate models where the dependent variable is binary, taking on values .

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

, the arithmetic mean of the

y

-data. The total prediction error would be

E_1 = \sum_^n

. If, however,

x

and a function relating

y

x

are known, for example a straight line

\hat_i = a + b x_i

, then the prediction error becomes

E_2 = \sum_^n

. The

coefficient of determination In statistics, the coefficient of determination, denoted ''R''2 or ''r''2 and pronounced "R squared", is the proportion of the variation in the dependent variable that is predictable from the independent variable(s). It is a statistic used ...

then becomes

r^2 = \frac = 1 - \frac

and is the fraction of variance of

y

that is explained by

x

. Its square root is Pearson's

product-moment correlation In statistics, the Pearson correlation coefficient (PCC, pronounced ) ― also known as Pearson's ''r'', the Pearson product-moment correlation coefficient (PPMCC), the bivariate correlation, or colloquially simply as the correlation coefficient ...

r

. There are several other correlation coefficients that have PRE interpretation and are used for variables of different scales: {, class="wikitable" , - ! 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 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. , yes , - , cardinal , binary , point biserial correlation , yes

References

Statistical forecasting