PRESS statistic

In statistics, the predicted residual error sum of squares (PRESS) statistic is a form of cross-validation used in regression analysis to provide a summary measure of the fit of a model to a sample of observations that were not themselves used to estimate the model. It is calculated as the sums of squares of the prediction residuals for those observations.^[1][2][3]

A fitted model having been produced, each observation in turn is removed and the model is refitted using the remaining observations. The out-of-sample predicted value is calculated for the omitted observation in each case, and the PRESS statistic is calculated as the sum of the squares of all the resulting prediction errors:^[4]

$\operatorname{PRESS} =\sum_{i=1}^n (y_i - \hat{y}_{i, -i})^2$

Given this procedure, the PRESS statistic can be calculated for a number of candidate model structures for the same dataset, with the lowest values of PRESS indicating the best structures. Models that are over-parameterised (over-fitted) would tend to give small residuals for observations included in the model-fitting but large residuals for observations that are excluded.

References

1. "Statsoft:StatSoft.com Electronic Statistics Textbook - Statistics Glossary". Retrieved August 2012. External link in |title= (help)
2. Allen, D. M. (1974), "The Relationship Between Variable Selection and Data Augmentation and a Method for Prediction," Technometrics, 16, 125–127
3. Tarpey, Thaddeus (2000) "A Note on the Prediction Sum of Squares Statistic for Restricted Least Squares", The American Statistician, Vol. 54, No. 2, May, pp. 116–118
4. "R Graphical Manual:Allen's PRESS (Prediction Sum-Of-Squares) statistic, aka P-square". Retrieved August 2012.