Estimation of influential points in any data set from coefficient of determination and its leave-one-out cross-validated counterpart

Abstract

Coefficient of determination (R (2)) and its leave-one-out cross-validated analogue (denoted by Q (2) or R cv (2) ) are the most frequantly published values to characterize the predictive performance of models. In this article we use R (2) and Q (2) in a reversed aspect to determine uncommon points, i.e. influential points in any data sets. The term (1 - Q (2))/(1 - R (2)) corresponds to the ratio of predictive residual sum of squares and the residual sum of squares. The ratio correlates to the number of influential points in experimental and random data sets. We propose an (approximate) F test on (1 - Q (2))/(1 - R (2)) term to quickly pre-estimate the presence of influential points in training sets of models. The test is founded upon the routinely calculated Q (2) and R (2) values and warns the model builders to verify the training set, to perform influence analysis or even to change to robust modeling.