Validation and verification of regression in small data sets

Abstract

Four different methods of using small data sets in multivariate modelling are compared w.r.t. predictive precision in the long-run. The modelling in this case concerns multivariate calibration: ŷ= f( X). The study consists of a Monte Carlo simulation within a large data base of real data; X=NIR reflectance spectra and y=protein percentage, measured in 922 whole maize plant samples. Small data sets (40–120 objects) were repeatedly selected at random from the data base, each time simulating the situation of having only a small set of samples available for estimating, optimizing and assessing the calibration model. The `true' apparent prediction error was each time controlled in the remaining data base. This was replicated 100 times in order to study the statistical performance of the four different validation methods. In each Monte Carlo replicate, the splitting of the available data set into calibration set and test set was compared to full cross validation. The results demonstrated that removing samples from an already limited set of available samples to an independent VALIDATION TEST SET seriously reduced the predictive performance of the calibrated models, and at the same time gave uncertain, systematically over-optimistic assessment of the models' predictive performance. Full CROSS VALIDATION gave improved predictive performance, and gave only slightly over-optimistic assessment of this predictive performance. Further removal of even more of the available samples for use in an independent VERIFICATION TEST SET gave in-the-long-run correct, although uncertain estimates of the predictive performance of the calibrated models, but this performance level had seriously deteriorated. Alternative verification of the model's predictive performance by the method of CROSS VERIFICATION gave results very similar to those of the cross validation. These results from real data correspond closely to previous findings for artificially simulated data. It appears that full cross validation is superior to both the use of independent validation test set and independent verification test set.