Small sample and selection bias effects in calibration under latent factor regression models

Abstract

AbstractWe study bias of predictors when a multivariate calibration procedure has been applied to relate a scalar y (concentration of an analyte, say) to a vector x (spectral intensities, say). The model for data is assumed to be of latent factor regression type, with multiple regression models and errors‐in‐variables models as special cases. The calibration procedures explicitly studied are OLSR, PLSR and PCR. Often a practical device to increase precision in the calibration is to select y more or less systematically, in order to achieve increased variation (overdispersion). However, this leads to biased coefficients in the predictor, possible to see when observed it y is regressed on predicted ŷ (x) for a separate validation set. Another bias effect is a sample size effect, increasing with reduced calibration sample size and with increasing dimension of x (absent when x is univariate). Formulae are given for these bias effects, both separately and in combination, and the formulae are illustrated and compared with simulation results. As a qualitative example, PLSR and PCR are less sensitive than OLSR to small samples, but equally sensitive to selection. Copyright © 2007 John Wiley & Sons, Ltd.