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

We 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. When y has been more or less systematically selected in the calibration in order to achieve increased variation (overdispersion), a practical device to increase precision, this leads to biased coefficients in the predictor, possible to see when observed 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.