Insights into multivariate calibration using errors-in-variables modeling

A {ital q}-vector of responses, y, is related to a {ital p}-vector of explanatory variables, x, through a causal linear model. In analytical chemistry, y and x might represent the spectrum and associated set of constituent concentrations of a multicomponent sample which are related through Beer`s law. The model parameters are estimated during a calibration process in which both x and y are available for a number of observations (samples/specimens) which are collectively referred to as the calibration set. For new observations, the fitted calibration model is then used as the basis for predicting the unknown values of the new x`s (concentrations) form the associated new y`s (spectra) in the prediction set. This prediction procedure can be viewed as parameter estimation in an errors-in-variables (EIV) framework. In addition to providing a basis for simultaneous inference about the new x`s, consideration of the EIV framework yields a number of insights relating to the design and execution of calibration studies. A particularly interesting result is that predictions of the new x`s for individual samples can be improved by using seemingly unrelated information contained in the y`s from the other members of the prediction set. Furthermore, motivated by this EIV analysis, this result can be extended beyond the causal modeling context to a broader range of applications of multivariate calibration which involve the use of principal components regression.