Latent variable multivariate regression modeling

The latent variable multivariate regression (LVMR) model is made up of two sets of variables, X and Y, both of which contain a latent variable structure plus random error. The wide applicability of this model is illustrated in this paper with several real examples. The chemometrics community has developed several empirical methods to estimate the latent structure in this model, including partial least squares regression (PLS) and principal components regression (PCR). However, the majority of the statistical work in this area relies on the standard or reduced rank regression models, thus ignoring the latent variable nature of the X data. Considering methods like PLS and PCR in the context of these models has led to some misleading conclusions. This paper reaffirms the claim made frequently in the chemometrics literature that the reason PLS and PCR have been successful is that they take into account the latent variable structure in the data. It is also shown through several examples that the LVMR model provides the means to model more effectively many datasets in applied science resulting in improved techniques for process monitoring, experimental design and prediction. The focus in this paper is on the general model rather than on parameter estimation methods.