A statistical framework for multivariate latent variable regression methods based on maximum likelihood

Abstract

A statistical framework is developed to contrast methods used for parameter estimation for a latent variable multivariate regression (LVMR) model. This model involves two sets of variables, X and Y, both with multiple variables and sharing a common latent structure with additive random errors. The methods contrasted are partial least squares (PLS) regression, principal component regression (PCR), reduced rank regression (RRR) and canonical co-ordinate regression (CCR). The framework is based on a constrained maximum likelihood analysis of the model under assumptions of multivariate normality. The constraint is that the estimates of the latent variables are restricted to be linear functions of the X variables, which is the form of the estimates for the methods being contrasted. The resulting framework is a continuum regression that goes from RRR to PCR depending on the ratio of error variances in the X and Y spaces. PLS does not arise as a member of the continuum; however, the method does offer some insight into why PLS would work well in practice. The constrained maximum likelihood result is also compared with the unconstrained maximum likelihood analysis to investigate the impact of the constraint. The results are illustrated on a simulated example. Copyright © 1999 John Wiley & Sons, Ltd.