GH Biplot in Reduced-Rank Regression Based on Partial Least Squares

Abstract

One of the challenges facing statisticians is to provide tools to enable researchers to interpret and present theirdata and conclusions in ways easily understood by the scientific community. One of the tools available for this purpose is amultivariate graphical representation called reduced rank regression biplot. This biplot describes how to construct a graphicalrepresentation in nonsymmetric contexts such as approximations by least squares in multivariate linear regression models ofreduced rank. However multicollinearity invalidates the interpretation of a regression coefficient as the conditional effect of aregressor, given the values of the other regressors, and hence makes biplots of regression coefficients useless. So it was, in thesearch to overcome this problem, Alvarez and Griffin [1], presented a procedure for coefficient estimation in a multivariateregression model of reduced rank in the presence of multicollinearity based on PLS (Partial Least Squares) and generalizedsingular value decomposition. Based on these same procedures, a biplot construction is now presented for a multivariateregression model of reduced rank in the presence of multicollinearity. This procedure, called PLSSVD GH biplot, provides auseful data analysis tool which allows the visual appraisal of the structure of the dependent and independent variables. Thispaper defines the procedure and shows several of its properties. It also provides an implementation of the routines in R andpresents a real life application involving data from the FAO food database to illustrate the procedure and its properties.