Evaluation of the predictive power of biplot axes to automate the construction and layout of biplots based on the accuracy of direct readings from common outputs of multivariate analyses: 1. application to principal component analysis

Predictive biplots, as developed by J.C. Gower and coworkers, can be a very useful tool to aid the interpretation of the outcomes of multivariate analyses. This paper covers a statistical methodology that enables the automation of the construction of predictive biplots, as well as an R function, AutoBiplots.PCA( ), which applies the methodology to principal components analysis. A case study based on the sensory analysis of coffees is used to illustrate the methodology as well as the outputs of the R function. The method relies on the definition of a variable's mean standard predictive error, mspe, as the degree of accuracy in the process of predicting the original values from the biplots, which is compared with a predefined tolerance value (T_axis) to decide if the correspondent biplot axis is drawn in the biplot. Standard predictive errors, spe, are calculated for each unit in relation to each biplot axis in each two‐dimensional plot and are compared with a predefined tolerance value (T_units) to decide which units shall be faced as outliers. The R function automates the process, enabling the user to decide on the degree of precision of the actual analysis. Besides providing a solution for the automatic production of predictive biplots, the methodology offers new insights for the interpretation of multivariate analyses outputs on the basis of a sound principle, the degree of precision of the analysis. This provides an automatic way for the selection of variables that explain latent dimensions and also helps in deciding on the number of important latent dimensions for model developments. Copyright © 2012 John Wiley & Sons, Ltd.