Multi-block regression based on combinations of orthogonalisation, PLS-regression and canonical correlation analysis

This paper reviews the basic ideas underlying two new multi-block techniques, the SO-PLS and the PO-PLS regression methods. A discussion is given about how the two methods are related to each other and to standard regression and analysis of variance (ANOVA). In particular the relation between SO-PLS and Type I ANOVA is underlined. It is shown how the sums of squares can be split according to blocks introduced and also how two-way ANOVA applied to cross-validated residuals, can be utilised for testing significance of the blocks. Different ways of interpreting the results for both methods are considered and illustrated by examples. Relations to other proposed methods and ideas are discussed.

•Multi-blocks: invariance, collinearity and different dimensionality.•Focus on incremental/additional and common variability.•SS decomposition and ANOVA obtained by CV-ANOVA.•Multivariate interaction.