A Bridge Between PLS Path Modeling and Multi-Block Data Analysis

Abstract

A situation where J blocks of variables X 1, …, X J are observed on the same set of individuals is considered in this paper. A factor analysis approach is applied to blocks instead of variables. The latent variables (LV’s) of each block should well explain their own block and at the same time the latent variables of same order should be as highly correlated as possible (positively or in absolute value). Two path models can be used in order to obtain the first order latent variables. The first one is related to confirmatory factor analysis: each LV related to one block is connected to all the LV’s related to the other blocks. Then, PLS path modeling is used with mode A and centroid scheme. Use of mode B with centroid and factorial schemes is also discussed. The second model is related to hierarchical factor analysis. A causal model is built by relating the LV’s of each block X j to the LV of the super-block X J + 1 obtained by concatenation of X 1, …, X J . Using PLS estimation of this model with mode A and path-weighting scheme gives an adequate solution for finding the first order latent variables. The use of mode B with centroid and factorial schemes is also discussed. The higher order latent variables are found by using the same algorithms on the deflated blocks. The first approach is compared with the MAXDIFF/MAXBET Van de Geer’s algorithm (1984) and the second one with the ACOM algorithm (Chessel and Hanafi, 1996). Sensory data describing Loire wines are used to illustrate these methods.