Analysis of multiblock and hierarchical PCA and PLS models

Multiblock and hierarchical PCA and PLS methods have been proposed in the recent literature in order to improve the interpretability of multivariate models. They have been used in cases where the number of variables is large and additional information is available for blocking the variables into conceptually meaningful blocks. In this paper we compare these methods from a theoretical or algorithmic viewpoint using a common notation and illustrate their differences with several case studies. Undesirable properties of some of these methods, such as convergence problems or loss of data information due to deflation procedures, are pointed out and corrected where possible. It is shown that the objective function of the hierarchical PCA and hierarchical PLS methods is not clear and the corresponding algorithms may converge to different solutions depending on the initial guess of the super score. It is also shown that the results of consensus PCA (CPCA) and multiblock PLS (MBPLS) can be calculated from the standard PCA and PLS methods when the same variable scalings are applied for these methods. The standard PCA and PLS methods require less computation and give better estimation of the scores in the case of missing data. It is therefore recommended that in cases where the variables can be separated into meaningful blocks, the standard PCA and PLS methods be used to build the models and then the weights and loadings of the individual blocks and super block and the percentage variation explained in each block be calculated from the results. © 1998 John Wiley & Sons, Ltd.