Clusterwise simultaneous component analysis for analyzing structural differences in multivariate multiblock data

Many studies yield multivariate multiblock data, that is, multiple data blocks that all involve the same set of variables (e.g., the scores of different groups of subjects on the same set of variables). The question then rises whether the same processes underlie the different data blocks. To explore the structure of such multivariate multiblock data, component analysis can be very useful. Specifically, 2 approaches are often applied: principal component analysis (PCA) on each data block separately and different variants of simultaneous component analysis (SCA) on all data blocks simultaneously. The PCA approach yields a different loading matrix for each data block and is thus not useful for discovering structural similarities. The SCA approach may fail to yield insight into structural differences, since the obtained loading matrix is identical for all data blocks. We introduce a new generic modeling strategy, called clusterwise SCA, that comprises the separate PCA approach and SCA as special cases. The key idea behind clusterwise SCA is that the data blocks form a few clusters, where data blocks that belong to the same cluster are modeled with SCA and thus have the same structure, and different clusters have different underlying structures. In this article, we use the SCA variant that imposes equal average cross-products constraints (ECP). An algorithm for fitting clusterwise SCA ECP solutions is proposed and evaluated in a simulation study. Finally, the usefulness of clusterwise SCA is illustrated by empirical examples from eating disorder research and social psychology.