Hierarchical relations between methods for simultaneous component analysis and a technique for rotation to a simple simultaneous structure

The present paper discusses several methods for (simultaneous) component analysis of scores of two or more groups of individuals on the same variables. Some existing methods are discussed, and a new method (SCA-S) is developed for simultaneous component analysis in such a way that for each set essentially the same component structure is found (SCA-S). This method is compared to alternative methods for analysing such data which employ the same component weights matrix (SCA-W) or the same pattern matrix (SCA-P) across data sets. Among these methods, SCA-W always explains the highest amount of variance, SCA-S the lowest, and SCA-P takes the position in between. These explained variances can be compared to the amount of variance explained by separate PCAs. Implications of such fit differences are discussed. In addition, it is shown how, for cases where SCA-S does not fit well, one can use SCA-W (and SCA-P) to find out if and how correlational structures differ. Finally, some attention is paid to facilitating the interpretation of an SCA-S solution. Like the other SCA methods, SCA-S has rotational freedom. This rotational freedom is exploited in a specially designed simple structure rotation technique for SCA-S. This technique is illustrated on an empirical data set.