Techniques for rotating two or more loading matrices to optimal agreement and simple structure: A comparison and some technical details

Matrices of factor loadings are often rotated to simple structure. When more than one loading matrix is available for the same variables, the loading matrices can be compared after rotating them all (separately) to simple structure. An alternative procedure is to rotate them to optimal agreement, and then compare them. In the present paper techniques are described that combine these two procedures. Specifically, five techniques that combine the ideals of rotation to optimal agreement and rotation to simple structure are compared on the basis of contrived and empirical data. For the contrived data, it is assessed to what extent the rotations recover the underlying common structure. For both the contrived and the empirical data it is studied to what extent the techniques give well matching rotated matrices, to what extent these have a simple structure, and to what extent the most prominent parts of the different loading matrices agree. It was found that the simple procedure of combining a Generalized Procrustes Analysis (GPA) with Varimax on the mean of the matched loading matrices performs very well on all criteria, and, for most purposes, offers an attractive compromise of rotation to agreement and simple structure. In addition to this comparison, some technical improvements are proposed for Bloxom's rotation to simple structure and maximum similarity.This research has been made possible by a fellowship from the Royal Netherlands Academy of Arts and Sciences to the author. The author is obliged to René van der Heijden for assistance in programming the procedures in the simulation study reported in this paper, and to Jos ten Berge, three anonymous reviewers and an associate editor for helpful comments on an earlier version of this paper.