A general method for computing hierarchical component structures by Goldberg's Bass-Ackwards method

Goldberg [Goldberg, L. R.(in press) Doing it all Bass-Ackwards: The development of hierarchical factor structures from the top down. Journal of Research in Personality] has recently described a novel method for computing hierarchical component structures via a "top down" design. He calls his method the "Bass Ackwards" approach to distinguish it from more common "bottom up" methods for obtaining hierarchical solutions. Using simple matrix equations, in this commentary I demonstrate that the end result of a Bass Ackwards analysis-that is, the correlations between component scores from different levels of a hierarchy-can be calculated without calculating the actual component scores. By avoiding component-score calculation, Goldberg's method (a) can be applied to any data set in which a correlation matrix is available, and (b) can be applied to factor scores rather than estimated factor scores. Goldberg suggests that the Bass Ackwards method represents a useful tool for elucidating the underlying dimensionality and latent structure (rotation in hyperspace) of a data set. Computer code is reported in the Appendix A of this article to help personality researchers more fully evaluate this claim in empirical and simulated data.