Dual extraction ofR-mode andQ-mode factor solutions

Abstract

It is mathematically possible to extract both R-mode and Q-mode factors simultaneously (RQ-mode factor analysis)by invoking the Eckhart-Young theorem. The resulting factors will be expressed in measures determined by the form of the scalings that have been applied to the original data matrix. Unless the measures for both solutions are meaningful for the problem at hand, the factor results may be misleading or uninterpretable. Correspondence analysis uses a symmetrical scaling of both rows and columns to achieve measures of proportional similarity between objects and variables. In the literature, the resulting similarity is a χ 2 distance appropriate for analysis of enumerated data, the original application of correspondence analysis. Justification for the use of this measure with interval or ratio data is unconvincing, but a minor modification of the scaling procedure yields the profile similarity, which is an appropriate measure. Symmetrical scaling of rows and columns is unnecessary for RQ-mode factor analysis. If the data are scaled so the minor product W'Wis the correlation matrix, the major product WW'is expressed in the Euclidean distances between objects. Therefore, RQ-mode factor analysis can be performed so that the Rmode is a principal components solution and the Qmode is a principal coordinates solution. For applications where the magnitudes of differences are important, this approach will yield more interpretable results than will correspondence analysis.