PARAFAC2: Extensions of a Procedure for “Explanatory” Factor‐Analysis and Multidimensional Scaling

Recently developed techniques for factor analysis and multidimensional scaling (PARAFAC‐INDSCAL) allow discovery of a unique orientation of axes and thus “more explanatory” factors or dimensions. A certain price is paid for these advances, however. In multidimensional scaling applications, the price is a restriction of the possible form of the solution to orthogonal “nonrelated” as opposed to oblique “perceptually related” dimensions. In factor analytic applications, the price is a limitation to those three‐way data sets showing “system‐variation” [Harshman, UCLA W.P.P. No. 16 (1970)]. An extension of PARAFAC to summed‐cross‐product matrices (e.g., covariance matrices) overcomes both of these limitations. For factor analysis with PARAFAC2 it is only necessary that the average effect of a given factor change, relative to the other factors, from one covariance matrix to the next. For multidimensional scaling, PARAFAC2 describes the perceptual space in terms of orthogonal dimensions only when this best fits the data. Otherwise, it will recover the stimulis projections on oblique dimensions and also give the angles between these dimensions. Advantages of PARAFAC2 are explored by comparing its results with those of PARAFAC. Both factor‐analytic and multidimensional scaling applications are demonstrated using data from speech physiology and perception.