Parafac2 is the most flexible Simultaneous Component Analysis (SCA) model that produces an essentially unique solution. In this paper, we discuss how Parafac2’s special sign indeterminacy affects applications of SCA, and we reveal how an external criterion variable can be used to ensure that estimated Parafac2 weights are meaningfully signed across the levels of the nesting mode. We present an example with real data from clinical psychology that illustrates the importance of Parafac2’s special sign indeterminacy, as well as the effectiveness of our proposed solution. We also discuss the implications of our results for general applications of SCA.

Some existing three-way factor analysis and MDS models incorporate Cattell's “Principle of Parallel Proportional Profiles”. These models can—with appropriate data—empirically determine a unique best fitting axis orientation without the need for a separate factor rotation stage, but they have not been general enough to deal with what Tucker has called “interactions” among dimensions. This article presents a proof of unique axis orientation for a considerably more general parallel profiles model which incorporates interacting dimensions. The model, Xk=AADk HBDk B', does not assume symmetry in the data or in the interactions among factors. A second proof is presented for the symmetrically weighted case (i.e., where ADk=BDk). The generality of these models allows one to impose successive restrictions to obtain several useful special cases, including PARAFAC2 and three-way DEDICOM.