, Volume 61, Issue 1, pp 133–154 | Cite as

Uniqueness proof for a family of models sharing features of Tucker's three-mode factor analysis and PARAFAC/candecomp

  • Richard A. Harshman
  • Margaret E. Lundy


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., whereADk=BDk). The generality of these models allows one to impose successive restrictions to obtain several useful special cases, including PARAFAC2 and three-way DEDICOM.

Key words

Parallel proportional profiles intrinsic axes DEDICOM PARAFAC2 Cattell trilinear models quadrilinear models factor rotation problem multidimensional scaling principal components oblique confactor 


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Copyright information

© The Psychometric Society 1996

Authors and Affiliations

  • Richard A. Harshman
    • 1
  • Margaret E. Lundy
    • 1
  1. 1.Psychology DepartmentUniversity of Western OntarioLondonCanada

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