DOI: 10.1007/bf02293554. More Factors than Subjects, Tests and Treatments: An Indeterminacy Theorem for Canonical Decomposition and Individual differences Scaling

More Factors than Subjects, Tests and Treatments: An Indeterminacy Theorem for Canonical Decomposition and Individual differences Scaling

10.1007/bf02293554

Crossref journal-article

Cambridge University Press (CUP)

Psychometrika (56)

Abstract

Some methods that analyze three-way arrays of data (including INDSCAL and CANDECOMP/PARAFAC) provide solutions that are not subject to arbitrary rotation. This property is studied in this paper by means of the “triple product” [A, B, C] of three matrices. The question is how well the triple product determines the three factors. The answer: up to permutation of columns and multiplication of columns by scalars—under certain conditions. In this paper we greatly expand the conditions under which the result is known to hold. A surprising fact is that the nonrotatability characteristic can hold even when the number of factors extracted is greater than every dimension of the three-way array, namely, the number of subjects, the number of tests, and the number of treatments.

Bibliography

Kruskal, J. B. (1976). More Factors than Subjects, Tests and Treatments: An Indeterminacy Theorem for Canonical Decomposition and Individual differences Scaling. Psychometrika, 41(3), 281â293.

Authors 1

Joseph B. Kruskal (first)

References 10 Referenced 78

1. Brockett, R. W. & Dobkin, D. On the number of multiplications required for matrix multiplications. Unpublished manuscript.
2. Brockett, R. W. & Dobkin, D. On the optimal evaluation of a set of bilinear forms. Unpublished manuscript.
{'journal-title': 'Linear algebra and its applications', 'article-title': 'Trilinear decomposition of three-way arrays: Rank and uniqueness in arithmetic complexity and in statistical models', 'author': 'Kruskal', 'key': 'S0033312300037005_CR8'} / Linear algebra and its applications / Trilinear decomposition of three-way arrays: Rank and uniqueness in arithmetic complexity and in statistical models by Kruskal
{'volume-title': 'Contributions to mathematical psychology', 'year': '1964', 'author': 'Tucker', 'key': 'S0033312300037005_CR10'} / Contributions to mathematical psychology by Tucker (1964)
10.1007/BF02310791
4. Harshman, R. A. Determination and proof of minimum uniqueness conditions for PARAFAC1. Working Papers in Phonetics No. 22, University of California at Los Angeles, 1972.
3. Carroll, J. D. Personal communication.
{'key': 'S0033312300037005_CR5', 'article-title': 'Foundations of the PARAFAC procedure: Models and conditions for an “explanatory” multimodel factor analysis', 'volume': '16', 'author': 'Harshman', 'year': '1970', 'journal-title': 'Working Papers in Phonetics'} / Working Papers in Phonetics / Foundations of the PARAFAC procedure: Models and conditions for an “explanatory” multimodel factor analysis by Harshman (1970)
10.1007/BF02165411
10.1007/978-1-4684-2001-2_4

Dates

Type	When
Created	19 years, 7 months ago (Jan. 25, 2006, 9:08 a.m.)
Deposited	6 months ago (Feb. 20, 2025, 8:32 p.m.)
Indexed	3 weeks, 6 days ago (July 30, 2025, 11:29 a.m.)
Issued	48 years, 11 months ago (Sept. 1, 1976)
Published	48 years, 11 months ago (Sept. 1, 1976)
Published Online	7 months, 3 weeks ago (Jan. 1, 2025)
Published Print	48 years, 11 months ago (Sept. 1, 1976)

Funders 0

None

BibTeX

@article{Kruskal_1976, title={More Factors than Subjects, Tests and Treatments: An Indeterminacy Theorem for Canonical Decomposition and Individual differences Scaling}, volume={41}, ISSN={1860-0980}, url={http://dx.doi.org/10.1007/bf02293554}, DOI={10.1007/bf02293554}, number={3}, journal={Psychometrika}, publisher={Cambridge University Press (CUP)}, author={Kruskal, Joseph B.}, year={1976}, month=sep, pages={281–293} }

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