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Determination of the Number of Independent Parameters of a Score Matrix from the Examination of Rank Orders

Published online by Cambridge University Press:  01 January 2025

Joseph F. Bennett
Joseph F. Bennett*
Affiliation:
Massachusetts Institute of Technology
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Abstract

Two ordinal consequences are drawn from the linear multiple-factor analysis model. First, the number R(s, d) of distinct ways in which s subjects can be ranked by linear functions of d factors is limited by the recursive expression R(s, d) = R(s−, d)+(s−1) R(s−, d−1). Second, every set S of d+2 subjects can be separated into two subsets S* and S − S* such that no linear function of d variables can rank all S* over all S − S*, and vice versa. When these results are applied to the hypothetical data of Thurstone's “box problem,” three independent parameters are found. Relations to Thurstone's suggestion for a non-correlational factor analysis are discussed.

Type
Original Paper
Information
Psychometrika , Volume 21 , Issue 4 , December 1956 , pp. 383 - 393
Copyright
Copyright © 1956 The Psychometric Society

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Footnotes

*

The research reported here was supported jointly by the Army, Navy, and Air Force under contract with the Massachusetts Institute of Technology. The manuscript was prepared while Dr. Bennett was a Fellow of the Center for Advanced Study in the Behavioral Sciences, Stanford, California. His untimely death on May 4, 1956, at the age of 29, is regretfully announced.

References

Bennett, J. F. A method for determing the dimensionality of a set of rank orders. Unpublished Ph.D. dissertation, University of Michigan, 1951.Google Scholar
Coombs, C. H. Psychological scaling without a unit of measurement. Psychol. Rev., 1950, 57, 145–158CrossRefGoogle ScholarPubMed
Thurstone, L. L. Multiple-factor analysis, Chicago: Univ. Chicago Press, 1947Google Scholar