, Volume 21, Issue 4, pp 383–393 | Cite as

Determination of the number of independent parameters of a score matrix from the examination of rank orders

  • Joseph F. Bennett


Two ordinal consequences are drawn from the linear multiple-factor analysis model. First, the numberR(s, d) of distinct ways in whichs subjects can be ranked by linear functions ofd factors is limited by the recursive expressionR(s, d)=R(s−1,d)+(s−1)R(s−1,d−1). Second, every setS ofd+2 subjects can be separated into two subsetsS* andS − S* such that no linear function ofd variables can rank allS* over allS − 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.


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  1. 1.
    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 Unpublished Ph.D. dissertation, University of Michigan, 1951.Google Scholar
  2. 2.
    Coombs, C. H. Psychological scaling without a unit of measurement.Psychol. Rev., 1950,57, 145–158.PubMedGoogle Scholar
  3. 3.
    Thurstone, L. L. Multiple-factor analysis. Chicago: Univ. Chicago Press, 1947.Google Scholar

Copyright information

© Psychometric Society 1956

Authors and Affiliations

  • Joseph F. Bennett
    • 1
  1. 1.Massachusetts Institute of TechnologyUSA

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