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

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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.

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