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