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

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.