Abstract

The principle of least squares is applied to the selection of an optimum set of <i>n</i> response values, or of (<i>n</i>-1) basic response differences, corresponding to <i>n</i> stimuli, from <i>m</i> observed response differences provided by the paired-comparison method of psychometric measurement. This method of selection is shown to be the most appropriate generalization, to the case of a usable but incomplete set, i.e., with <i>n</i>-1≦<i>m</i><½<i>n</i>(<i>n</i>-1), of observed response differences, of a corrected form of the algorithm of J. P. Guilford. Unlike the Guilford algorithm, however, direct application of the least-squares principle implies response values which are independent of the <i>a priori</i> ordering of the <i>n</i> stimuli, even in the incomplete-set case.

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  1. L. L. Thurstone, Am. J. Psychol. 38, 368–389 (1927)
  2. F. Kottler, unpublished work.
  3. J. P. Guilford, Psyclhomtetric Metllods (McGraw-Hill Book Company, Inc., New York, 1936), pp. 217–243.
  4. It can be shown, however, that the Guilford-type solution is equivalent to the new method only if these apparent redundancies are not discarded. Their retention characterizes the "corrected form of the algorithm of J. P. Guilford" mentioned in the abstract. of this paper.
  5. A convenient method for the inversion of integer-element matrices, using row transformations only, is given by J. Barkley Rosser in J. Research Natl. Bur. Standards 49, 349 (No. 5, November, 1952).
  6. This departure, for derivation only, is temporary.
  7. P. S. Dwyer, Ann. Math. Stat. (1)15, 82 (March, 1944).
  8. An alternative procedure, which will not be further elaborated here, for avoiding the impasse resulting from singularity of AA, where A′ is not augmented as described above, is to factor A into a product of two rank-(n-1) matrices B and M, where M consists of (n-1) rows, each specifying a differencing operation, and then to solve for the (n-1)-element vector represented by the matrix product Mr; this procedure involves the inversion of the (n-1) × (n-l) symmetric matrix BB, which is generally nonsingular. An interesting feature of this solution for Mr, entirely equivalent to that for r, is its interpretation as the correct way to calculate, according to the least-squares principle, a set of (n-1) "basic differences," the objective of the Guilford "differencing" algorithm.
  9. In this case, the new method degenerates into the Kottler "summation algorithm."
  10. G. C. Higgins and R. N. Wolfe, J. Opt. Soc. Am. 45, 121 (1955).

Dwyer, P. S.

P. S. Dwyer, Ann. Math. Stat. (1)15, 82 (March, 1944).

Guilford, J. P.

J. P. Guilford, Psyclhomtetric Metllods (McGraw-Hill Book Company, Inc., New York, 1936), pp. 217–243.

It can be shown, however, that the Guilford-type solution is equivalent to the new method only if these apparent redundancies are not discarded. Their retention characterizes the "corrected form of the algorithm of J. P. Guilford" mentioned in the abstract. of this paper.

Higgins, G. C.

G. C. Higgins and R. N. Wolfe, J. Opt. Soc. Am. 45, 121 (1955).

Kottler, F.

F. Kottler, unpublished work.

Thurstone, L. L.

L. L. Thurstone, Am. J. Psychol. 38, 368–389 (1927)

Wolfe, R. N.

G. C. Higgins and R. N. Wolfe, J. Opt. Soc. Am. 45, 121 (1955).

Other (10)

L. L. Thurstone, Am. J. Psychol. 38, 368–389 (1927)

F. Kottler, unpublished work.

J. P. Guilford, Psyclhomtetric Metllods (McGraw-Hill Book Company, Inc., New York, 1936), pp. 217–243.

It can be shown, however, that the Guilford-type solution is equivalent to the new method only if these apparent redundancies are not discarded. Their retention characterizes the "corrected form of the algorithm of J. P. Guilford" mentioned in the abstract. of this paper.

A convenient method for the inversion of integer-element matrices, using row transformations only, is given by J. Barkley Rosser in J. Research Natl. Bur. Standards 49, 349 (No. 5, November, 1952).

This departure, for derivation only, is temporary.

P. S. Dwyer, Ann. Math. Stat. (1)15, 82 (March, 1944).

An alternative procedure, which will not be further elaborated here, for avoiding the impasse resulting from singularity of AA, where A′ is not augmented as described above, is to factor A into a product of two rank-(n-1) matrices B and M, where M consists of (n-1) rows, each specifying a differencing operation, and then to solve for the (n-1)-element vector represented by the matrix product Mr; this procedure involves the inversion of the (n-1) × (n-l) symmetric matrix BB, which is generally nonsingular. An interesting feature of this solution for Mr, entirely equivalent to that for r, is its interpretation as the correct way to calculate, according to the least-squares principle, a set of (n-1) "basic differences," the objective of the Guilford "differencing" algorithm.

In this case, the new method degenerates into the Kottler "summation algorithm."

G. C. Higgins and R. N. Wolfe, J. Opt. Soc. Am. 45, 121 (1955).

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