Abstract

The principle of least squares is applied to the selection of an optimum set of n response values, or of (n−1) basic response differences, corresponding to n stimuli, from m 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 n-1 m<12n(n-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 a priori ordering of the n stimuli, even in the incomplete-set case.

© 1955 Optical Society of America

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References

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  1. L. L. Thurstone, Am. J. Psychol. 38, 368–389 (1927).
    [CrossRef]
  2. F. Kottler, unpublished work.
  3. J. P. Guilford, Psychometric Methods (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. Standards49, 349 (No. 5, November, 1952).
    [CrossRef]
  6. This departure, for derivation only, is temporary.
  7. P. S. Dwyer, Ann. Math. Stat. ( 1)15, 82 (March, 1944).
    [CrossRef]
  8. An alternative procedure, which will not be further elaborated here, for avoiding the impasse resulting from singularity of A′A, 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−1) symmetric matrix B′B, 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).
    [CrossRef]

1955 (1)

1944 (1)

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

1927 (1)

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

Barkley Rosser, J.

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. Standards49, 349 (No. 5, November, 1952).
[CrossRef]

Dwyer, P. S.

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

Guilford, J. P.

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

Higgins, G. C.

Kottler, F.

F. Kottler, unpublished work.

Thurstone, L. L.

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

Wolfe, R. N.

Am. J. Psychol. (1)

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

Ann. Math. Stat. (1)

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

J. Opt. Soc. Am. (1)

Other (7)

An alternative procedure, which will not be further elaborated here, for avoiding the impasse resulting from singularity of A′A, 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−1) symmetric matrix B′B, 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.”

F. Kottler, unpublished work.

J. P. Guilford, Psychometric Methods (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. Standards49, 349 (No. 5, November, 1952).
[CrossRef]

This departure, for derivation only, is temporary.

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Figures (12)

Fig. 1
Fig. 1

Relationship of preference ratio to response difference in units of standard normal deviate.

Fig. 2
Fig. 2

Matrices of preference ratios and of corresponding response differences (standard normal deviates).

Fig. 3
Fig. 3

A typical skew symmetric matrix of response differences for an incomplete set of paired comparisons.

Fig. 4
Fig. 4

An illustration of the Guilford method of assigning psychometric scale values on the basis of incomplete paired-comparison data, and an examination of the ability of these values to predict the observed response differences.

Fig. 5
Fig. 5

Operational and data matrices involved in treatment of the illustrative example by the method of minimizing sums of squares of observed-minus-computed response differences.

Fig. 6
Fig. 6

Matrices d (in skew symmetric form), dA, AA, and (AA)−1 of the least-squares solution of the illustrative example.

Fig. 7
Fig. 7

Response values assigned by the least-squares solution, and comparison of the observed and computed response differences.

Fig. 8
Fig. 8

Comparison of response values assigned by the Guilford method and the new method.

Fig. 9
Fig. 9

Results of Guilford-method assignment of response values resulting from two sets of judgments of the same stimuli: filled circles are assignment from first set of judgments; open circles are assignment from second set of judgments.

Fig. 10
Fig. 10

Results of new-method assignment of response values resulting from two sets of judgments of the same stimuli; filled circles are assignment from first set of judgments; open circles are assignment from second set of judgments.

Fig. 11
Fig. 11

Relationship of subjective judgments of definition to objective measurements when subjective assignments are computed by the Guilford method.

Fig. 12
Fig. 12

Relationship of subjective judgments of definition to objective measurements when subjective assignments are computed by the new method.

Equations (5)

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A r d .
( d - A r ) ( d - A r ) .
A A r - A d = 0 ,
r = ( A A ) - 1 ( A d ) ,
A ( d - A r ) = 0.