A Nonmetrical Method for Paired Comparisons

Abstract

In this paper, the authors propose a new nonmetrical algorithm to construct an interval scale from the numerical data obtained by paired comparisons. In the algorithm, the band function model is introduced, and, with respect to the function's average characteristics, a monotone increasing relation is assumed between the numerical data and the difference of a psychological scale on the attribute of the compared objects.
The algorithm to construct an interval scale is based on an iterative method of calculus, and has the following advantages.
1) Not necessarily all results about the comparison of each pair of objects are required.
2) The amount of fazziness or scattering included in the results of the paired comparisons can be known from the width of the dead zone introduced in the algorithm.
3) An interval scale can be constructed more precisely by this algorithm than by a usual paired comparison method.
Through the numerical experiments using artificial data, the fundamental characteristics of the algorithm are examined.
For testing its practical utility, a case of subjective comparisons of weights by human hands is taken up as an example, and the algorithm is applied to reconstructing their real weights. The reconstructed weights show good agreement with the real weights measured by a direct balance.