Journal of the Japan Statistical Society, Japanese Issue

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The fourth method of quantification proposed by Hayashi (1952, 1961) has been widely used in various fields of applied statistics throughout Japan as a powerful tool of classification or multidimensional scaling.
The purpose of this paper is to compare this with a competitor, principal coordinates analysis introduced by Torgerson (1952) and expounded by Gower (1966), by four kinds of criteria; faithfulness of representation, identifiability, positivity of eigenvalues and the speed of decrease of eigenvalues when arranged in order of magnitude. Both the second and the third criteria are of theoretical character but the others are empirical and more or less subjective. Two illustrating examples are given to show the performances of the two methods when the similarities between objects under consideration are defined in several different ways. Principal coordinates analysis seems to be superior to the fourth method except by the third criterion.

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A t-(v, k, λ_t) design on a finite point set S of the cardinality v is a family Σof k-subsets of S such that every t-subset of S is contained in exactly λ_t members of Σ. Here by using the finite affine geometry, we show the existence of a 3-(2^t, 2^d, λ₃) design with λ₃=φ(t-3, d-3, 2) for every t>d_??_2.

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