We derive ratio scales from paired comparison judgments in a reciprocal matrix A. When the judgments are consistent, we have a principal eigenvalue structure which is preserved when A is perturbed. Mathematical conditions are given on the size of the perturbations to produce a good approximation to A by a matrix W of ratios formed from the derived scale. Goodness is analyzed for both metric and order properties. The results of this paper point strongly to the fact that the eigenvalue process is the intrinsic solution to the problem of deriving a ratio scale, thus one does not need to invent extraneous normative criteria to solve an inconsistent problem.
In the present paper, we discuss a latent structure analysis for assessing learning structures of acquiring two kinds of skill. This discussion presents a “pairwise” assessment procedure for explaining the learning structure of acquiring the skills concerned. We propose a latent class model for this objective. This model explains prerequisite and transfererence relations between the skills. A parameter estimation procedure is derived by use of the EM algorithm. A numerical example is also included to illustrate the estimation procedure.
Solutions of Correspondence Analysis (CA) are obtained with three artificial two-way contingency tables. Two new concepts are introduced to clarify the structure of the problem. One is dual space representation, which is closely related to the representation proposed by Tenenhaus and Young (1985). Another is orthogonal polynomial systems for finite sums. Further, these suggest possible procedures for obtaining solutions of other method of multivariate data analysis.
This paper proposes a mutidimensional scaling method for asymmetric proximity data whose diagonal entries are not all equal. For a suggested model, necessary and sufficient conditions and algebraic solutions are presented. The method consisting of two algorithms is developed, which gives multidimensional configuration, unidimensional scale of any psychological quantities and the degree of asymmetry of the data. The quantities should be interpreted through data analysis or examined with any hypothesis. A simulation study using artificial data demonstrates that the method works successfully, showing sufficient robustness and capability to recover the original structure. Illustrative applications are presented. Extensions of the model are argued, and properties of the model and the algorithms are discussed with another simulation study.