A note on the determination of configuration and weights for a class of individual scaling models

Consider the typical problem in individual scaling, namely finding a common configuration and weights for each individual from the given interpoint distances or scalar products. Within the STRAIN framework it is shown that the problem of determining weights for a given configuration can be posed as a standard quadratic programming problem. A set of necessary conditions for an optimal configuration to satisfy are given. A closed form expression for the configuration is obtained for the one dimensional case and an approach is given for the two dimensional case.     

Multidimensional successive categories scaling: A maximum likelihood method

A single-step maximum likelihood estimation procedure is developed for multidimensional scaling of dissimilarity data measured on rating scales. The procedure can fit the euclidian distance model to the data under various assumptions about category widths and under two distributional assumptions. The scoring algorithm for parameter estimation has been developed and implemented in the form of a computer program. Practical uses of the method are demonstrated with an emphasis on various advantages of the method as a statistical procedure.The research reported here was partly supported by Grant A6394 to the author by Natural Sciences and Engineering Research Council of Canada. Portions of this research were presented at the Psychometric Society meeting in Uppsala, Sweden, in June, 1978. MAXSCAL-2.1, a program to perform the computations discussed in this paper may be obtained from the author. Thanks are due to Jim Ramsay for his helpful comments.     

On quantifying different types of categorical data

In quantifying categorical data, constraints play an important role in characterizing the outcome. In the Guttman-type quantification of contingency tables and multiple-choice data (incidence data), the trivial solution due to the marginal constraints is typically removed before quantification; this removal, however, has the effect of distorting the shape of the total space. Awareness of this is important for the interpretation of the quantified outcome. The present study provides some relevant formulas for those cases that are affected by the trivial solution and those cases that are not. The characterization of the total space used by the Guttman-type quantification and pertinent discussion are presented.This study was supported by a grant from The Natural Sciences and Engineering Research Council of Canada to S. Nishisato.     

Constrained latent class analysis: Simultaneous classification and scaling of discrete choice data

A reparameterization of a latent class model is presented to simultaneously classify and scale nominal and ordered categorical choice data. Latent class-specific probabilities are constrained to be equal to the preference probabilities from a probabilistic ideal-point or vector model that yields a graphical, multidimensional representation of the classification results. In addition, background variables can be incorporated as an aid to interpreting the latent class-specific response probabilities. The analyses of synthetic and real data sets illustrate the proposed method.The authors thank Yosiho Takane, the editor and referees for their valuable suggestions. Authors are listed in reverse alphabetical order.     

A new solution to the problem of finding all numerical solutions to ordered metric structures

A new algorithm is used to test and describe the set of all possible solutions for any linear model of an empirical ordering derived from techniques such as additive conjoint measurement, unfolding theory, general Fechnerian scaling and ordinal multiple regression. The algorithm is computationally faster and numerically superior to previous algorithms.This research was supported in part by NIGMS grant GM-01231 to the University of Michigan. Authors' names are in alphabetic order.     

Piecewise method of reciprocal averages for dual scaling of multiple-choice data

The proposed method handles the classical method of reciprocal averages (MRA) in a piecewise (item-by-item) mode, whereby one can deal with smaller matrices and attain faster convergence to a solution than the MRA. A new concept the principle of constant proportionality is introduced to provide an interesting interpretation for scaling multiple-choice data a la Guttman. A small example is presented for discussion of the technique.This study was supported by the Natural Sciences and Engineering Research Council Canada Grant (No. A4581) to S. Nishisato. The authors are indebted to reviewers for valuable comments.     

A technique for the nonmetric analysis of paired comparisons of psychological intervals

A nonmetric coordinate adjustment technique is developed which determines scale values for objects whose interobject intervals (differences in subjective value) have been directly compared. In Monte Carlo simulations, the degree of metric determinancy of the scale values is shown to be quite high even when the amount of error is relatively high. This robustness under high-error conditions permitted the analysis of individual subject data in experiments on the direct comparison of loudness differences and loudness ratios where only one judgment per interval comparison was obtained per subject.This research was supported by a grant from the National Research Council of Canada.