Parameter estimation for the thurstone case III model

A Monte Carlo study was conducted to investigate the ability of three estimation criteria to recover the parameters of Case V and Case III models from comparative judgment data. Significant differences in recovery are shown to exist.This research was supported by grant SOC76-20517 from the National Science Foundation. The authors acknowledge with appreciation comments received on an earlier draft by Harold Lindman, Joseph Zinnes, and anonymous reviewers.     

A multidimensional scaling model for the size-weight illusion

The kinds of individual differences in perceptions permitted by the weighted euclidean model for multidimensional scaling (e.g., INDSCAL) are much more restricted than those allowed by Tucker's Three-mode Multidimensional Scaling (TMMDS) model or Carroll's Idiosyncratic Scaling (IDIOSCAL) model. Although, in some situations the more general models would seem desirable, investigators have been reluctant to use them because they are subject to transformational indeterminacies which complicate interpretation. In this article, we show how these indeterminacies can be removed by constructing specific models of the phenomenon under investigation. As an example of this approach, a model of the size-weight illusion is developed and applied to data from two experiments, with highly meaningful results. The same data are also analyzed using INDSCAL. Of the two solutions, only the one obtained by using the size-weight model allows examination of individual differences in the strength of the illusion; INDSCAL can not represent such differences. In this sample, however, individual differences in illusion strength turn out to be minor. Hence the INDSCAL solution, while less informative than the size-weight solution, is nonetheless easily interpretable.This paper is based on the first author's doctoral dissertation at the Department of Psychology, University of Illinois at Urbana-Champaign. The aid of Professor Ledyard R Tucker is gratefully acknowledged.     

Comparing latent distributions

We consider the problem of comparingm latent population distributions when the observed values are scores on a test battery with binary items. The latent densities are assumed to be normal densities, and we consider a test for equality of the means as well as a test equality of the variances. In addition, we consider a longitudinal model, where the test battery has been applied to the same individuals at different points in time. This model allows for correlations between the latent variable at different time points, and methods are discussed for estimating the correlation coefficient.This work was supported in part by a grant from the Danish Social Science Research Council.     

An individual differences additive model: An alterating least squares method with optimal scaling features

An individual differences additive model is discussed which represents individual differences in additivity by differential weighting of additive factors. A procedure for estimating the model parameters for various data measurement characteristics is developed. The procedure is evaluated using both Monte Carlo and real data. The method is found to be very useful in describing certain types of developmental change in cognitive structure, as well as being numerically robust and efficient.The work reported here was partly supported by Grant A6394 to the first author by the Natural Sciences and Engineering Research Council of Canada.     

Assessment of social skills using a Thurstone Equal-Appearing Interval scale

Although Likert ratings of social skill are widely used in behavioral assessment, such ratings may have limited replicability potential and social validity. The present study compared the reliability of Likert and Thurstone Equal-Appearing Interval scales. The Thurstone-derived scale was slightly more reliable than its Likert counterpart. Moreover, the empirical and clinical potential of Thurstone scaling substantially exceeds that of Likert and other rating methods.     

Axiomatic foundations of a three-set guttman simplex model with applicability to longitudinal data

In this paper the usual two-set Guttman simplex model is extended to three sets. The axiomatic foundations of this extention are presented. Two cases are discussed. In Case 1 there is a three-set joint order, while in Case 2 there is a two-set joint order consistent across all levels of the third set. Case 2 represents the first clear formulation of a longitudinal developmental scale. The model is discussed in terms of its most straightforward application, longitudinal developmental data, and in terms of other possible applications.This research was supported in part by NSF grant BNS 8108145.     

Meaning and Method in the Social Sciences<Superscript>1</Superscript>

Academia’s mathematical metaphysics are briefly explored en route to an elaboration of the qualitatively rigorous requirements underpinning the calibration and unambiguous interpretation of quantitative instrumentation in any science. Of particular interest are Gadamer’s emphases on number as the paradigm of the noetic, on the role of play in interpretation, and on Hegel’s sense of method as the activity of the thing itself that thought experiences. These point toward and overlap with (1) Latour’s study of the metrological social networks through which technological phenomena are brought into language as modes of being that can be understood, and (2) the way that Rasch’s models for measurement comprise a potential beginning for metaphysically astute, qualitatively and quantitatively integrated, mathematical methods in the social sciences. The paper closes with observations on the general problem that is philosophy, the need to remain open to multiplicities of meaning even as clear understandings are sought and obtained.     

Constrained multidimensional unfolding of confusion matrices: Goal point and slide vector models

Two constrained multidimensional unfolding models, the goal point and slide vector models, are proposed for analyzing confusion matrices. In both models, the row and column stimuli are expressed as two sets of points in a low-dimensional space, where the difference vector connecting a column point to the corresponding row point indicates the change in the stimulus representation through a cognitive process. The difference vector is constrained by the hypothesis that the trend in the representational change is invariant across stimuli: the goal point model constrains all difference vectors to point toward a single point, and the slide vector model constrains all difference vectors to be parallel to each other. In both models the coordinates of points are estimated by the maximum-likelihood method. Examples illustrate that the two models allow us to examine hypotheses about invariant trends in representational changes and to grasp such trends from the resulting configurations.     

Robust canonical discriminant analysis

A method for robust canonical discriminant analysis via two robust objective loss functions is discussed. These functions are useful to reduce the influence of outliers in the data. Majorization is used at several stages of the minimization procedure to obtain a monotonically convergent algorithm. An advantage of the proposed method is that it allows for optimal scaling of the variables. In a simulation study it is shown that under the presence of outliers the robust functions outperform the ordinary least squares function, both when the underlying structure is linear in the variables as when it is nonlinear. Furthermore, the method is illustrated with empirical data.The research of the first author was supported by the Netherlands Organization of Scientific Research (NWO grant 560-267-029).