Describing the elephant: Structure and function in multivariate data

There is a unity underlying the diversity of models for the analysis of multivariate data. Essentially, they constitute a family models, most generally nonlinear, for structural/functional relations between variables drawn from a behavior domain.     

Differentiability of Kruskal's stress at a local minimum

It is shown that Kruskal's multidimensional scaling loss function is differentiable at a local minimum. Or, to put it differently, that in multidimensional scaling solutions using Kruskal's stress distinct points cannot coincide.     

Temporal reproduction

A signal appeared for a certain time period. After the period elapsed, pigeons had to begin and complete a sequence of 15 responses in a time window ranging from the signal duration to 50% longer. Sessions involved as many as 10 different signal durations occurring in a random sequence. The times produced by pigeons often were in the same ranges as those that have been found with adult human subjects. The average times were described equally well as linear or power functions of signal duration. However, instead of the overestimation of durations usually found when animals have timed the duration of antecedent stimuli, the linear functions suggested that the pigeons underestimated the durations of their own behavior. The birds showing the strongest control when the conditions involved eight or 10 different duration requirements revealed the constant coefficients of variation that support Weber's law and scalar timing theory. Scalar timing in temporal differentiation appears to depend on non-ambiguous information about the duration required for reinforcement and on a high degree of sensitivity to the duration requirement.     

An entropy approach to the scaling of ordinal categorical data

This paper studies the problem of scaling ordinal categorical data observed over two or more sets of categories measuring a single characteristic. Scaling is obtained by solving a constrained entropy model which finds the most probable values of the scales given the data. A Kullback-Leibler statistic is generated which operationalizes a measure for the strength of consistency among the sets of categories. A variety of data of two and three sets of categories are analyzed using the entropy approach.This research was partially supported by the Air Force Office of Scientific Research under grant AFOSR 83-0234. The support by the Air Force through grant AFOSR-83-0234 is gratefully acknowledged. The comments of the editor and referees have been most helpful in improving the paper, and in bringing several additional references to our attention.     

肯定性“重要度“语义量词的多级估量模糊集模型建立及其应用

在人员选拔和心理素质评价研究中,需要通过被研究者在特定的问题、事件或情境上所作出的评价,来了解他们的心理活动特点。为了确保评价的可靠性,在量表设定上多采用自然语言的回答方式,如是或否,或极重要、很重要、较重要、稍重要等.等级性自然语言可表征人们评价事物的心理量,称语义量词.但这些语义量词不是一个数值,而是一个边界不确定的数量范围,难以作数量化分析,一直是心理测量方法学研究的重要课题。马谋超提出的模糊评判的“多级估量法”［‘l为等级性自然语言的数量化研究提供了重要的理论依据。优秀军事指挥员心理选拔与…     

Multidimensional scaling of hierarchical sorting data applied to facial expressions

A new algorithm for multidimensional scaling analysis of sorting data and hierarchical-sorting data is tested by applying it to facial expressions of emotion. We construct maps in “facial expression space” for two sets of still photographs: the I-FEEL series (expressions displayed spontaneously by infants and young children), and a subset of the Lightfoot series (posed expressions, all from one actress). The analysis avoids potential artefacts by fitting a map directly to the subject's judgments, rather than transforming the data into a matrix of estimated dissimilarities as an intermediate step. The results for both stimulus sets display an improvement in the extent to which they agree with existing maps. Some points emerge about the limitations of sorting data and the need for caution when interpreting MDS configurations derived from them.     

On the Psychological Significance of the Euclidean Representation of Implicit Personality Theory

The overall purpose of the present study was to investigate the psychological significance of the Euclidean representation of implicit personality theory (IPT) by testing the predictive usefulness of this representation. Multidimensional scaling analyses of 18 subjects' estimates of assumed trait co-occurences (IPT) gave Euclidean trait-distance matrices which were used to predict the subjects' trait-to-trait inferences. The results showed that: (1) The Euclidean representations of subjects' IPTs can be used to predict ( p < 0.10) most subjects' (14 of 18) trait-to-trait inferences; (2) The similarity between the Euclidean representation of subjects' IPTs was positively correlated (r_s = 0.43, p < 0.001) with the similarity between subjects' trait-to-trait inferences. These results suggest that the Euclidean representation of IPT has some predictive value, and thus, may at least to some extent, be justified and considered as psychologically significant.     

Uniqueness proof for a family of models sharing features of Tucker's three-mode factor analysis and PARAFAC/candecomp

Some existing three-way factor analysis and MDS models incorporate Cattell's “Principle of Parallel Proportional Profiles”. These models can—with appropriate data—empirically determine a unique best fitting axis orientation without the need for a separate factor rotation stage, but they have not been general enough to deal with what Tucker has called “interactions” among dimensions. This article presents a proof of unique axis orientation for a considerably more general parallel profiles model which incorporates interacting dimensions. The model, X_k=A^AD_k H^BD_k B', does not assume symmetry in the data or in the interactions among factors. A second proof is presented for the symmetrically weighted case (i.e., where^AD_k=^BD_k). The generality of these models allows one to impose successive restrictions to obtain several useful special cases, including PARAFAC2 and three-way DEDICOM. We want to express appreciation for the contributions of several colleagues: Jos M. F. ten Berge and Henk A. L. Kiers carefully went through more than one version of this article, found an important error, and contributed many improvements. J. Douglas Carroll and Shizuhiko Nishisato acted with unusual editorial preserverance and flexibility, thereby making possible the successful completion of a difficult assessment and revision process. Joseph B. Kruskal has long provided crucial mathematical insights and inspiration to those working in this area, but this is particularly true for us. His contributions to this specific article include early discussion of basic questions and careful examination of some earlier attempted proofs, finding them to be invalid. The anonymous reviewers also made useful suggestions. Some portions of this work were supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada.     

A new procedure for detection of crossing DIF

The purpose of this paper is to present a hypothesis testing and estimation procedure, Crossing SIBTEST, for detecting crossing DIF. Crossing DIF exists when the difference in the probabilities of a correct answer for the two examinee groups changes signs as ability level is varied. In item response theory terms, crossing DIF is indicated by two crossing item characteristic curves. Our new procedure, denoted as Crossing SIBTEST, first estimates the matching subtest score at which crossing occurs using least squares regression analysis. A Crossing SIBTEST statistic then is used to test the hypothesis of crossing DIF. The performance of Crossing SIBTEST is evaluated in this study.This research was partially supported by a grant from the Law School Admission Council and by National Science Foundation Mathematics Grant NSF-DMS-94-04327. The research reported here is collaborative in every respect and the order of authorship is alphabetical. The authors thank Jeff Douglas and Louis Roussos for their useful comments and discussions.