Confidence regions for INDSCAL using the jackknife and bootstrap techniques

Bootstrap and jackknife techniques are used to estimate ellipsoidal confidence regions of group stimulus points derived from INDSCAL. The validity of these estimates is assessed through Monte Carlo analysis. Asymptotic estimates of confidence regions based on a MULTISCALE solution are also evaluated. Our findings suggest that the bootstrap and jackknife techniques may be used to provide statements regarding the accuracy of the relative locations of points in space. Our findings also suggest that MULTISCALE asymptotic estimates of confidence regions based on small samples provide an optimistic view of the actual statistical reliability of the solution. The authors wish to thank Geert DeSoete, Richard A. Harshman, William Heiser, Jon Kettenring, Joseph B. Kruskal, Jacqueline Meulman, James O. Ramsay, John W. Tukey, Paul A. Tukey, and Mike Wish. Sharon L. Weinberg is a consultant at AT&T Bell Laboratories, Murray Hill, New Jersey 07974.     

Sequential estimation in multidimensional scaling

The concept of sequential estimation is introduced in multidimensional scaling (MDS). The sequential estimation method developed in this paper refers to continually updating estimates of a configuration as new observations are added. This method has a number of advantages, such as a locally optimal design of the experiment can be easily constructed, and dynamic experimentation is made possible. Using artificial data, the performance of our sequential method is illustrated.We are indebted to anonymous reviewers for their suggestions. In addition, we thank Dr. Frank Critchley for his helpful comments on our Q/S algorithm.     

Spatial versus tree representations of proximity data

In this paper we investigated two of the most common representations of proximities, two-dimensional euclidean planes and additive trees. Our purpose was to develop guidelines for comparing these representations, and to discover properties that could help diagnose which representation is more appropriate for a given set of data. In a simulation study, artificial data generated either by a plane or by a tree were scaled using procedures for fitting either a plane (KYST) or a tree (ADDTREE). As expected, the appropriate model fit the data better than the inappropriate model for all noise levels. Furthermore, the two models were roughly comparable: for all noise levels, KYST accounted for plane data about as well as ADDTREE accounted for tree data. Two properties of the data proved useful in distinguishing between the models: the skewness of the distribution of distances, and the proportion of elongated triangles, which measures departures from the ultrametric inequality, Applications of KYST and ADDTREE to some twenty sets of real data, collected by other investigators, showed that most of these data could be classified clearly as favoring either a tree or a two-dimensional representation.A portable PASCAL program implementing the Sattath and Tversky [1977] ADDTREE algorithm is available from J. Corter, Department of Psychology, Stanford University, Stanford, California 94305.     

Principal component analysis with external information on both subjects and variables

A method for structural analysis of multivariate data is proposed that combines features of regression analysis and principal component analysis. In this method, the original data are first decomposed into several components according to external information. The components are then subjected to principal component analysis to explore structures within the components. It is shown that this requires the generalized singular value decomposition of a matrix with certain metric matrices. The numerical method based on the QR decomposition is described, which simplifies the computation considerably. The proposed method includes a number of interesting special cases, whose relations to existing methods are discussed. Examples are given to demonstrate practical uses of the method.The work reported in this paper was supported by grant A6394 from the Natural Sciences and Engineering Research Council of Canada to the first author. Thanks are due to Jim Ramsay, Haruo Yanai, Henk Kiers, and Shizuhiko Nishisato for their insightful comments on earlier versions of this paper. Jim Ramsay, in particular, suggested the use of the QR decomposition, which simplified the presentation of the paper considerably.     

A general solution for a class of weakly constrained linear regression problems

This paper contains a globally optimal solution for a class of functions composed of a linear regression function and a penalty function for the sum of squared regression weights. Global optimality is obtained from inequalities rather than from partial derivatives of a Lagrangian function. Applications arise in multidimensional scaling of symmetric or rectangular matrices of squared distances, in Procrustes analysis, and in ridge regression analysis. The similarity of existing solutions for these applications is explained by considering them as special cases of the general class of functions addressed.The author is obliged to Henk Kiers and Willem Heiser for helpful comments.     

Was euclid an unnecessarily sophisticated psychologist?

A survey of the current state of multidimensional scaling using the city-block metric is presented. Topics include substantive and theoretical issues, recent algorithmic developments and their implications for seemingly straightforward analyses, isometries with other metrics, links to graph-theoretic models, and future prospects.Presented as the 1991 Psychometric Society Presidential Address. I am indebted to Doug Carroll, John Daws, Jan de Leeuw, Geert De Soete, Wayne DeSarbo, Eric Holman, Larry Hubert, Chingis Izmailov, Joe Kruskal, Rob Nosofsky, Akinori Okada, Roger Shepard, Auke Tellegen, and Wijbrandt van Schuur for many helpful comments on this research and to Yuko Minowa for bibliographic assistance. Doug Carroll's comments on an early draft were especially useful. Parts of this research were supported by a grant from AT&T to the University of Illinois. Some of the computational results reported here were obtained during the early 1970's at the Institute for Mathematical Studies in the Social Sciences at Stanford. Grateful acknowledgment is made to the Institute's staff and its Co-Directors, Richard C. Atkinson and Patrick Suppes.     

A simple multivariate probabilistic model for preferential and triadic choices

Multidimensional probabilistic models of behavior following similarity and choice judgements have proven to be useful in representing multidimensional percepts in Euclidean and non-Euclidean spaces. With few exceptions, these models are generally computationally intense because they often require numerical work with multiple integrals. This paper focuses attention on a particularly general triad and preferential choice model previously requiring the numerical evaluation of a 2n-fold integral, wheren is the number of elements in the vectors representing the psychological magnitudes. Transforming this model to an indefinite quadratic form leads to a single integral. The significance of this form to multidimensional scaling and computational efficiency is discussed.The authors would like to thank Jean-Claude Falmagne and Norman Johnson for suggestions and advice concerning quadratic forms.     

Effects on indscal of non-orthogonal perceptions of object space dimensions

A general question is raised concerning the possible consequences of employing the very popular INDSCAL multidimensional scaling model in cases where the assumptions of that model may be violated. Simulated data are generated which violate the INDSCAL assumption that all individuals perceive the dimensions of the common object space to be orthogonal. INDSCAL solutions for these various sets of data are found to exhibit extremely high goodness of fit, but systematically distorted object spaces and negative subject weights. The author advises use of Tucker's three-mode model for multidimensional scaling, which can account for non-orthogonal perceptions of the object space dimensions. It is shown that the INDSCAL model is a special case of the three-mode model.     

A procedure for ordering object pairs consistent with the multidimensional unfolding model

A procedure for ordering object (stimulus) pairs based on individual preference ratings is described. The basic assumption is that individual responses are consistent with a nonmetric multidimensional unfolding model. The method requires data where a numerical response is independently generated for each individual-object pair. In conjunction with a nonmetric multidimensional scaling procedure, it provides a vehicle for recovering meaningful object configurations.The author wishes to thank Jack Hoadley, Larry Mayer, Sheldon Newhouse, Stuart Rabinowitz, Forrest Young, and three anonymous reviewers for their useful suggestions.     

Thurstone's discriminal processes fifty years later

Four issues are discussed concerning Thurstone's discriminal processes: the distributions governing the representation, the nature of the response decision rules, the relation of the mean representation to physical characteristics of the stimulus, and factors affecting the variance of the representation. A neural schema underlying the representation is proposed which involves samples in time of pulse trains on individual neural fibers, estimators of parameters of the several pulse trains, samples of neural fibers, and an aggregation of the estimates over the sample. The resulting aggregated estimate is the Thurstonian representation. Two estimators of pulse rate, which is monotonic with signal intensity, are timing and counting ratios and two methods of aggregation are averaging and maximizing. These lead to very different predictions in a speed-accuracy experiment; data indicate that both estimators are available and the aggregation is by averaging. Magnitude estimation data are then used both to illustrate an unusual response rule and to study the psychophysical law. In addition, the pattern of variability and correlation of magnitude estimates on successive trials is interpreted in terms of the sample size over which the aggregation takes place. Neural sample size is equated with selective attention, and is an important factor affecting the variability of the representation. It accounts for the magical number seven phenomenon in absolute identification and predicts the impact of nonuniform distributions of intensities on the absolute identification of two frequencies. 1977 Psychometric Society Presidential Address. This work was supported in part by a grant of the National Science Foundation to Harvard University. I wish to express my appreciation to S. Burbeck, D. M. Green, M. Shaw, and B. Wandell for their useful comments on an earlier draft of this paper.