The problem of the additive constant and eigenvalues in metric multidimensional scaling

This paper is concerned with the additive constant problem in metric multidimensional scaling. First the influence of the additive constant on eigenvalues of a scalar product matrix is discussed. The second part of this paper is devoted to the introduction of a new formulation of the additive constant problem. A solution is given for fixed dimensionality, by maximizing a normalized index of fit with a gradient method. An experimental computation has shown that the author's solution is accurate and easy to follow.     

The additive constant problem in multidimensional scaling

The problem of choosing the correct additive constant to convert relative interstimulus distances to absolute interstimulus distances in multidimensional scaling is investigated. An artificial numerical example is constructed, and various trial values of the constant are inserted to demonstrate the effect on the multidimensional map of making a variety of incorrect choices. Finally, a general solution to the problem, suggested by Dr. Ledyard R Tucker, is presented; each of the computational steps in this solution is set down for easy reference.This study was supported in part by Office of Naval Research Contract N6onr-270-20 and by National Science Foundation Grant G-642 to Princeton University.     

A generalized majorization method for least souares multidimensional scaling of pseudodistances that may be negative

The usual convergence proof of the SMACOF algorithm model for least squares multidimensional scaling critically depends on the assumption of nonnegativity of the quantities to be fitted, called the pseudodistances. When this assumption is violated, erratic convergence behavior is known to occur. Three types of circumstances in which some of the pseudodistances may become negative are outlined: nonmetric multidimensional scaling with normalization on the variance, metric multidimensional scaling including an additive constant, and multidimensional scaling under the city-block distance model. A generalization of the SMACOF method is proposed to resolve the difficulty that is based on the same rationale frequently involved in robust fitting with least absolute residuals.I am grateful to Patrick Groenen and Rian van Blokland-Vogelesang for their help with some of the computations, and to the anonymous referees for their very useful comments.     

Nonmetric multidimensional scaling: A monte carlo study of the basic parameters

Metric determinacy of nonmetric multidimensional scaling was investigated as a function of the number of points being scaled, the amount of error in the data being scaled, and the accuracy of estimation of the Minkowski distance function parameters, dimensionality and the r-constant. It was found that nonmetric scaling may provide better models if (1) the true structure is of low dimensionality, (2) the dimensionality of recovered structure is not less than the dimensionality of the true structure, (3) degree of error is low, and (4) the degrees of freedom ratio is greater than about 2.5. It was also found that (5) accurate estimation of the Minkowski constant leads to a better model only if the dimensionality has been properly estimated.This report is based on a thesis submitted in partial fulfillment of the degree of Master of Arts at the University of North Carolina, April, 1970. The thesis is an outgrowth of earlier work done with Forrest W. Young. The author is indebted to Forrest W. Young, Norman Cliff, and Lyle V. Jones for their assistance in the preparation of this report. This report was supported in part by PHS research grant No. M-10006 from the National Institute of Mental Health, Public Health Service.     

An empirical evaluation of multidimensional successive intervals

The multidimensional method of successive intervals and the method of complete triads are applied to similarity judgments of Munsell colors varying in brightness, saturation, and hue. Both methods yield configurations that correlate highly with the Munsell color structure. This validation of these scaling methods in an area of known dimensionality indicates their applicability for exploration in areas of unknown dimensionality. This study was supported in part by the Office of Naval Research contract N6onr-270-20 with Princeton University and also in part by funds from the National Science Foundation. The research was carried out when the author was an Educational Testing Service Psychometric Fellow at Princeton University.     

Nonmetric multidimensional scaling: Recovery of metric information

The degree of metric determinancy afforded by nonmetric multidimensional scaling was investigated as a function of the number of points being scaled, the true dimensionality of the data being scaled, and the amount of error contained in the data being scaled. It was found 1) that if the ratio of the degrees of freedom of the data to that of the coordinates is sufficiently large then metric information is recovered even when random error is present; and 2) when the number of points being scaled increases the stress of the solution increases even though the degree of metric determinacy increases.This report was supported in part by a PHS research grant No. M-10006 from the National Institute of Mental Health, Public Health Service, and in part by a Science Development grant No. GU-2059, from the National Science Foundation. The author is indebted to Charles R. Sherman for his assistance in gathering the data and for his critical re-writing of sections of this report. The assistance of Lyle V. Jones in his critical readings and comments is also deeply appreciated.     

Multidimensional scaling of nominal data: The recovery of metric information with alscal

Multidimensional scaling has recently been enhanced so that data defined at only the nominal level of measurement can be analyzed. The efficacy of ALSCAL, an individual differences multidimensional scaling program which can analyze data defined at the nominal, ordinal, interval and ratio levels of measurement, is the subject of this paper. A Monte Carlo study is presented which indicates that (a) if we know the correct level of measurement then ALSCAL can be used to recover the metric information presumed to underlie the data; and that (b) if we do not know the correct level of measurement then ALSCAL can be used to determine the correct level and to recover the underlying metric structure. This study also indicates, however, that with nominal data ALSCAL is quite likely to obtain solutions which are not globally optimal, and that in these cases the recovery of metric structure is quite poor. A second study is presented which isolates the potential cause of these problems and forms the basis for a suggested modification of the ALSCAL algorithm which should reduce the frequency of locally optimal solutions.     

Multidimensional scaling: I. Theory and method

Multidimensional scaling can be considered as involving three basic steps. In the first step, a scale of comparative distances between all pairs of stimuli is obtained. This scale is analogous to the scale of stimuli obtained in the traditional paired comparisons methods. In this scale, however, instead of locating each stimulus-object on a given continuum, the distances between each pair of stimuli are located on a distance continuum. As in paired comparisons, the procedures for obtaining a scale of comparative distances leave the true zero point undetermined. Hence, a comparative distance is not a distance in the usual sense of the term, but is a distance minus an unknown constant. The second step involves estimating this unknown constant. When the unknown constant is obtained, the comparative distances can be converted into absolute distances. In the third step, the dimensionality of the psychological space necessary to account for these absolute distances is determined, and the projections of stimuli on axes of this space are obtained. A set of analytical procedures was developed for each of the three steps given above, including a least-squares solution for obtaining comparative distances by the complete method of triads, two practical methods for estimating the additive constant, and an extension of Young and Householder's Euclidean model to include procedures for obtaining the projections of stimuli on axes from fallible absolute distances.This study was carried out while the author was an Educational Testing Service Psychometric Fellow at Princeton University. The author expresses his appreciation to his thesis adviser, Dr. H. Gulliksen, for his guidance throughout the study and to Dr. B. F. Green, Jr., for valuable assistance on several of the derivations.     

An Examination Of Some Factors Related To Using Different Minkowski Models In Non-Metric Multidimensional Scaling

Great interest in non-metric multidimensional scaling has resulted in a number of computer programs to derive solutions. This study examined the effect upon stress of data generated under five metrics and recovered under all five metrics. MDSCAL-5M. TORSCA-9, and POLYCON-II were used to analyse these data. POLYCON-II was the most accurate, although none of the programs was highly successful. In most cases recovery with the Euclidian metric provided, if not the best, very close to the best recovery regardless of the true metric. This study also raised the question of the advisability of using different metric models in nonmetric multidimensional scaling and found that even very different Minkowski metrics are quite similar in the way they rank order dissimilarities.     

Determining the dimensionality in spatial representations of semantic concepts

When multidimensional scaling solutions are used to study semantic concepts, the dimensionality of the optimal configuration has to be determined. Several strategies have been proposed to choose the appropriate dimensionality. In the present paper, the traditional dimensionality choice criteria were evaluated and compared to a method based on the prediction of an external criterion. Two studies were conducted in which typicality of an exemplar within a semantic concept was predicted from its distance to the concept centroid. In contrast to the low-dimensional solutions selected by the traditional methods, predictions of an external criterion improved with additional dimensions up till dimensionalities that were much higher than what is common in the literature. This suggests that traditional methods underestimate the richness of semantic concepts as revealed in spatial representations derived from similarity measures.     

Random effects diagonal metric multidimensional scaling models

By assuming a distribution for the subject weights in a diagonal metric (INDSCAL) multidimensional scaling model, the subject weights become random effects. Including random effects in multidimensional scaling models offers several advantages over traditional diagonal metric models such as those fitted by the INDSCAL, ALSCAL, and other multidimensional scaling programs. Unlike traditional models, the number of parameters does not increase with the number of subjects, and, because the distribution of the subject weights is modeled, the construction of linear models of the subject weights and the testing of those models is immediate. Here we define a random effects diagonal metric multidimensional scaling model, give computational algorithms, describe our experiences with these algorithms, and provide an example illustrating the use of the model and algorithms.We would like to thank J. Douglas Carroll for early consultation of this research, and Robert I. Jennrich for commenting on an earlier draft of this paper and for help on the computational algorithms. James O. Ramsay and Forrest W. Young were instrumental in providing the example data. This work was supported in part by National Institute of Mental Health grant 1 R43 MH57559-01. We would also like to thank the anonymous referees for comments that helped to clarify our work.     

Constant curvature Riemannian scaling

A multidimensional scaling algorithm is proposed for fitting distances to constant curvature Riemannian spaces. Examples are given and potential applications are discussed. Some general properties of Riemannian spaces are also discussed. It is argued that some restriction, such as that of constant curvature, is necessary to obtain simple unique solutions in Riemannian spaces.     

Determining the Dimensionality of Multidimensional Scaling Representations for Cognitive Modeling

Multidimensional scaling models of stimulus domains are widely used as a representational basis for cognitive modeling. These representations associate stimuli with points in a coordinate space that has some predetermined number of dimensions. Although the choice of dimensionality can significantly influence cognitive modeling, it is often made on the basis of unsatisfactory heuristics. To address this problem, a Bayesian approach to dimensionality determination, based on the Bayesian Information Criterion (BIC), is developed using a probabilistic formulation of multidimensional scaling. The BIC approach formalizes the trade-off between data-fit and model complexity implicit in the problem of dimensionality determination and allows for the explicit introduction of information regarding data precision. Monte Carlo simulations are presented that indicate, by using this approach, the determined dimensionality is likely to be accurate if either a significant number of stimuli are considered or a reasonable estimate of precision is available. The approach is demonstrated using an established data set involving the judged pairwise similarities between a set of geometric stimuli. Copyright 2001 Academic Press.     

The integration of multidimensional scaling and multivariate analysis with optimal transformations

The recent history of multidimensional data analysis suggests two distinct traditions that have developed along quite different lines. In multidimensional scaling (MDS), the available data typically describe the relationships among a set of objects in terms of similarity/dissimilarity (or (pseudo-)distances). In multivariate analysis (MVA), data usually result from observation on a collection of variables over a common set of objects. This paper starts from a very general multidimensional scaling task, defined on distances between objects derived from one or more sets of multivariate data. Particular special cases of the general problem, following familiar notions from MVA, will be discussed that encompass a variety of analysis techniques, including the possible use of optimal variable transformation. Throughout, it will be noted how certain data analysis approaches are equivalent to familiar MVA solutions when particular problem specifications are combined with particular distance approximations.This research was supported by the Royal Netherlands Academy of Arts and Sciences (KNAW). An earlier version of this paper was written during a stay at McGill University in Montréal; this visit was supported by a travel grant from the Netherlands Organization for Scientific Research (NWO). I am grateful to Jim Ramsay and Willem Heiser for their encouragement and helpful suggestions, and to the Editor and referees for their constructive comments.     

The tunneling method for global optimization in multidimensional scaling

This paper focuses on the problem of local minima of the STRESS function. It turns out that unidimensional scaling is particularly prone to local minima, whereas full dimensional scaling with Euclidean distances has a local minimum that is global. For intermediate dimensionality with Euclidean distances it depends on the dissimilarities how severe the local minimum problem is. For city-block distances in any dimensionality many different local minima are found. A simulation experiment is presented that indicates under what conditions local minima can be expected. We introduce the tunneling method for global minimization, and adjust it for multidimensional scaling with general Minkowski distances. The tunneling method alternates a local search step, in which a local minimum is sought, with a tunneling step in which a different configuration is sought with the same STRESS as the previous local minimum. In this manner successively better local minima are obtained, and experimentation so far shows that the last one is often a global minimum.This paper is based on the 1994 Psychometric Society's outstanding thesis award of the first author. The authros would like to thank Robert Tijssen of the CWTS Leiden for kindly making available the co-citation data of the Psychometric literature. This paper is an extended version of the paper presented at the Annual Meeting of the Psychometric Society at Champaign-Urbana, Illin., June 1994.     

Dimensional and metric structures in multidimensional stimuli

The present experiments investigated two characteristics of subjects’ multidimensional representations: their dimensional organization and metric structure, for both analyzable and integral stimuli. In Experiment 1, subjects judged the dissimilarity between all pairs of stimuli differing in brightness and size (analyzable stimuli), while in Experiment 2, subjects made dissimilarity judgments for stimuli varying in width height, and area shape (integral stimuli). For the brightness size stimuli, the findings that (a) brightness judgments were independent of size (and vice versa) and (b) the best fitting scaling solution was one that depicted an orthogonal structure are strong evidence that subjects perceived brightness size as a dimensionally organized structure. In contrast, for the rectangle stimuli, neither width height nor area shape contributed additively to overall dissimilarity. The results of the metric fitting were more equivocal. For all stimulus sets, the Euclidean metric yielded scaling solutions with lower stress values than the city block metric. When bidimensional ratings were regressed on unidimensional ratings, the city block metric yielded a slightly higher correlation coefficient than the Euclidean metric for brightness size stimuli. The two rules of combination were equivalent for the width-height stimuli, but the Euclidean metric provided a better approximation for the area shape stimuli. The results were discussed in terms of how subjects integrate physical dimensions for the case of integral stimuli and the superiority of dimensional vs. metric structure as an indicator of stimulus analyzability.     

MULTIDIMENSIONAL SCALING OF GEOMETRICAL FIGURES

In two experiments with geometrical figures as stimuli, constructed in order to generate perceptual variation of roundness and height, four multidimensional scaling methods were compared, two metric and two nonmetric. The methods gave similar solutions. The fact that the solutions could be interpreted in accordance with expectations, points to the validity of the methods.     

Interactive scaling with individual subjects

A metric multidimensional scaling (MDS) procedure based on computer-subject interaction is developed, and an experiment designed to validate the procedure is presented. The interactive MDS system allows generalization of current MDS systems in two directions: (a) very large numbers of stimuli may be scaled; and (b) the scaling is performed with individual subjects, facilitating the investigation of individual as well as group processes. The experiment provided positive support for the interactive MDS system. Specifically, (a) individual data are amenable to meaningful interpretation, and they provide a tentative basis for quantitative investigation; and (b) grouped data provide meaningful interpretive and quantitative results which are equivalent to results from standard paired-comparisons methods.This report was supported in part by a PHS research grant, No. M-10006, from the National Institute of Mental Health, in part by a Science Development grant, No. GU-2059, from the National Science Foundation, both granted to the Psychometric Laboratory at the University of North Carolina, and in part by a PHS research grant, No. MH-16474, from the National Institute of Mental Health, Public Health Service, granted to the second author. The major portion of this research was performed while the second author was the L. L. Thurstone Distinguished Fellow at the Psychometric Laboratory of the University of North Carolina while on leave from the University of Southern California. The authors are indebted to Amnon Rapoport and Thomas S. Wallsten for their critical evaluations of an earlier version of this report. While this paper was entirely a cooperative effort on the part of both authors, the first author was primarily responsible for the algorithms, and the second for developing the mathematical model.     

The Dimensionality of Locus of Control Among French Students

To learn more about the dimensionality of locus of control, I developed a new internal-external (I-E) scale for French students. Four criteria were used for the construction of the scale: causal explanation and orientation of behavioral outcomes, situational contents, and control ideology. Questionnaires were administered to 200 male and female undergraduates in psychology. A principal-components analysis and a nonmetric multidimensional scaling were used. The hypothesis of the unidimensionality of locus of control was confirmed.     

Solving implicit equations in psychometric data analysis

Many data analysis problems in psychology may be posed conveniently in terms which place the parameters to be estimated on one side of an equation and an expression in these parameters on the other side. A rule for improving the rate of convergence of the iterative solution of such equations is developed and applied to four problems: the principal axis communality problem, individual differences multidimensional scaling,L _P norm multiple regression, andL _P norm factor analysis of a data matrix. The rule results in substantially faster solutions or in solutions where none would be possible without the rule.This work was supported by National Research Council of Canada grant APA 320 to the author.