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.     

Speech perception by budgerigars (Melopsittacus undulatus): spoken vowels

Discrimination of natural, sustained vowels was studied in 5 budgerigars. The birds were trained using operant conditioning procedures on a same-different task, which was structured so that response latencies would provide a measure of stimulus similarity. These response latencies were used to construct similarity matrices, which were then analyzed by multidimensional scaling (MDS) procedures. MDS produced spatial maps of these speech sounds where perceptual similarity was represented by spatial proximity. The results of the three experiments suggest that budgerigars perceive natural, spoken vowels according to phonetic categories, find the acoustic differences among different talkers less salient than the acoustic differences among vowel categories, and use formant frequencies in making these complex discriminations.     

Distance stability analysis in multidimensional scaling using the jackknife method

Stability or sensitivity analysis is an important topic in data analysis that has received little attention in the application of multidimensional scaling (MDS), for which the only available approaches are given in terms of a coordinate‐based analytical jackknife methodology. Although in MDS the prime interest is in assessing the stability of the points in the configuration, this methodology may be influenced by imprecisions resulting from the inherently necessary Procrustes method. This paper proposes an analytical distance‐based jackknife procedure to study stability and cross‐validation in MDS in terms of the jackknife distances, which is not influenced by the Procrustes method. For each object, the corresponding jackknife estimated points are considered as naturally clustered points, and stability and cross‐validation are analysed in terms of the MDS distances arising from the jackknife procedure, on the basis of a weighted cluster‐MDS algorithm. A jackknife‐relevant configuration is also proposed for cross‐validation in terms of coordinates, in a cluster‐MDS framework.     

Scaling techniques for modeling directional knowledge

A common way for researchers to model or graphically portray spatial knowledge of a large environment is by applying multidimensional scaling (MDS) to a set of pairwise distance estimations. We introduce two MDS-like techniques that incorporate people’s knowledge of directions instead of (or in addition to) their knowledge of distances. Maps of a familiar environment derived from these procedures were more accurate and were rated by participants as being more accurate than those derived from nonmetric MDS. By incorporating people’s relatively accurate knowledge of directions, these methods offer spatial cognition researchers and behavioral geographers a sharper analytical tool than MDS for studying cognitive maps.     

Normative location theory: Placement in continuous space

This paper selectively reviews an area of operations research, refered to as normative location theory, that deals with the placement of objects in continuous space. The computational procedures discussed in this literature require initially that a certain set objects be located a priori; the placement of the objects in a second set is then determined in such a way that the total “interaction” among the fixed and variable objects is minimized. A number of strategies appropriate for different distance functions are surveyed and a numerical example is given illustrating one particular iterative algorithm. Although the current versions of these location methods depend upon more than the nonmetric information contained in the proximity measures defined for all pairs of objects, they can be generalized, and consequently, several possible connections to nonmetric multidimensional scaling are pointed out.     

Experimenting With Perceptual Change Strategies

Nonmetric multidimensional scaling procedures are employed in this experimental evaluation of perceptual changes resulting from differences in the presentation of information about stimuli. Longitudinal MDS techniques are found to be useful in monitoring perceptual changes produced by various communication alternatives. Communication strategies aimed at changing the relationships among stimuli are more effective than those designed to change the perceptual space structure. Furthermore, providing comparative information about groups of stimuli has more apparent impact than providing information about a single stimulus.     

Cluster differences scaling with a within-clusters loss component and a fuzzy successive approximation strategy to avoid local minima

Cluster differences scaling is a method for partitioning a set of objects into classes and simultaneously finding a low-dimensional spatial representation ofK cluster points, to model a given square table of dissimilarities amongn stimuli or objects. The least squares loss function of cluster differences scaling, originally defined only on the residuals of pairs of objects that are allocated to different clusters, is extended with a loss component for pairs that are allocated to the same cluster. It is shown that this extension makes the method equivalent to multidimensional scaling with cluster constraints on the coordinates. A decomposition of the sum of squared dissimilarities into contributions from several sources of variation is described, including the appropriate degrees of freedom for each source. After developing a convergent algorithm for fitting the cluster differences model, it is argued that the individual objects and the cluster locations can be jointly displayed in a configuration obtained as a by-product of the optimization. Finally, the paper introduces a fuzzy version of the loss function, which can be used in a successive approximation strategy for avoiding local minima. A simulation study demonstrates that this strategy significantly outperforms two other well-known initialization strategies, and that it has a success rate of 92 out of 100 in attaining the global minimum.     

Scale properties of the Holtzman inkblots

Scaling techniques were presently applied to perceptions of inkblots, to empirically delineate the relationship between their stimulus properties and the nature of verbal associations elicited in projective testing. The Holtzman Inkblot Technique (HIT) was administered in group form to a relatively diverse group of college students. Subjects also individually rated the similarity of pairs of the HIT inkblots. Similarity judgments were analyzed via a multidimensional scaling (MDS) approach which recovered dimensions of variations among blots. The MDS procedures also captured variation across subjects in their utilization of blot stimulus properties. MDS solutions generally reflected differences among the blots along dimensions of physical characteristics of the blots. Differences in responsiveness of subjects to these characteristics appeared to reliably reflect meaningful substantive distinctions among subjects, many of which were not captured by traditional HIT variables. Implications were discussed in terms of further MDS applications and possible re-evaluation of HIT variables or procedures.     

Multidimensional Scaling in Counseling Research and Practice

The authors briefly describe the use of multidimensional scaling (MDS) in counseling. They discuss types of data that have been analyzed using MDS, kinds of MDS analyses, and the interpretation of MDS results and review MDS research relevant to vocational, family, group, and individual counseling. They also briefly discuss potential applications in counseling practice.     

ConPar: a method for identifying groups of concordant subject proximity matrices for subsequent multidimensional scaling analyses

A common representation of data within the context of multidimensional scaling (MDS) is a collection of symmetric proximity (similarity or dissimilarity) matrices for each of M subjects. There are a number of possible alternatives for analyzing these data, which include: (a) conducting an MDS analysis on a single matrix obtained by pooling (averaging) the M subject matrices, (b) fitting a separate MDS structure for each of the M matrices, or (c) employing an individual differences MDS model. We discuss each of these approaches, and subsequently propose a straightforward new method (CONcordance PARtitioning—ConPar), which can be used to identify groups of individual-subject matrices with concordant proximity structures. This method collapses the three-way data into a subject×subject dissimilarity matrix, which is subsequently clustered using a branch-and-bound algorithm that minimizes partition diameter. Extensive Monte Carlo testing revealed that, when compared to K-means clustering of the proximity data, ConPar generally provided better recovery of the true subject cluster memberships. A demonstration using empirical three-way data is also provided to illustrate the efficacy of the proposed method.     

A Counseling Practitioner's Primer to the Use of Multidimensional Scaling

Applications of multidimensional scaling (MDS) to counseling practice are discussed, and the author's use of MDS as a tool in a counseling training group is described.     

Avoiding the dangers of averaging across subjects when using multidimensional scaling

Ashby, Maddox and Lee (Psychological Science, 5 (3) 144) argue that it can be inappropriate to fit multidimensional scaling (MDS) models to similarity or dissimilarity data that have been averaged across subjects. They demonstrate that the averaging process tends to make dissimilarity data more amenable to metric representations, and conduct a simulation study showing that noisy data generated using one distance metric, when averaged, may be better fit using a different distance metric. This paper argues that a Bayesian measure of MDS models has the potential to address these difficulties, because it takes into account data-fit, the number of dimensions used by an MDS representation, and the precision of the data. A method of analysis based on the Bayesian measure is demonstrated through two simulation studies with accompanying theoretical analysis. In the first study, it is shown that the Bayesian analysis rejects those MDS models showing better fit to averaged data using the incorrect distance metric, while accepting those that use the correct metric. In the second study, different groups of simulated ‘subjects’ are assumed to use different underlying configurations. In this case, the Bayesian analysis rejects MDS representations where a significant proportion of subjects use different configurations, or when their dissimilarity judgments contain significant amounts of noise. It is concluded that the Bayesian analysis provides a simple and principled means for systematically accepting and rejecting MDS models derived from averaged data.     

Using a multidimensional scaling approach to investigate the underlying basis of ease of learning judgments

Jönsson, F. U. & Lindström, B. R. (2009) Using a multidimensional scaling approach to investigate the underlying basis of ease of learning judgments. Scandinavian Journal of Psychology, 51, 103–108. Before studying a material it is of strategic importance to first assess its difficulty, so called Ease of Learning (EOL) judgments. A multidimensional scaling (MDS) procedure was used to investigate the underlying basis of EOL judgments for 24 nouns, which to the authors’ knowledge has not been done before. In addition, Judgments of Learning (JOL) followed by a free recall test was performed. The MDS analysis indicated that EOL judgments for the nouns are based on multiple cues (dimensions), namely word length, frequency, and concreteness. Moreover, the concreteness values of the nouns, as judged by an independent group, were correlated with both the JOLs and the concreteness dimension from the MDS analysis. This indicates that EOLs and JOLs for single words are based, to some extent, on the same cues.     

Identification Model Based on the Maximum Information Entropy Principle

A new theoretical approach to stimulus identification is proposed through a probabilistic multidimensional model based on the maximum information entropy principle. The approach enables us to derive the multidimensional scaling (MDS) choice model, without appealing to Luce's choice rule and without defining a similarity function. It also clarifies the relationship between the MDS choice model and the optimal version of the identification model based on Ashby's general recognition theory; it is shown theoretically that the identification model derived from the new approach includes these two models as special cases. Finally, as an application of our approach, a model of similarity judgment is proposed and compared with Ashby's extended similarity model. Copyright 2001 Academic Press.     

A stochastic multidimensional scaling procedure for the empirical determination of convex indifference curves for preference/choice analysis

The vast majority of existing multidimensional scaling (MDS) procedures devised for the analysis of paired comparison preference/choice judgments are typically based on either scalar product (i.e., vector) or unfolding (i.e., ideal-point) models. Such methods tend to ignore many of the essential components of microeconomic theory including convex indifference curves, constrained utility maximization, demand functions, et cetera. This paper presents a new stochastic MDS procedure called MICROSCALE that attempts to operationalize many of these traditional microeconomic concepts. First, we briefly review several existing MDS models that operate on paired comparisons data, noting the particular nature of the utility functions implied by each class of models. These utility assumptions are then directly contrasted to those of microeconomic theory. The new maximum likelihood based procedure, MICROSCALE, is presented, as well as the technical details of the estimation procedure. The results of a Monte Carlo analysis investigating the performance of the algorithm as a number of model, data, and error factors are experimentally manipulated are provided. Finally, an illustration in consumer psychology concerning a convenience sample of thirty consumers providing paired comparisons judgments for some fourteen brands of over-the-counter analgesics is discussed.     

Crime scene actions and offender characteristics in a sample of Finnish stranger rapes

The crime scene actions employed by offenders in stranger rapes were analysed in relation to offender characteristics. Data were drawn from an official police database and consisted of stranger rapes occurring in Finland between 1992 and 2001 (n = 100). The structure of dichotomous variables derived through a content analysis of crime scene actions and offender characteristics were analysed with non‐metric multidimensional scaling (MDS). The first analysis revealed three separate action themes, with thematic emphases on Hostility, Involvement or Theft. The MDS‐solution for offender characteristics suggested four themes: Conventional; Psychiatric/Elderly; Criminal/Violent; and Criminal/Property. Each case was assigned to one of the themes or as a hybrid in order to analyse the associations between action themes and characteristics. The only significant association was found between the action theme, Theft and characteristics theme Criminal/Property. The results are discussed in relation to previous research. Copyright © 2004 John Wiley & Sons, Ltd.