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 multidimensional scaling analysis of 16 complex sounds

The INDSCAL multidimensional scaling model was used to investigate the distinctive features involved in the perception of 16 complex nonspeech sounds. The signals differed along four physical dimensions: fundamental frequency, waveform, formant frequency, and number of formants. Scaling results indicated that subjects’ similarity ratings could be accounted for by three psychological or perceptual dimensions. A statistically reliable correspondence was observed between these perceptual dimensions and the physical characteristics of fundamental frequency, waveform, and a combination of the two formant parameters. These results were further explored with Johnson’s (1967) hierarchical clustering analysis. Large differences in featural saliency occurred in the group data with fundamental accounting for more variability than the remaining dimensions. Further analysis of individual subject data revealed large individual differences in featural saliency. These differences were related to past musical experience of the subject and to earlier findings using similar signals. It was concluded that (1) the INDSCAL model provides a useful method for the analysis of auditory perception in the nonspeech mode, and (2) featural saliency in such sounds is likely to be determined by an unspecified attentional mechanism.     

The <Emphasis Type="Italic">K</Emphasis>-INDSCAL Model for Heterogeneous Three-Way Dissimilarity Data

A weighted Euclidean distance model for analyzing three-way dissimilarity data (stimuli by stimuli by subjects) for heterogeneous subjects is proposed. First, it is shown that INDSCAL may fail to identify a common space representative of the observed data structure in presence of heterogeneity. A new model that removes the rotational invariance of the classical multidimensional scaling problem and specifies K common homogeneous spaces is proposed. The model, called mixture INDSCAL in K classes, or briefly K-INDSCAL, still includes individual saliencies. However, the large number of parameters in K-INDSCAL may produce instability of the estimates and therefore a parsimonious model will also be discussed. The parameters of the model are estimated in a least-squares fitting context and an efficient coordinate descent algorithm is given. The usefulness of K-INDSCAL is demonstrated by both artificial and real data analyses.     

Individual differences in perceived similarity and preference for visual art: A multidimensional scaling analysis

Previous studies have shown that individuals may attend to different dimensions in making an overall judgment of similarity between complex stimuli. The present study investigated the nature of differences in the perceived similarity of reproductions of paintings by the use of multidimensional scaling techniques. Using the INDSCAL model, a group of art-trained students are shown to differ significantly from a group of nonart students in terms of their differential weighing of a set of common dimensions. The same subjects’ preferences are examined in relation to these differences by use of the PREFMAP hierarchy of models. While the simplest (vector) model was found to be appropriate for almost all subjects, large differences in vector direction and average subject ideal-point location are found. Implications for future studies of responses to art are discussed.     

Dimensions of Picture Identification Test need associations

To find a simple and more general structure for need association matrices obtained from Picture Identification Test (PIT) results, PIT need association matrices of eight college student groups were analyzed by INDSCAL, a multidimensional scaling technique. A common three dimensional structure was found. Based on the scale locations of the 22 needs, the dimensions were labeled the Combative, the Personal, and the Competitive dimensions. The INDSCAL model was used as a target for six other comparison groups including male and female English college students, male addicts and controls, and male delinquents and controls. There was enough similarity between the dimensional structure of the need associations of the comparison and target groups to assume that all groups shared the same basic structure of need associations. There were enough differences, however, to suggest that personality characteristics of groups may be related to deviations in the associative structure of their motivation systems. Applications of the techniques to clinical and personality evaluations of groups and individuals were discussed.     

Multidimensional scaling reveals two dimensions of thermal pain

The Individual Differences Scaling (INDSCAL) model of multidimensional scaling was used to explore the dimensions of thermal pain. The observers made 66 similarity judgments to all pairs of 12 different thermal stimulus intensities ranging from zero to noxious. Analysis of the data revealed a two-dimensional group stimulus space. The major dimension ordered the stimuli with respect to their intensity. This quantitative, strength-of-sensation dimension may be interpreted as indicating how weak or strong a stimulus feels, apart from any secondary qualities of warmth or pain. The second dimension was related to the qualitative aspects of the stimuli. This bipolar dimension contained two attributes: a pain attribute ranging from just detectable warmth to painful, and a warm-hot attribute running from just detectable warmth to hot. This study demonstrates the utility of the INDSCAL approach to the understanding of pain and offers a new technique for answering the age-old question concerning the number and qualities of the dimensions underlying the pain experience.     

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.     

An empirical comparison between some models for multidimensional scaling

Abstract.— The results of three methods for multidimensional scaling—Torgerson's metric analysis, a non-metric method (TORSCA), and a method dealing with individual differences in multidimensional scaling (INDSCAL)—were compared in experiments on rhythm experience and on perception of sound quality. The INDSCAL analysis seemed to be the most adequate method for treating the data in these experiments.     

Comparison of perceptual dimensions uncovered from infant-cry signals using the method of pair-comparisons and the semantic differential

Since infant cry sounds may provide caregivers with information concerning an infant's needs, several studies have examined the perceptual features of cry signals. This tradition is followed in the present study, which utilises two techniques; the method of pair-comparisons (coupled to INDSCAL) and the Semantic Differential (SD). In each case 24 infant cries (six each of pain, hunger, pleasure and birth) are used as stimuli. Data were collected from mothers; 39 rated each cry on 50 SD scales, and 26 provided ratings of similarity for every possible pairwise combination of cries. The similarity of the perceptual spaces described by the two techniques is demonstrated by a canonical analysis which compared the SD factor matrix with the INDSCAL dimension-weight matrix. Two of three possible common dimensions were uncovered. This similarity is also indicated by a standard multiple regression analysis by which the INDSCAL dimensions were described in terms of the SD scales. However, it appears the SD emphasised emotional characteristics of the cry signals whereas INDSCAL emphasised physical attributes.     

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.     

Transformation of a three-mode multidimensional scaling solution to indscal form

Relations between Tucker's three-mode multidimensional scaling and Carroll and Chang's INDSCAL are discussed. The possibility is raised that it may be profitable to attempt to transform a three-mode solution to the general form of an INDSCAL solution. Operationally, this involves transforming the three-mode core matrix so that each section is, as nearly as possible, a diagonal matrix. A technique is developed for accomplishing such a transformation, and is applied to two sets of data from the literature. Results indicate that the process is both feasible and valuable, providing useful information on the relative appropriateness of the two models.     

Dimensions of the perception of art: verbal scales and similarity judgements

Abstract.— An INDSCAL analysis was performed on subjects' similarity judgements of all pairs of 12 landscape paintings. Three principal INDSCAL dimensions were identified by analysing responses to 24 unidimensional rating scales derived from previous experiments. The present findings, together with those of previously published studies, suggest that three of the principal dimensions of perception of art appear to be (1) Hedonic-Representational, (2) Clarity, (3) A Dynamic factor involving activity, balance and symmetry. It is suggested that although Complexity may be an important factor influencing responses to simple stimuli, its rôle in the perception of paintings is small, and has frequently been over-emphasised.     

Similarity and preference judgments of musical stimuli

Two experiments were performed comparing similarity (technical-structural) and preference personal-affective) judgments of jazz improvizations (Experiment 1) and of classical, pop-rock, and jazz music (Experiment 2). A multidimensional scaling paradigm (INDSCAL) was used for pairwise comparisons of the 12 musical pieces in each experiment. Each 2 minute piece was subdivided into 10 second segments in order to reduce fatigue or boredom and provide better sampling across comparison trials. In general, similarity dimensions reflected greater agreement among the subjects that did preference dimensions and were hierarchically ordered. The dominant dimensions for both similarity and preference judgments of jazz improvizations were: Tempo, Dominant Instrument (Horns-guitar), and Articulation. The dominant dimensions for similarity judgments involving all three genres were: Classical-Contemporary, Jazz-Rock, and Tempo. The dominant dimensions for preference judments were: Rock-Classical, Jazz-Classical, and Tempo. In general, the preference judgments of musically sophisticated subjects were less constrained by salient dimensions.     

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.     

Multidimensional scaling of individual differences for an individual person with a hand-held calculator

An extension to the INDSCAL method of individual differences scaling was developed. The method derives measures of deviation from a “standard” observer, who is defined in terms of the INDSCAL subject weights. These weights are calculated for one individual person at a time. The method is implemented in a small hand-held calculator. The calculator is used for data acquisition, and it interacts with the user in a conversational manner through an alpha-numeric display. The present implementation of the method scales 84 judgments of pair similarity among seven objects and derives subject weights on two dimensions. The method can be extended to sets of up to 16 objects and up to six dimensions. The method is a feasible, cost-effective approach to classification of individuals on complex perceptual attributes within a single 1-h session.     

Bilingualism and the perception of professional concepts

A selected set of professional concepts was subjected to analysis through two separate multidimensional scaling techniques, the INDSCAL and TORSCA models, to evaluate the intergroup perceptual differences of four experimental groups, made up of unilingual French, unilingual English and bilingual students. The linguistic relativism thesis provided the research hypotheses on the relationship between language access and usage and concept perception. The multidimensional scaling techniques were applied to the matrix of subjects' similarity judgments on pairs of concepts, thus enabling the identification of three dimensions. The dimensions were labelled as conjunctive, relational and disjunctive, and were assumed to be related to the criteria employed by the subjects in their similarity rankings. An analysis of variance of the individual saliences on each dimension provided evidence of linguistic relativism for both the relational and disjunctive dimensions. These findings support the contention that unilingual speakers of separate languages differ from each other and from bilingual speakers in their perception of professional concepts.     

Dimensions in the perception of architecture:I. Identification and interpretation of dimensions of similarity

Pairwise similarity-dissimilarity ratings of 20 slides of buildings representing major architectural styles were subjected to an INDSCAL analysis. Four major similarity dimensions were identified. Subsequently, the stimuli were rated on two batteries of scales: (1) collative and affective ratings, and (2) stylistic and technical ratings. Canonical correlations and redundancy indices revealed close relations between the multidimensional stimulus locations and the scale and factor spaces. The findings of the three experiments suggest that the four principal dimensions in the perception of architecture appear to be (1)Clarity, (2)Hedonic Tone/Arousal, (3) Uncertainty, and (4) Familiarity.     

Nonmetric maximum likelihood multidimensional scaling from directional rankings of similarities

A maximum likelihood estimation procedure is developed for multidimensional scaling when (dis)similarity measures are taken by ranking procedures such as the method of conditional rank orders or the method of triadic combinations. The central feature of these procedures may be termed directionality of ranking processes. That is, rank orderings are performed in a prescribed order by successive first choices. Those data have conventionally been analyzed by Shepard-Kruskal type of nonmetric multidimensional scaling procedures. We propose, as a more appropriate alternative, a maximum likelihood method specifically designed for this type of data. A broader perspective on the present approach is given, which encompasses a wide variety of experimental methods for collecting dissimilarity data including pair comparison methods (such as the method of tetrads) and the pick-M method of similarities. An example is given to illustrate various advantages of nonmetric maximum likelihood multidimensional scaling as a statistical method. At the moment the approach is limited to the case of one-mode two-way proximity data, but could be extended in a relatively straightforward way to two-mode two-way, two-mode three-way or even three-mode three-way data, under the assumption of such models as INDSCAL or the two or three-way unfolding models.The first author's work was supported partly by the Natural Sciences and Engineering Research Council of Canada, grant number A6394. Portions of this research were done while the first author was at Bell Laboratories. MAXSCAL-4.1, a program to perform the computations described in this paper can be obtained by writing to: Computing Information Service, Attention: Ms. Carole Scheiderman, Bell Laboratories, 600 Mountain Ave., Murray Hill, N.J. 07974. Thanks are due to Yukio Inukai, who generously let us use his stimuli in our experiment, and to Jim Ramsay for his helpful comments on an earlier draft of this paper. Confidence regions in Figures 2 and 3 were drawn by the program written by Jim Ramsay. We are also indebted to anonymous reviewers for their suggestions.