A direct approach to individual differences scaling using increasingly complex transformations

A family of models for the representation and assessment of individual differences for multivariate data is embodied in a hierarchically organized and sequentially applied procedure called PINDIS. The two principal models used for directly fitting individual configurations to some common or hypothesized space are the dimensional salience and perspective models. By systematically increasing the complexity of transformations one can better determine the validities of the various models and assess the patterns and commonalities of individual differences. PINDIS sheds some new light on the interpretability and general applicability of the dimension weighting approach implemented by the commonly used INDSCAL procedure.     

An analysis of individual differences in similarity judgments about complex random forms

A variation of the method of graded dichotomies was used to obtain judgments of interstimulus similarity for all pairings of 21 12-turn random shapes. Using the Tucker-Messick procedure, a six-dimensional spatial model, representing individual variation in performance on the task, was derived. Six “idealized individuals” (IIs) were isolated. A five-dimensional Euclidean spatial model was obtained for each II using the Shepard-Kruskal scaling technique. The resulting psychological dimensions of perceived form were related to two sets of physical measures on the forms: the factor space suggested by Brown and Owen and a set of more basic physical measures. These basic measures allowed interpretation of most of the psychological dimensions and revealed differences between IIs in patterns of cue utilization.     

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.     

A spatial analysis of the discriminability of forms in noise

Multidimensional scaling (MDS) techniques were used to study the form characteristics which transcend noise in a discrimination task. Three dimensions of random polygons were identified and were psychophysically related to quantifted form measures.     

The Problem of the Condition of Arousal of the Pure Sensation of Cutaneous Pain

In order to develop scales representing curiosity-arousing components of pictures (slides) of either common objects or random shapes, a total of 56 college student Ss rated both sets of material on nine dimensions. Factor analyses indicated that the nine dimensions could be reduced to two factors. In order to test the ability of these factors to predict memorial performance and exploratory behavior, an additional 80 undergraduates were then tested for recognition or free looking time (FLT) of either object or random shape slides. Separate regression equations, using FLT and recognition performance as criterion measures, were computed. The present results not only indicate stability in visual exploratory scaling plus redundancy among various attributes of visual stimuli, but also suggest that behavior can be considered as elicited by two separate attributes of visual stimuli.     

Multidimensional scaling of measures of distance between partitions

The techniques of multidimensional scaling were used to study the numerical behavior of twelve measures of distance between partitions, as applied to partition lattices of four different sizes. The results offer additional support for a system of classifying partition metrics, as proposed by Boorman (1970), and Boorman and Arabie (1972). While the scaling solutions illuminated differences between the measures, at the same time the particular data with which the measures were concerned offered a basis both for counterexamples to some common assumptions about multidimensional scaling and for some conjectures as to the nature of scaling solutions. The implications of the latter findings for selected examples from the literature are considered. In addition, the methods of partition data analysis discussed here are applied to an example using sociobiological data. Finally, an argument is made against undue emphasis upon interpreting dimensions in nonmetric scaling solutions.     

The psychophysical dimensions of similarity among random shapes

Six psychological dimensions were recovered from similarity judgments of 20 random forms using Kruskal’s nonmetric scaling technique. These were related to 10 physical factors describing the forms by using canonical correlation analysis and rotations of the resulting vectors. Four factors accounted for 2/3 of the variance of the intersection of the physical and psychological spaces for each of 11 Ss. The factors are described as Complexity, Curvature, Curvature Dispersion, and Straight-length Dispers ion. The problem of the proper psychological distance metric is discussed, and individual differences among Ss are analyzed. The results lead to the conclusion that the four major factors found may be good representations of processes occurring in the perceptual system.     

Multidimensional scaling of form: A psychophysical analysis

Both metric and nonmetric multidimensional scaling methods were used to analyze similarity estimates when random polygons were used as stimulus patterns. Three dimensions, dispersion, jaggedness, and elongation were obtained with both analyses and were related to physical measures of the patterns.     

A TABLE OF EXPECTED STRESS VALUES FOR RANDOM RANKINGS IN NONMETRIC MULTIDIMENSIONAL SCALING

The results of applying a nonmetric multidimensional scaling algorithm to sets of pseudo random data are used as the basis for constructing, by means of regression techniques, a table which will assist a user to decide whether empirically obtained data sets are the result of a random process. The table covers the range from 12 to 48 objects (points) for one to five recovered dimensions.     

Memory for multidimensional source information

Source memory may comprise recollection of multiple features of the encoding episode. To analyze the simultaneous representation and retrieval of those multiple features, a multinomial memory model is presented that measures memory for crossed dimensions of source information. The first experiment investigated the validity of the new model. The model showed an excellent statistical fit to empirical data, and the parameters of multidimensional source memory were sensitive to manipulations of source similarity on distinct dimensions. The second experiment used the model to test the hypothesis that source memory for individual context attributes is stochastically related in the case of conscious recollection but independent in the case of familiarity-based recognition judgments. The prediction was supported by the introduction of a "remember"-"know" distinction in a multidimensional source memory test.     

Visual form discrimination: Multidimensional analyses

Nonmetric multidimensional scaling techniques were used to evaluate discrimination latency as a similarity measure. A three dimensional Euclidian solution was found for a sample of random polygons. The dimensions were labeled compactness, jaggedness, and elongation. The spatial configuration was highly similar to results obtained from similarity estimates and the dimensions were systematically related to physical shape measures.     

The Multidimensional Scaling Of A Set Of Artistic Drawings : Perceived Structure And Scale Correlates

Product-centered research on creativity approaches the criterion problem of what is to be the referent for creativity through the analysis of tangible products such as a r t objects, writing, or scientific achievements. The present research is concerned with the evaluation and study of artist drawings contributed by sophomore students a t the Rhode Island School of Design. Multi-dimensional scaling methods were applied to similarity judgments obtained from art experts on two separate sets of 26 drawings. Three similarity dimensions accounted for the interstimulus distances for each set of drawings. Although no statistical test was available, the dimensions from the two seta appeared to correspond. Scale values of 4 drawings common to the two sets were consistent, and the dimensions appeared to define very similar stimulus characteristics. It was concluded that multidimensional scaling procedures provided a means for differentiating among a set of complex, esthetic products. Scale values of drawings on the three dimensions also correlated differentially with cognitive and achievement measures available on the students, suggesting that product dimensions identified via similarity judgments were related to characteristics of individuals producing the products. Hypotheses were developed as to the psychological meaning of the three product dimensions.     

A general parametric model of similarity and subjective difference

The field of multidimensional scaling is dominated by models that lack inherent parameters. Correcting parameters have been introduced, e.g. INDSCAL, to increase power of prediction. Although a nonparametirc model with correcting parameters may exhibit a very good fit to data, a parametric model is intrinsically superior. The general parametric model proposed here yields measures of both absolute and relative subjective differences (dissimilarity) in addition to similarity. It is basically unidimensioanal. Rules for combining values of attributes into a single multidimensional value may be applied either to the input or to the output of the model. One of the resulting functions is a generalization of the Eisler-Ekman similarity function. A special case of another function is identical to the Minkowski class of distance functions (including INDSCAL). The model is not limited to pairwise relations. It yields unitary measures for any number of objects.     

Alphabetic confusion: A test of models for individuals

An earlier article reported extensive analyses of confusion data compiled from group averages (Townsend, 1971). The present study provided for essentially the same analyses with different long-term data obtained with two individuals, the primary intent being to examine the ability of recognition and scaling models to explain data at the individual level (which the recognition models purport to describe) and to compare the confusion characteristics of the English uppercase alphabet between the two Ss and between the individual Ss and the group-averaged data. The choice and overlap models were superior to the all-or-none model in predicting the empirical confusion matrices and tended to explain the data structure in a similar manner. Multidimensional scaling analysis again supported a Euclidean metric and suggested four or five underlying stimulus dimensions. However, as before, there were no overriding intuitively appealing psychological dimensions corresponding to these, and possible reasons are discussed. The choice and overlap models appeared to fit as well or better at the individual level than at the group level and the all-or-none model to fit worse. In the present study, probability correct was fit even better by the all-or-none model than in the group study and replicated the result of being better here than the overlap and choice models. Individuals and the group were consistent in their sensory confusions as represented by similarity parameters in the choice and overlap models but differed in their response biases. A simple measure of physical similarity explained 50% of the variance of the similarity structure in the confusion data.     

A monte carlo investigation of the statistical significance of Kruskal's nonmetric scaling procedure

Recent advances in computer based psychometric techniques have yielded a collection of powerful tools for analyzing nonmetric data. These tools, although particularly well suited to the behavioral sciences, have several potential pitfalls. Among other things, there is no statistical test for evaluating the significance of the results. This paper provides estimates of the statistical significance of results yielded by Kruskal's nonmetric multidimensional scaling. The estimates, obtained from attempts to scale many randomly generated sets of data, reveal the relative frequency with which apparent structure is erroneously found in unstructured data. For a small number of points (i.e., six or seven) it is very likely that a good fit will be obtained in two or more dimensions when in fact the data are generated by a random process. The estimates presented here can be used as a bench mark against which to evaluate the significance of the results obtained from empirically based nonmetric multidimensional scaling.A preliminary version of this paper was presented at the International Federation for Information Processing Congress 68 in Edinburgh, Scotland, August 5–10, 1968.     

Analysis of individual differences in multidimensional scaling via an n-way generalization of “Eckart-Young” decomposition

An individual differences model for multidimensional scaling is outlined in which individuals are assumed differentially to weight the several dimensions of a common “psychological space”. A corresponding method of analyzing similarities data is proposed, involving a generalization of “Eckart-Young analysis” to decomposition of three-way (or higher-way) tables. In the present case this decomposition is applied to a derived three-way table of scalar products between stimuli for individuals. This analysis yields a stimulus by dimensions coordinate matrix and a subjects by dimensions matrix of weights. This method is illustrated with data on auditory stimuli and on perception of nations.     

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.     

Conceptual coherence affects phonological activation of context objects during object naming

In 4 picture-word interference experiments, speakers named a target object that was presented with a context object. Using auditory distractors that were phonologically related or unrelated either to the target object or the context object, the authors assessed whether phonological processing was confined to the target object or not. Phonological activation of the context objects was reliably observed if the target and context objects were embedded in a conceptually coherent scene (e.g., if the picture showed a mouse eating some cheese), regardless of whether the target was cued by its thematic role (agent vs. patient) or by color. However, this activation dissipated if the two objects were presented in an arbitrary object array (e.g., if the cheese was presented along with a finger). These findings suggest that conceptual coherence among multiple objects affects the information flow in the conceptual-lexical system during speech planning.     

Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis

Multidimensional scaling is the problem of representingn objects geometrically byn points, so that the interpoint distances correspond in some sense to experimental dissimilarities between objects. In just what sense distances and dissimilarities should correspond has been left rather vague in most approaches, thus leaving these approaches logically incomplete. Our fundamental hypothesis is that dissimilarities and distances are monotonically related. We define a quantitative, intuitively satisfying measure of goodness of fit to this hypothesis. Our technique of multidimensional scaling is to compute that configuration of points which optimizes the goodness of fit. A practical computer program for doing the calculations is described in a companion paper.