WINDSORT: A fully automated technique for collecting sort data for multidimensional scaling analysis

WINDSORT is a microcomputer program that uses an automated sorting procedure to collect the preprocess proximity data commonly analyzed using multidimensional scaling. The traditional manual sorting task is less tedious, more enjoyable, and far easier for subjects to perform than the tasks that are used in alternative data-collection techniques. However, complete automation of the sorting task has not previously been fully successful WINDSORT uses a form of the hierarchical sorting technique which is reputed to yield richer data than single-sort techniques A maximum of 45 stimuli can be scaled using WINDSORT. Resulting output includes dissimilarity matrices which are ready to analyze using MDS.     

Differentiability of Kruskal's stress at a local minimum

It is shown that Kruskal's multidimensional scaling loss function is differentiable at a local minimum. Or, to put it differently, that in multidimensional scaling solutions using Kruskal's stress distinct points cannot coincide.     

Restricted multidimensional scaling models

A class of multidimensional scaling models are developed wherein certain parameters may be fixed as known constants, or proportional to one another. Traditional multidimensional scaling can be obtained as a special case by fixing only the orientation and origin of a configuration. Methods of obtaining least-square estimates of the parameters via nonlinear programming are discussed, and an effective computer program is developed to implement application of the models to data. Several well-known data sets are reanalyzed under various restricted models, and the results demonstrate the possibility of achieving insight not attainable under the traditional approach. The potential distortion arising from inadequate model specification is discussed, and the importance of substantive theory to multidimensional scaling research is emphasized.     

Design considerations for multidimensional scaling

This paper begins with a short tutorial on multidimensional scaling. The focus of the remainder of the paper is on the proper designing of research that will use multidimensional scaling analysis techniques and includes suggestions for the interpretation of results.     

A multidimensional scaling program for distinctiveness and similarity based on stimulus classification and information measurement

When observers classify a set of multidimensional items on the basis of similarity, they generate information measures that are isomorphic to item vectors whose lengths correspond to the judged distinctiveness of each item and whose angles define the position of each item in a euclidean similarity space. The PROSCALE computer program calculates the item vectors and spatial positions from similarity classification data and then generates and rotates orthogonal dimensions of the similarity space. PROSCALE can also use variables associated with each item to generate oblique dimensions that span the space. PROSCALE carries out multidimensional similarity scaling or unidimensional magnitude estimation scaling on as many as 70 items for as many as 50 observers.     

Multidimensional successive categories scaling: A maximum likelihood method

A single-step maximum likelihood estimation procedure is developed for multidimensional scaling of dissimilarity data measured on rating scales. The procedure can fit the euclidian distance model to the data under various assumptions about category widths and under two distributional assumptions. The scoring algorithm for parameter estimation has been developed and implemented in the form of a computer program. Practical uses of the method are demonstrated with an emphasis on various advantages of the method as a statistical procedure.The research reported here was partly supported by Grant A6394 to the author by Natural Sciences and Engineering Research Council of Canada. Portions of this research were presented at the Psychometric Society meeting in Uppsala, Sweden, in June, 1978. MAXSCAL-2.1, a program to perform the computations discussed in this paper may be obtained from the author. Thanks are due to Jim Ramsay for his helpful comments.     

当代大学生自比型价值观数据的潜在结构

本研究对采自3796名在校大学生的Rokeach Value Survey自比型数据进行了因素分析和多维尺度分析。因素分析从两组价值观选项中分别获得6个双极因素,虽然内容各不相同,但均体现出个人指向-亲社会指向的特点。多维尺度分析所获得的两组选项的2维空间距离分布结果也体现类似特点,结果显示终极性价值观可分为四类,工具性价值观可分为五类。当前大学生价值观中存在着舒适的物质生活、兴奋的生活、幸福、快乐和自尊以及雄心壮志的、勇敢的和诚实的等个人取向内容占优势的可能性。在两种分析方法中,多维尺度分析更适合对自比型价值观数据潜在结构的探究。     

Simultaneous classification and multidimensional scaling with external information

For the exploratory analysis of a matrix of proximities or (dis)similarities between objects, one often uses cluster analysis (CA) or multidimensional scaling (MDS). Solutions resulting from such analyses are sometimes interpreted using external information on the objects. Usually the procedures of CA, MDS and using external information are carried out independently and sequentially, although combinations of two of the three procedures (CA and MDS, or multidimensional scaling and using external information) have been proposed in the literature. The present paper offers a procedure that combines all three procedures in one analysis, using a model that describes a partition of objects with cluster centroids represented in a low-dimensional space, which in turn is related to the information in the external variables. A simulation study is carried out to demonstrate that the method works satisfactorily for data with a known underlying structure. Also, to illustrate the method, it is applied to two empirical data sets.     

A note on the use of directional statistics in weighted euclidean distances multidimensional scaling models

The weighted euclidean distances model in multidimensional scaling (WMDS) represents individual differences as dimension saliences which can be interpreted as the orientations of vectors in a subject space. It has recently been suggested that the statistics of directions would be appropriate for carrying out tests of location with such data. The nature of the directional representation in WMDS is reviewed and it is argued that since dimension saliences are almost always positive, the directional representations will usually be confined to the positive orthant. Conventional statistical techniques are appropriate to angular representations of the individual differences which will yield angles in the interval (0, 90) so long as dimension saliences are nonnegative, a restriction which can be imposed. Ordinary statistical methods are also appropriate with several linear indices which can be derived from WMDS results. Directional statistics may be applied more fruitfully to vector representations of preferences.     

Ideal Point Discriminant Analysis Revisited with a Special Emphasis on Visualization

Ideal point discriminant analysis is a classification tool which uses highly intuitive multidimensional scaling procedures. However, in the last paper, Takane wrote about it. He concludes that the interpretation is rather intricate and calls that a weakness of the model. We summarize the conditions that provide an easy interpretation and show that in maximum dimensionality they can be obtained without any loss. For reduced dimensionality, it is conjectured that loss is minor which is examined using several data sets. This research was conducted while the author was sponsored by the Netherlands Organisation for Scientific Research (NWO), Innovational Grant, no. 452-06-002.     

A theoretical framework for testing the additive difference model for dissimilarities data: Representing gambles as multidimensional stimuli

Implicit within the acceptance of most multidimensional scaling models as accurate representations of an individual's cognitive structure for a set of complex stimuli, is the acceptance of the more general Additive Difference Model (ADM). A theoretical framework for testing the ordinal properties of the ADM for dissimilarities data is presented and is illustrated for a set of three-outcome gambles. Paired comparison dissimilarity judgments were obtained for two sets of gambles. Judgments from one set were first analyzed using the ALSCAL individual differences scaling model. Based on four highly interpretable dimensions derived from this analysis, a predicted set of dimensions were obtained for each subject for the second set of gambles. The ordinal properties of the ADM necessary for interdimensional additivity and intradimensional subtractivity were then tested for each subject's second set of data via a new computer-based algorithm, ADDIMOD. The tests indicated that the ADM was rejected. Although violations of the axioms were significantly less than what would be expected by chance, for only one subject was the model clearly supported. It is argued that while multidimensional scaling models may be useful as data reduction techniques, they do not reflect the perceptual processes used by individuals to form judgments of similarity. Implications for further study of multidimensional scaling models are offered and discussed.     

Perhaps Unidimensional Is Not Unidimensional

Miller (1956) identified his famous limit of 7 ± 2 items based in part on absolute identification—the ability to identify stimuli that differ on a single physical dimension, such as lines of different length. An important aspect of this limit is its independence from perceptual effects and its application across all stimulus types. Recent research, however, has identified several exceptions. We investigate an explanation for these results that reconciles them with Miller’s work. We find support for the hypothesis that the exceptional stimulus types have more complex psychological representations, which can therefore support better identification. Our investigation uses data sets with thousands of observations for each participant, which allows the application of a new technique for identifying psychological representations: the structural forms algorithm of Kemp and Tenenbaum (2008) . This algorithm supports inferences not possible with previous techniques, such as multidimensional scaling.     

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.     

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.     

On-line multidimensional scaling programs with interactive graphical display

The beginnings of a system of interactive multidimensional scaling programs with real-time display of the graphical output have been established on the Honeywell DDP-224 computer. Two programs have been completed: (1) MDPREF—a computer program for multidimensional analysis of preference data—has been converted from the GE-635 to run interactively on the DDP-224 computer. Its solution is printed from a typewriter, and the configuration of stimuli are displayed on a scope in two-dimensional view. (2) ROTATE—an on-line rotation program—enables the user to rotate the configuration in three dimensions within a higher dimensional space.     

Anatomy of a stimulus domain: The relation between multidimensional and unidimensional scaling of noise bands

Theoretically, there are many possible relationships between multidimensional scaling and unidimensional scalings of the same stimulus domain. In particular, it is uncertain what will happen if the number of psychological “dimensions” exceeds the number of physical variables. The multidimensional scaling of noise bands, unidimensionally a relatively well-understood domain, was done to explore these problems. In correspondence with the number of physical variables, a two-dimensional configuration was found to give a satisfactory account of the judgments of magnitude of stimulus difference. Axes of loudness, volume, and density were found to fit the configuration with a high degree of precision, lending support to the metric value of numbers produced in magnitude estimation. Pitch, or frequency, also had a simple relationship to the configuration, but was not an axis or dimension. Therefore, the usual conceptualization of judgments of overall similarity as the result of combining difference on separate dimensions is questioned. It is suggested that multidimensional configurations may sometimes correspond to internal representations of general importance.     

The analysis of proximities: Multidimensional scaling with an unknown distance function. II

The first in the present series of two papers described a computer program for multidimensional scaling on the basis of essentially nonmetric data. This second paper reports the results of two kinds of test applications of that program. The first application is to artificial data generated by monotonically transforming the interpoint distances in a known spatial configuration. The purpose is to show that the recovery of the original metric configuration does not depend upon the particular transformation used. The second application is to measures of interstimulus similarity and confusability obtained from some actual psychological experiments.     

Multidimensional Unfolding by Nonmetric Multidimensional Scaling of Spearman Distances in the Extended Permutation Polytope

A multidimensional unfolding technique that is not prone to degenerate solutions and is based on multidimensional scaling of a complete data matrix is proposed: distance information about the unfolding data and about the distances both among judges and among objects is included in the complete matrix. The latter information is derived from the permutation polytope supplemented with the objects, called the preference sphere. In this sphere, distances are measured that are closely related to Spearman's rank correlation and that are comparable among each other so that an unconditional approach is reasonable. In two simulation studies, it is shown that the proposed technique leads to acceptable recovery of given preference structures. A major practical advantage of this unfolding technique is its relatively easy implementation in existing software for multidimensional scaling.     

Homogeneity analysis withk sets of variables: An alternating least squares method with optimal scaling features

Homogeneity analysis, or multiple correspondence analysis, is usually applied tok separate variables. In this paper we apply it to sets of variables by using sums within sets. The resulting technique is called OVERALS. It uses the notion of optimal scaling, with transformations that can be multiple or single. The single transformations consist of three types: nominal, ordinal, and numerical. The corresponding OVERALS computer program minimizes a least squares loss function by using an alternating least squares algorithm. Many existing linear and nonlinear multivariate analysis techniques are shown to be special cases of OVERALS. An application to data from an epidemiological survey is presented.This research was partly supported by SWOV (Institute for Road Safety Research) in Leidschendam, The Netherlands.