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.     

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.     

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.     

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.     

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.     

Confidence regions for multidimensional scaling analysis

Techniques are developed for surrounding each of the points in a multidimensional scaling solution with a region which will contain the population point with some level of confidence. Bayesian credibility regions are also discussed. A general theorem is proven which describes the asymptotic distribution of maximum likelihood estimates subject to identifiability constraints. This theorem is applied to a number of models to display asymptotic variance-covariance matrices for coordinate estimates under different rotational constraints. A technique is described for displaying Bayesian conditional credibility regions for any sample size.The research reported here was supported by grant number APA 320 to the author by the National Research Council of Canada.     

Three case studies in the Bayesian analysis of cognitive models

Bayesian statistical inference offers a principled and comprehensive approach for relating psychological models to data. This article presents Bayesian analyses of three influential psychological models: multidimensional scaling models of stimulus representation, the generalized context model of category learning, and a signal detection theory model of decision making. In each case, the model is recast as a probabilistic graphical model and is evaluated in relation to a previously considered data set. In each case, it is shown that Bayesian inference is able to provide answers to important theoretical and empirical questions easily and coherently. The generality of the Bayesian approach and its potential for the understanding of models and data in psychology are discussed.     

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.     

Empirically indistinguishable multidimensional IRT and locally dependent unidimensional item response models

Multidimensionality is a core concept in the measurement and analysis of psychological data. In personality assessment, for example, constructs are mostly theoretically defined as unidimensional, yet responses collected from the real world are almost always determined by multiple factors. Significant research efforts have concentrated on the use of simulated studies to evaluate the robustness of unidimensional item response models when applied to multidimensional data with a dominant dimension. In contrast, in the present paper, I report the result from a theoretical investigation that a multidimensional item response model is empirically indistinguishable from a locally dependent unidimensional model, of which the single dimension represents the actual construct of interest. A practical implication of this result is that multidimensional response data do not automatically require the use of multidimensional models. Circumstances under which the alternative approach of locally dependent unidimensional models may be useful are discussed.     

Scale Alignment in the Between-Item Multidimensional Partial Credit Model

In between-item multidimensional item response models, it is often desirable to compare individual latent trait estimates across dimensions. These comparisons are only justified if the model dimensions are scaled relative to each other. Traditionally, this scaling is done using approaches such as standardization—fixing the latent mean and standard deviation to 0 and 1 for all dimensions. However, approaches such as standardization do not guarantee that Rasch model properties hold across dimensions. Specifically, for between-item multidimensional Rasch family models, the unique ordering of items holds within dimensions, but not across dimensions. Previously, Feuerstahler and Wilson described the concept of scale alignment, which aims to enforce the unique ordering of items across dimensions by linearly transforming item parameters within dimensions. In this article, we extend the concept of scale alignment to the between-item multidimensional partial credit model and to models fit using incomplete data. We illustrate this method in the context of the Kindergarten Individual Development Survey (KIDS), a multidimensional survey of kindergarten readiness used in the state of Illinois. We also present simulation results that demonstrate the effectiveness of scale alignment in the context of polytomous item response models and missing data.     

Investigating With IRT and MDS Approaches Translation and Adaptation of Rating Scales for Spanish-Speaking Populations

The goal of this study is to investigate how features of a rating scale developed for English-speaking populations interact with Spanish-speaking respondents' response styles and functional categories of judgment. A sample of 400 Spanish-speaking students took a translated scale and a scaling task developed to measure response sets and functional categories of judgment, respectively. Three response set models—extreme response, central tendency, and acquiescence—under two conditions—base and revised with respondents' functional categories—were studied with item response theory and multidimensional scaling methods. Revising the number of scale categories with the number of salient functional categories statistically improved fit of the base models. Multidimensional scaling results showed scale content features interacting with response styles and functional categories. Translation of rating scales requires adapting scale features to characteristics of target languages, such as salient response styles and respondents' functional categories of judgment.     

Restricted multidimensional scaling models for asymmetric proximities

Restricted multidimensional scaling models [Bentler & Weeks, 1978] allowing constraints on parameters, are extended to the case of asymmetric data. Separate functions are used to model the symmetric and antisymmetric parts of the data. The approach is also extended to the case in which data are presumed to be linearly related to squared distances. Examples of several models are provided, using journal citation data. Possible extensions of the models are considered. This research was supported in part by USPHS Grant 0A01070, P. M. Bentler, principal investigator, and NIMH Grant MH-24819, E. J. Anthony and J. Worland, principal investigators. The authors wish to thank E. W. Holman and several anonymous reviewers for their valuable suggestions concerning this research.     

THE DIMENSIONALITY PARADOX IN COMPARATIVE JUDGMENT: A RESOLUTION

S jöberg , L. The dimensionality paradox in comparative judgment: a resolution. Scand. J. Psychol ., 1968, 9, 97–108. — It has often been possible to erect unidimensional scales although stimuli have been clearly multidimensional. This fact is surprising since one expects the multidimensional structure to be reflected in data. It is suggested that this structure might be recovered from dispersions of comparative judgments. The suggestion is shown to be valid on three sets of paired comparison data. Implications for uni- and multidimensional scaling are discussed.     

Estimating standard errors in feature network models

Feature network models are graphical structures that represent proximity data in a discrete space while using the same formalism that is the basis of least squares methods employed in multidimensional scaling. Existing methods to derive a network model from empirical data only give the best‐fitting network and yield no standard errors for the parameter estimates. The additivity properties of networks make it possible to consider the model as a univariate (multiple) linear regression problem with positivity restrictions on the parameters. In the present study, both theoretical and empirical standard errors are obtained for the constrained regression parameters of a network model with known features. The performance of both types of standard error is evaluated using Monte Carlo techniques.     

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.     

Statistical consistency and hypothesis testing for nonmetric multidimensional scaling

The properties of nonmetric multidimensional scaling (NMDS) are explored by specifying statistical models, proving statistical consistency, and developing hypothesis testing procedures. Statistical models with errors in the dependent and independent variables are described for quantitative and qualitative data. For these models, statistical consistency often depends crucially upon how error enters the model and how data are collected and summarized (e.g., by means, medians, or rank statistics). A maximum likelihood estimator for NMDS is developed, and its relationship to the standard Shepard-Kruskal estimation method is described. This maximum likelihood framework is used to develop a method for testing the overall fit of the model.     

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.     

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.     

Spatial,non-spatial and hybrid models for scaling

In this paper, hierarchical and non-hierarchical tree structures are proposed as models of similarity data. Trees are viewed as intermediate between multidimensional scaling and simple clustering. Procedures are discussed for fitting both types of trees to data. The concept of multiple tree structures shows great promise for analyzing more complex data. Hybrid models in which multiple trees and other discrete structures are combined with continuous dimensions are discussed. Examples of the use of multiple tree structures and hybrid models are given. Extensions to the analysis of individual differences are suggested.1976 Psychometric Society Presidential Address.While people too numerous to list here have contributed ideas, inspiration, and other help, I particularly wish to acknowledge the contributions of Sandra Pruzansky, without whom this paper could not have been written. I would also like to acknowledge the past contributions of my long-time colleague Jih-Jie Chang, without whose help I probably would not have beenasked to write it.