Some statistical considerations in multidimensional scaling

Some shortcomings of current methods of estimating the magnitude of perceived difference are considered. A statistical model for perceived difference is derived which avoids these difficulties and employs judgments of ratios of differences as data. Three estimators of squared difference are developed.This study was conducted while the author was a Psychometric Fellow at Princeton University and Educational Testing Service and is part of a dissertation presented in candidacy for the degree of doctor of philosophy. This research was supported by Office of Naval Research Contract Nonr 1858 and by National Science Foundation Grant GB3402. Extensive use was made of the computing facilities of Princeton University supported in part by National Science Foundation Grant NSF-GP579. The author wishes to express his appreciation to Prof. H. Gulliksen, Prof. F. Geldard, Dr. C. Helm, and Dr. F. Lord for their comments and encouragement.     

Some small sample results for maximum likelihood estimation in multidimensional scaling

Some aspects of the small sample behavior of maximum likelihood estimates in multidimensional scaling are investigated by Monte Carlo. An investigation of Model M2 in the MULTISCALE program package shows that the chi-square test of dimensionality requires a correction of tabled chi-square values to be unbiased. A formula for this correction in the case of two dimensions is estimated. The power of the test of dimensionality is acceptable with as few as two replications for 15 stimuli and as few as five replications for 10 stimuli. The biases in the exponent and standard error estimates in this model are also investigated.The research reported here was supported by grant number APA 320 to the author by the National Science and Engineering Research Council of Canada.     

Robust multidimensional scaling

A method for multidimensional scaling that is highly resistant to the effects of outliers is described. To illustrate the efficacy of the procedure, some Monte Carlo simulation results are presented. The method is shown to perform well when outliers are present, even in relatively large numbers, and also to perform comparably to other approaches when no outliers are present.This research was supported by Grant A8351 from the Natural Sciences and Engineering Research Council of Canada to Ian Spence.     

Sequential estimation in multidimensional scaling

The concept of sequential estimation is introduced in multidimensional scaling (MDS). The sequential estimation method developed in this paper refers to continually updating estimates of a configuration as new observations are added. This method has a number of advantages, such as a locally optimal design of the experiment can be easily constructed, and dynamic experimentation is made possible. Using artificial data, the performance of our sequential method is illustrated.We are indebted to anonymous reviewers for their suggestions. In addition, we thank Dr. Frank Critchley for his helpful comments on our Q/S algorithm.     

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.     

Pairwise nonmetric multidimensional scaling

A method of nonmetric multidimensional scaling is described which minimizes pairwise departures from monotonicity. The procedure is relatively simple, both conceptually and computationally. Experience to date suggests that it produces solutions comparable to those of other methods.     

Completely nonmetric multidimensional scaling

Most of the distance models underlying multidimensional scaling assume that if a stimulus y is between the stimuli x and z on each dimension, then x and z should be the farthest apart of the three stimuli. An iterative algorithm is described that uses only this betweenness prediction to infer the ordering of a set of stimuli on each of one or two dimensions. Applied to previously published semantic similarity data, this algorithm produced two-dimensional configurations that were similar in appearance to Euclidean configurations but generally involved fewer violations of the betweenness prediction.     

Maximum likelihood estimation in multidimensional scaling

A variety of distributional assumptions for dissimilarity judgments are considered, with the lognormal distribution being favored for most situations. An implicit equation is discussed for the maximum likelihood estimation of the configuration with or without individual weighting of dimensions. A technique for solving this equation is described and a number of examples offered to indicate its performance in practice. The estimation of a power transformation of dissimilarity is also considered. A number of likelihood ratio hypothesis tests are discussed and a small Monte Carlo experiment described to illustrate the behavior of the test of dimensionality in small samples.The research reported here was supported by grant number APA 320 to the author by the National Research Council of Canada.     

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.     

Constrained multidimensional scaling inN spaces

A gradient method is used to obtain least squares estimates of parameters of them-dimensional euclidean model simultaneously inN spaces, given the observation of all pairwise distances ofn stimuli for each space. The procedure can estimate an additive constant as well as stimulus projections and the metric of the reference axes of the configuration in each space. Each parameter in the model can be fixed to equal some a priori value, constrained to be equal to any other parameter, or free to take on any value in the parameter space. Two applications of the procedure are described.     

The additive constant problem in multidimensional scaling

The problem of choosing the correct additive constant to convert relative interstimulus distances to absolute interstimulus distances in multidimensional scaling is investigated. An artificial numerical example is constructed, and various trial values of the constant are inserted to demonstrate the effect on the multidimensional map of making a variety of incorrect choices. Finally, a general solution to the problem, suggested by Dr. Ledyard R Tucker, is presented; each of the computational steps in this solution is set down for easy reference.This study was supported in part by Office of Naval Research Contract N6onr-270-20 and by National Science Foundation Grant G-642 to Princeton University.     

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.     

An individual differences model for multidimensional scaling

A quantitative system is presented to permit the determination of separate multidimensional perceptual spaces for individuals having different viewpoints about stimulus interrelationships. The structure of individual differences in the perception of stimulus relationships is also determined to provide a framework for ascertaining the varieties of consistent individual viewpoints and their relationships with other variables.This research was supported in part by the National Institute of Mental Health, United States Public Health Service, under Research Grants M-2878 and M-4186 to Educational Testing Service, in part by Educational Testing Service, and in part by the Office of Naval Research under Contract Nonr-1834(39) and the University of Illinois. The authors wish to thank Drs. Harold Gulliksen and Douglas N. Jackson for their helpful comments and Miss Henrietta Gallagher for supervising the computations. Portions of this paper were presented at the American Psychological Association meetings in Chicago, September 1960.This paper was written while Dr. Messick was a Fellow at the Center for Advanced Study in the Behavioral Sciences.     

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.     

A multidimensional scaling approach to mental multiplication

Adults consistently make errors in solving simple multiplication problems. These errors have been explained with reference to the interference between similar problems. In this paper, we apply multidimensional scaling (MDS) to the domain of multiplication problems, to uncover their underlying similarity structure. A tree-sorting task was used to obtain perceived dissimilarity ratings. The derived representation shows greater similarity between problems containing larger operands and suggests that tie problems (e.g., 7 x 7) hold special status. A version of the generalized context model (Nosofsky, 1986) was used to explore the derived MDS solution. The similarity of multiplication problems made an important contribution to producing a model consistent with human performance, as did the frequency with which such problems arise in textbooks, suggesting that both factors may be involved in the explanation of errors.     

Nonmetric multidimensional scaling with clustering of subjects

A new nonmetric multidimensional scaling method is devised to analyze three-way data concerning inter-stimulus similarities obtained from many subjects. It is assumed that subjects are classified into a small number of clusters and that the stimulus configuration is specific to each cluster. Under this assumption, the classification of subjects and the scaling used to derive the configurations for clusters are simultaneously performed using an alternating least-squares algorithm. The monotone regression of ordinal similarity data, the scaling of stimuli and the K -means clustering of subjects are iterated in the algorithm. The method is assessed using a simulation and its practical use is illustrated with the analysis of real data. Finally, some extensions are considered.