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.     

Monte Carlo studies in nonmetric scaling

In response to Arabie several random ranking studies are compared and discussed. Differences are typically very small, however it is noted that those studies which used arbitrary configurations tend to produce slightly higher stress values. The choice of starting configuration is discussed and we suggest that the use of a principal components decomposition of the doubly centered matrix of dissimilarities, or some transformation thereof, will yield an initial configuration which is superior to a randomly chosen one.This research was supported by the National Research Council of Canada (Grant No. A8351) and by the National Institute of Mental Health (Grant Nos. MH10006 and MH26504). The authorship order has been determined by Monte Carlo methods.     

An individual differences additive model: An alterating least squares method with optimal scaling features

An individual differences additive model is discussed which represents individual differences in additivity by differential weighting of additive factors. A procedure for estimating the model parameters for various data measurement characteristics is developed. The procedure is evaluated using both Monte Carlo and real data. The method is found to be very useful in describing certain types of developmental change in cognitive structure, as well as being numerically robust and efficient.The work reported here was partly supported by Grant A6394 to the first author by the Natural Sciences and Engineering Research Council of Canada.     

A scalar product model for the multidimensional scaling of choice

A multidimensional scaling analysis is presented for replicated layouts of pairwise choice responses. In most applications the replicates will represent individuals who respond to all pairs in some set of objects. The replicates and the objects are scaled in a joint space by means of an inner product model which assigns weights to each of the dimensions of the space. Least squares estimates of the replicates' and objects' coordinates, and of unscalability parameters, are obtained through a manipulation of the error sum of squares for fitting the model. The solution involves the reduction of a three-way least squares problem to two subproblems, one trivial and the other solvable by classical least squares matrix factorization. The analytic technique is illustrated with political preference data and is contrasted with multidimensional unfolding in the domain of preferential choice.The present work was initiated at Oregon Research Institute under National Institute of Mental Health Grant MH 12972. It was reformulated and completed while the first author was a Visiting Research Fellow at Educational Testing Service.Presently at the Department of Mathematics, University of Toronto.     

Multidimensional scaling of nominal data: The recovery of metric information with alscal

Multidimensional scaling has recently been enhanced so that data defined at only the nominal level of measurement can be analyzed. The efficacy of ALSCAL, an individual differences multidimensional scaling program which can analyze data defined at the nominal, ordinal, interval and ratio levels of measurement, is the subject of this paper. A Monte Carlo study is presented which indicates that (a) if we know the correct level of measurement then ALSCAL can be used to recover the metric information presumed to underlie the data; and that (b) if we do not know the correct level of measurement then ALSCAL can be used to determine the correct level and to recover the underlying metric structure. This study also indicates, however, that with nominal data ALSCAL is quite likely to obtain solutions which are not globally optimal, and that in these cases the recovery of metric structure is quite poor. A second study is presented which isolates the potential cause of these problems and forms the basis for a suggested modification of the ALSCAL algorithm which should reduce the frequency of locally optimal solutions.     

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.     

A dual scaling analysis for paired compositions

A paired composition is a response (upon a dependent variable) to the ordered pair <j, k> of stimuli, treatments, etc. The present paper develops an alternative analysis for the paired compositions layout previously treated by Bechtel's [1967] scaling model. The alternative model relaxes the previous one by including row and column scales that provide an expression of bias foreach pair of objects. The parameter estimation and hypothesis testing procedures for this model are illustrated by means of a small group analysis, which represents a new approach to pairwise sociometrics and personality assessment.This study was supported by Grant Nos. MH 12972, MH 10822, and MH 15506 from the National Institute of Mental Health, U. S. Public Health Service. Computing assistance was obtained from the Health Sciences Computing Facility, UCLA, sponsored by N.I.H. special research resources Grant RR-3.The motivation of this effort has been enhanced by Gerald Patterson and his associates, who have kindly provided the illustrative data at the end of the paper. The author would also like to express his appreciation to Wei-Ching Chang of the Oregon Research Institute and the University of Oregon for his substantive contributions to the paper, and to William Chaplin and Mark Layman of the Oregon Research Institute for the programming of the data analytic method. FORTRAN IV program listings for this analysis are available upon request to the author.     

A stochastic multidimensional scaling procedure for the empirical determination of convex indifference curves for preference/choice analysis

The vast majority of existing multidimensional scaling (MDS) procedures devised for the analysis of paired comparison preference/choice judgments are typically based on either scalar product (i.e., vector) or unfolding (i.e., ideal-point) models. Such methods tend to ignore many of the essential components of microeconomic theory including convex indifference curves, constrained utility maximization, demand functions, et cetera. This paper presents a new stochastic MDS procedure called MICROSCALE that attempts to operationalize many of these traditional microeconomic concepts. First, we briefly review several existing MDS models that operate on paired comparisons data, noting the particular nature of the utility functions implied by each class of models. These utility assumptions are then directly contrasted to those of microeconomic theory. The new maximum likelihood based procedure, MICROSCALE, is presented, as well as the technical details of the estimation procedure. The results of a Monte Carlo analysis investigating the performance of the algorithm as a number of model, data, and error factors are experimentally manipulated are provided. Finally, an illustration in consumer psychology concerning a convenience sample of thirty consumers providing paired comparisons judgments for some fourteen brands of over-the-counter analgesics is discussed.     

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.     

Re-evaluation of monte carlo studies in nonmetric multidimensional scaling

Critical examination is made of the recent controversy over the value of Monte Carlo techniques in nonmetric multidimensional scaling procedures. The case is presented that the major relevance of Monte Carlo studies is not for the local minima problem but for the meaningfulness of the obtained solutions.     

Probabilistic multidimensional scaling: Complete and incomplete data

Simple procedures are described for obtaining maximum likelihood estimates of the location and uncertainty parameters of the Hefner model. This model is a probabilistic, multidimensional scaling model, which assigns a multivariate normal distribution to each stimulus point. It is shown that for such a model, standard nonmetric and metric algorithms are not appropriate. A procedure is also described for constructing incomplete data sets, by taking into consideration the degree of familiarity the subject has for each stimulus. Maximum likelihood estimates are developed both for complete and incomplete data sets. This research was supported by National Science Grant No. SOC76-20517. The first author would especially like to express his gratitude to the Netherlands Institute for Advanced Study for its very substantial help with this research.     

ConPar: a method for identifying groups of concordant subject proximity matrices for subsequent multidimensional scaling analyses

A common representation of data within the context of multidimensional scaling (MDS) is a collection of symmetric proximity (similarity or dissimilarity) matrices for each of M subjects. There are a number of possible alternatives for analyzing these data, which include: (a) conducting an MDS analysis on a single matrix obtained by pooling (averaging) the M subject matrices, (b) fitting a separate MDS structure for each of the M matrices, or (c) employing an individual differences MDS model. We discuss each of these approaches, and subsequently propose a straightforward new method (CONcordance PARtitioning—ConPar), which can be used to identify groups of individual-subject matrices with concordant proximity structures. This method collapses the three-way data into a subject×subject dissimilarity matrix, which is subsequently clustered using a branch-and-bound algorithm that minimizes partition diameter. Extensive Monte Carlo testing revealed that, when compared to K-means clustering of the proximity data, ConPar generally provided better recovery of the true subject cluster memberships. A demonstration using empirical three-way data is also provided to illustrate the efficacy of the proposed method.     

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.     

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.     

Optimal scaling of paired comparison and rank order data: An alternative to guttman's formulation

A formulation, which is different from Guttman's is presented. The two formulations are both called the optimal scaling approach, and are proven to provide identical scale values. The proposed formulation has at least two advantages over Guttman's. Namely, (i) the former serves to clarify close relations of the optimal scaling approach to those of Slater and the vector model of preferential choice, and (ii) in addition to the stimulus scale values, it provides scores for the subjects, which indicate the degrees of response consistency (transitivity), relative to the optimum solution. The method is assumption-free and capable of multidimensional analysis.This study was partly supported by the National Research Council Grant (No. A4581) to S. Nishisato. The author is indebted to Dr. Bert F. Green, Jr., Mr. Tomoichi Ishizuka, and anonymous reviewers for their valuable comments on an earlier draft.     

Single subject incomplete designs for nonmetric multidimensional scaling

Monte Carlo procedures were used to investigate the properties of a nonmetric multidimensional scaling algorithm when used to scale an incomplete matrix of dissimilarities. Various recommendations for users who wish to scale incomplete matrices are made: (a) recovery was found to be satisfactory provided that the “degrees of freedom” ratio exceeded 3.5, irrespective of error level; (b) cyclic designs were found to provide best recovery, although random patterns of deletion performed almost as well; and (c) strongly locally connected designs, specifically overlapping cliques, were generally inferior. These conclusions are based on 837 scaling solutions and are applicable to stimulus sets containing more than 30 objects.     

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.