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.     

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.     

Effects of conditionality on INDSCAL and ALSCAL weights

The role of conditionality in the INDSCAL and ALSCAL procedures is explained. The effects of conditionality on subject weights produced by these procedures is illustrated via a single set of simulated data. Results emphasize the need for caution in interpreting subject weights provided by these techniques.     

A multidimensional scaling model for the size-weight illusion

The kinds of individual differences in perceptions permitted by the weighted euclidean model for multidimensional scaling (e.g., INDSCAL) are much more restricted than those allowed by Tucker's Three-mode Multidimensional Scaling (TMMDS) model or Carroll's Idiosyncratic Scaling (IDIOSCAL) model. Although, in some situations the more general models would seem desirable, investigators have been reluctant to use them because they are subject to transformational indeterminacies which complicate interpretation. In this article, we show how these indeterminacies can be removed by constructing specific models of the phenomenon under investigation. As an example of this approach, a model of the size-weight illusion is developed and applied to data from two experiments, with highly meaningful results. The same data are also analyzed using INDSCAL. Of the two solutions, only the one obtained by using the size-weight model allows examination of individual differences in the strength of the illusion; INDSCAL can not represent such differences. In this sample, however, individual differences in illusion strength turn out to be minor. Hence the INDSCAL solution, while less informative than the size-weight solution, is nonetheless easily interpretable.This paper is based on the first author's doctoral dissertation at the Department of Psychology, University of Illinois at Urbana-Champaign. The aid of Professor Ledyard R Tucker is gratefully acknowledged.     

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.     

A monte carlo investigation of recovery of structure by alscal

A Monte Carlo study was carried out to investigate the ability of ALSCAL to recover true structure inherent in simulated proximity measures. The nature of the simulated data varied according to (a) number of stimuli, (b) number of individuals, (c) number of dimensions, and (d) level of random error. Four aspects of recovery were studied: (a) SSTRESS, (b) recovery of true distances, (c) recovery of stimulus dimensions, and (d) recovery of individual weights. Results indicated that all four measures were rather strongly affected by random error. Also, SSTRESS improved with fewer stimuli in more dimensions, but the other three indices behaved in the opposite fashion. Most importantly, it was found that the number of individuals, over the range studied, did not have a substantial effect on any of the four measures of recovery. Practical implications and suggestions for further research are discussed.The authors wish to thank Drs. Forrest W. Young, Paul D. Isaac and Thomas E. Nygren, who provided many helpful comments during this project.     

External analysis with three-mode principal component models

Through external analysis of two-mode data one attempts to map the elements of one mode (e.g., attributes) as vectors in a fixed space of the elements of the other mode (e.g., stimuli). This type of analysis is extended to three-mode data, for instance, when the ratings are made by more individuals. It is described how alternating least squares algorithms for three-mode principal component analysis (PCA) are adapted to enable external analysis, and it is demonstrated that these techniques are useful for exploring differences in the individuals' mappings of the attribute vectors in the fixed stimulus space. Conditions are described under which individual differences may be ignored. External three-mode PCA is illustrated with data from a person perception experiment, designed after two studies by Rosenberg and his associates whose results were used as external information.We gratefully acknowledge the assistance of Piet Brouwer in implementing the external analysis options in the TUCKALS programs.     

Internal multidimensional unfolding about a single-ideal—A probabilistic solution

A solution is presented for an internal multidimensional unfolding problem in which all the judgments of a rectangular proximity matrix are a function of a single-ideal object. The solution is obtained by showing that when real and ideal objects are represented by normal distributions in a multidimensional Euclidean space, a vector of distances among a single-ideal and multiple real objects follows a multivariate quadratic form in normal variables distribution. An approximation to the vector's probability density function (PDF) is developed which allows maximum likelihood (ML) solutions to be estimated. Under dependent sampling, the likelihood function contains information about the parametric distances among real object pairs, permitting the estimation of single-ideal solutions and leading to more robust multiple-ideal solutions. Tests for single- vs. multiple-ideal solutions and dependent vs. independent sampling are given. Properties of the proposed model and parameter recovery are explored. Empirical illustrations are also provided.     

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.     

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 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.     

Principal component analysis of three-mode data by means of alternating least squares algorithms

A new method to estimate the parameters of Tucker's three-mode principal component model is discussed, and the convergence properties of the alternating least squares algorithm to solve the estimation problem are considered. A special case of the general Tucker model, in which the principal component analysis is only performed over two of the three modes is briefly outlined as well. The Miller & Nicely data on the confusion of English consonants are used to illustrate the programs TUCKALS3 and TUCKALS2 which incorporate the algorithms for the two models described.     

Recovery of structure in incomplete data by alscal

A Monte Carlo study was carried out in order to investigate the ability of ALSCAL to recover true structure inherent in simulated proximity measures when portions of the data are missing. All sets of simulated proximity measures were based on 30 stimuli and three dimensions, and selection of missing elements was done randomly. Properties of the simulated data varied according to (a) the number of individuals, (b) the level of random error, (c) the proportion of missing data, and (d) whether the same entries or different entries were deleted for each individual. Results showed that very accurate recovery of true distances, stimulus coordinates, and weight vectors could be achieved with as much as 60% missing data as long as sample size was sufficiently large and the level of random error was low.     

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.     

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 model and algorithm for multidimensional scaling with external constraints on the distances

A method for externally constraining certain distances in multidimensional scaling configurations is introduced and illustrated. The approach defines an objective function which is a linear composite of the loss function of the point configurationX relative to the proximity dataP and the loss ofX relative to a pseudo-data matrixR. The matrixR is set up such that the side constraints to be imposed onX's distances are expressed by the relations amongR's numerical elements. One then uses a double-phase procedure with relative penalties on the loss components to generate a constrained solutionX. Various possibilities for constructing actual MDS algorithms are conceivable: the major classes are defined by the specification of metric or nonmetric loss for data and/or constraints, and by the various possibilities for partitioning the matricesP andR. Further generalizations are introduced by substitutingR by a set ofR matrices,R _i ,i=1, ...r, which opens the way for formulating overlapping constraints as, e.g., in patterns that are both row- and column-conditional at the same time.     

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.     

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.     

Using a multidimensional scaling approach to investigate the underlying basis of ease of learning judgments

Jönsson, F. U. & Lindström, B. R. (2009) Using a multidimensional scaling approach to investigate the underlying basis of ease of learning judgments. Scandinavian Journal of Psychology, 51, 103–108. Before studying a material it is of strategic importance to first assess its difficulty, so called Ease of Learning (EOL) judgments. A multidimensional scaling (MDS) procedure was used to investigate the underlying basis of EOL judgments for 24 nouns, which to the authors’ knowledge has not been done before. In addition, Judgments of Learning (JOL) followed by a free recall test was performed. The MDS analysis indicated that EOL judgments for the nouns are based on multiple cues (dimensions), namely word length, frequency, and concreteness. Moreover, the concreteness values of the nouns, as judged by an independent group, were correlated with both the JOLs and the concreteness dimension from the MDS analysis. This indicates that EOLs and JOLs for single words are based, to some extent, on the same cues.