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.     

Joint-space analysis of “pick-any” data: Analysis of choices from an unconstrained set of alternatives

Social and naturally occurring choice phenomena are often of the pick-any type in which the number of choices made by a subject as well as the set of alternatives from which they are chosen is unconstrained. These data present a special analytical problem because the meaning of non-choice among pick-any choice data is always ambiguous: A non-chosen alternative may be either unacceptable, or acceptable but not considered, or acceptable and considered but not chosen. A model and scaling method for these data are introduced, allowing for this ambiguity of non-choice. Subjects are represented as points whose coordinates are proportional to the centroids of the points representing their choices. Alternatives are represented at points whose coordinates are proportional to the centroids of the points representing subjects who have chosen them. This centroid scaling technique estimates multidimensional joint spaces from the pick-any data.I am indebted to John Baird, Clyde Coombs, David Eames, John Hunter, Michael J. Levine, Elliot Noma, Robert Z. Norman, William S. Roy, Joseph Schwartz, Daniel Velleman, the editor, and anonymous reviewers for ideas and suggestions that have been incorporated into this work. Conferences organized by Hans J. Hummel for the Deutsche Forschungsgemeinschaft (1977) and by Samuel Leinhardt for the National Science Foundation (1975) were instrumental in the development of this work.     

The joint analysis of direct ratings,pairwise preferences,and dissimilarities

In studies involving judgments of similarity or dissimilarity, a variety of other variables may also be measured. Examples might be direct ratings of the stimuli, pairwise preference judgments, and physical measurements of the stimuli with respect to various properties. In such cases, there are important advantages to joint analyses of the dissimilarity and collateral variables. A variety of models are described for relating these and algorithms described for fitting these to data. A number of hypothesis tests are developed and an example offered.The research reported here was supported by grant number APA 320 to the author by the National Research Council of Canada.     

Three notes on ALSCAL

It is reported that (1) a new coordinate estimation routine is superior to that originally proposed for ALSCAL; (2) an oversight in the interval measurement level case has been found and corrected; and (3) a new initial configuration routine is superior to the original.     

Avoiding degeneracy in multidimensional unfolding by penalizing on the coefficient of variation

Multidimensional unfolding methods suffer from the degeneracy problem in almost all circumstances. Most degeneracies are easily recognized: the solutions are perfect but trivial, characterized by approximately equal distances between points from different sets. A definition of an absolutely degenerate solution is proposed, which makes clear that these solutions only occur when an intercept is present in the transformation function. Many solutions for the degeneracy problem have been proposed and tested, but with little success so far. In this paper, we offer a substantial modification of an approach initiated bythat introduced a normalization factor based on thevariance in the usual least squares loss function. Heiser unpublishedthesis, (1981) and showed that the normalization factor proposed by Kruskal and Carroll was not strong enough to avoid degeneracies. The factor proposed in the present paper, based on the coefficient of variation, discourages or penalizes nonmetric transformations of the proximities with small variation, so that the procedure steers away from solutions with small variation in the interpoint distances. An algorithm is described for minimizing the re-adjusted loss function, based on iterative majorization. The results of a simulation study are discussed, in which the optimal range of the penalty parameters is determined. Two empirical data sets are analyzed by our method, clearly showing the benefits of the proposed loss function.The authors would like to thank the editor, an associate editor, and three reviewers for their valuable comments and suggestions to improve the quality of this work.This revised article was published online in August 2005 with the PDF paginated correctly.     

A note on item information in any direction for the multidimensional three-parameter logistic model

The purpose of this note is twofold: (a) to present the formula for the item information function (IIF) in any direction for the Multidimensional 3-Parameter Logistic (M3-PL) model and (b) to give the equation for the location of maximum item information (θ_max) in the direction of the item discrimination vector. Several corollaries are given. Implications for future research are discussed.This research was supported in part by an Educational Testing Service (ETS) Harold T. Gulliksen Psychometric Research Fellowship to the author.This revised article was published online in August 2005 with the PDF paginated correctly.     

Multidimensional unfolding analyses of ranking data for groups

One probabilistic version of Coombs' unfolding model called the MMUR (Marginalization model for the Multidimensional Unfolding analysis of Ranking data) is extended to treat ranking data for groups. One favorable feature of the model is that it can both take into consideration individual differences without estimating the subject parameters and capture the differences between the groups in a systematic manner. Another advantage lies in the fact that one can see the group differences in the geometrical point configuration, since the model shows how the ideal points of the groups differ from each other in space. Four applications are provided which demonstrate that the model is useful for clarifying systematic differences in this type of data.     

Choice model analysis of the “pick any/n” type of binary data

In the “pick any/n” method, subjects are asked to choose any number of items from a list of n items according to some criterion. This kind of data can be analyzed as a special case of either multiple-choice data or successive categories data where the number of response categories is limited to two. An item response model was proposed for the latter case, which is a combination of an unfolding model and a choice model. The marginal maximum-likelihood estimation method was developed for parameter estimation to avoid incidental parameters, and an expectation-maximization algorithm used for numerical optimization. Two examples of analysis are given to illustrate the proposed method, which we call MAXSC.     

Multidimensional adaptive testing

Maximum likelihood and Bayesian procedures for item selection and scoring of multidimensional adaptive tests are presented. A demonstration using simulated response data illustrates that multidimensional adaptive testing (MAT) can provide equal or higher reliabilities with about one-third fewer items than are required by one-dimensional adaptive testing (OAT). Furthermore, holding test-length constant across the MAT and OAT approaches, substantial improvements in reliability can be obtained from multidimensional assessment. A number of issues relating to the operational use of multidimensional adaptive testing are discussed.     

Gleaning in the field of dual scaling

Some historical background and preliminary technical information are first presented, and then a number of hidden, but important, methodological aspects of dual scaling are illustrated and discussed: normed versus projected weights, the amount of information accounted for by each solution, a perfect solution to the problem of multidimensional unfolding, multidimensional quantification space, graphical display, number-of-option problems, option standardization versus item standardization, and asymmetry of symmetric (dual) scaling. Contrary to the common perception that dual scaling and similar quantification methods are now mathematically transparent, the present study demonstrates how much more needs to be clarified for routine use of the method to arrive at valid conclusions. Data analysis must be carried out in such a way that common sense, intuition and sound logic will prevail.Presidential Address delivered at the Annual Meeting of the Psychometric Society, Banff Centre for Conferences, Banff, Alberta, Canada, June 27–30, 1996. The work has been supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada. I am grateful to Ira Nishisato for his comments, Ingram Olkin and Yoshio Takane for important references, and Liqun Xu for computational help.