Parameter estimation for the thurstone case III model

A Monte Carlo study was conducted to investigate the ability of three estimation criteria to recover the parameters of Case V and Case III models from comparative judgment data. Significant differences in recovery are shown to exist.This research was supported by grant SOC76-20517 from the National Science Foundation. The authors acknowledge with appreciation comments received on an earlier draft by Harold Lindman, Joseph Zinnes, and anonymous reviewers.     

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.     

A new solution to the additive constant problem in metric multidimensional scaling

A new solution to the additive constant problem in metric multidimensional scaling is developed. This solution determines, for a given dimensionality, the additive constant and the resulting stimulus projections on the dimensions of a Euclidean space which minimize the sum of squares of discrepancies between the formal model for metric multidimensional scaling and the original data. A modification of Fletcher-Powell style functional iteration is used to compute solutions. A scale free index of the goodness of fit is developed to aid in selecting solutions of adequate dimensionality from multiple candidates.This research is based in part on the author's Ph.D. dissertation at the University of Illinois at Urbana-Champaign. Computer time was provided by the Campus Computing Network of the University of California, Los Angeles.     

A new solution to the problem of finding all numerical solutions to ordered metric structures

A new algorithm is used to test and describe the set of all possible solutions for any linear model of an empirical ordering derived from techniques such as additive conjoint measurement, unfolding theory, general Fechnerian scaling and ordinal multiple regression. The algorithm is computationally faster and numerically superior to previous algorithms.This research was supported in part by NIGMS grant GM-01231 to the University of Michigan. Authors' names are in alphabetic order.     

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.     

The problem of the additive constant and eigenvalues in metric multidimensional scaling

This paper is concerned with the additive constant problem in metric multidimensional scaling. First the influence of the additive constant on eigenvalues of a scalar product matrix is discussed. The second part of this paper is devoted to the introduction of a new formulation of the additive constant problem. A solution is given for fixed dimensionality, by maximizing a normalized index of fit with a gradient method. An experimental computation has shown that the author's solution is accurate and easy to follow.     

A solution to the many attitudes problem

Philosophical Studies - According to noncognitivism, normative beliefs are just desire-like attitudes. While noncognitivists have devoted great effort to explaining the nature of normative belief,...     

Optimal scaling by alternating length-constrained nonnegative least squares, with application to distance-based analysis

An important feature of distance-based principal components analysis, is that the variables can be optimally transformed. For monotone spline transformation, a nonnegative least-squares problem with a length constraint has to be solved in each iteration. As an alternative algorithm to Lawson and Hanson (1974), we propose the Alternating Length-Constrained Non-Negative Least-Squares (ALC-NNLS) algorithm, which minimizes the nonnegative least-squares loss function over the parameters under a length constraint, by alternatingly minimizing over one parameter while keeping the others fixed. Several properties of the new algorithm are discussed. A Monte Carlo study is presented which shows that for most cases in distance-based principal components analysis, ALC-NNLS performs as good as the method of Lawson and Hanson or sometimes even better in terms of the quality of the solution. Supported by The Netherlands Organization for Scientific Research (NWO) by grant nr. 030-56403 for the “PIONEER” project “Subject Oriented Multivariate Analysis” to the third author. We would like to thank the anonymous referees for their valuable remarks that have improved the quality of this paper.     

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.     

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.     

A modest solution to the problem of religious disagreement

Many assume that the best, and perhaps only, way to address the so-called Problem of Evil (PoE) is to claim that God does not do evil, but that God merely allows evil. This assumption depends on two claims: (i) the doing-allowing distinction exists and (ii) the doing-allowing distinction is morally significant. In this paper I try to undermine both of these claims. Against (i) I argue that some of the most influential analyses of the doing-allowing distinction face grave difficulties and that these difficulties are only exacerbated when applied to God. Against (ii) I argue that broadly Kantian considerations give a better explanation of the moral differences in paradigm cases than the doing-allowing distinction and, when applied to God, effectively dissolve the moral significance of the doing-allowing distinction. The upshot of this is that those who claim God does evil are no worse off than those who claim God allows evil regarding the PoE.     

An entropy approach to the scaling of ordinal categorical data

This paper studies the problem of scaling ordinal categorical data observed over two or more sets of categories measuring a single characteristic. Scaling is obtained by solving a constrained entropy model which finds the most probable values of the scales given the data. A Kullback-Leibler statistic is generated which operationalizes a measure for the strength of consistency among the sets of categories. A variety of data of two and three sets of categories are analyzed using the entropy approach.This research was partially supported by the Air Force Office of Scientific Research under grant AFOSR 83-0234. The support by the Air Force through grant AFOSR-83-0234 is gratefully acknowledged. The comments of the editor and referees have been most helpful in improving the paper, and in bringing several additional references to our attention.     

A general first-order solution to the ramification problem with cycles

We provide a solution to the ramification problem that integrates findings of different axiomatic approaches to ramification from the last ten to fifteen years. For the first time, we present a solution that: (1) is independent of a particular time structure, (2) is formulated in classical first-order logic, (3) treats cycles – a notoriously difficult aspect – properly, and (4) is assessed against a state-transition semantics via a formal correctness proof.This is achieved as follows: We introduce indirect effect laws that enable us to specify ramifications that are triggered by activation of a formula rather than just an atomic effect. We characterise the intended models of these indirect effect laws by a state-transition semantics. Afterwards, we show how to compile a class of indirect effect laws into first-order effect axioms that then solve the ramification and frame problems. We finally prove the resulting effect axioms sound and complete with respect to the semantics defined earlier.     

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.     

A note on the determination of configuration and weights for a class of individual scaling models

Consider the typical problem in individual scaling, namely finding a common configuration and weights for each individual from the given interpoint distances or scalar products. Within the STRAIN framework it is shown that the problem of determining weights for a given configuration can be posed as a standard quadratic programming problem. A set of necessary conditions for an optimal configuration to satisfy are given. A closed form expression for the configuration is obtained for the one dimensional case and an approach is given for the two dimensional case.     

Detecting chance: A solution to the null sensitivity problem in subliminal priming

In many paradigms, the persuasiveness of subliminal priming relies on establishing that stimuli are undetectable. The standard significance test approach is ill-suited as null results may reflect either truly undetectable stimuli or a lack of power to resolve weakly detectable stimuli. We present a novel statistical model as an alternative. The model provides for estimates of the probability that each individual is truly at chance. Researchers may select individuals for whom there are sufficiently high probabilities of true undetectability. The model is hierarchical, and estimation is done within the Bayesian framework.     

A general solution of the weighted orthonormal procrustes problem

A general solution for weighted orthonormal Procrustes problem is offered in terms of the least squares criterion. For the two-demensional case. this solution always gives the global minimum; for the general case, an algorithm is proposed that must converge, although not necessarily to the global minimum. In general, the algorithm yields a solution for the problem of how to fit one matrix to another under the condition that the dimensions of the latter matrix first are allowed to be transformed orthonormally and then weighted differentially, which is the task encountered in fitting analogues of the IDIOSCAL and INDSCAL models to a set of configurations.The authors are grateful to the Editor and the anonymous reviewers for their helpful comments on an earlier draft of this paper.