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.     

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.     

An individual difference model for the multidimensional analysis of preference data

A model for the analysis of paired comparison data is presented which combines features of the BTL-model with features of the Unfolding model. The model is metric, mathematically tractable, and has an exact algebraic solution. Since it is multidimensional and allows for individual differences, it is thought to be more realistic for some choice situations than either the Thurstone model or the BTL-model. No claim is made that the present model will be appropriate for all conceivable choice situations. Rather, it is argued that the fact that it is explicitly falsifiable is a point in its favor.This work was supported by a grant of the United States Public Health Service (1 R03 MH19139 01 MSM) and a David Ross (XR) grant of the Purdue Research Foundation (PRF 2132), both of which are gratefully acknowledged. We also owe thanks to Dr. Terry Cooper, whose careful scrutiny of an earlier draft of this paper allowed us to reduce the number of errors from n to n-1.     

The prediction of political evaluation and voting preference: A multidimensional analysis

Two separate studies, using college students in a political behavior class, tested the relationship between evaluation and voting preferences on the one hand and the dimensions emerging from a multidimensional scaling of similarity judgments for a set of nationally known political figures. Both studies found that political evaluation and voting preferences were highly predictable from three major dimensions underlying political perception. In Study I (n = 64), the subjects' self-ratings of liberal-conservatism yielded individual differences in the predictability of activity and potency from the political dimensions as well as differences in the relationships among evaluation, activity, and potency. In Study II (n = 51), the subjects' own authoritarianism indicated differences in the manner in which political figures were perceived in the multidimensional space.     

Group cohesion: A new multidimensional model

Based upon a careful perusal of all major theoretical reviews and most available controlled, empirical studies from the last 30 years, a five factor model of group cohesion is proposed. These factors are: attraction and bonding; support and caring; listening and empathy; self-disclosure and feedback; and process performance and goal attainment. To understand more fully the complex phenomenon of group cohesion, certain preconditions to the cohesive therapy group must be considered. The most significant of these are: selection of suitable participants; a balanced composition of the group; and effective orientation, training, and contracting. Important early group conditions are: resolving conflict and rebellion; constructive norming and culture building; and reducing avoidance and defensiveness. This new, generic, multidimensional model of group cohesion should prove of great value both to the clinician and the researcher in the group psychotherapy field and related areas.The author is indebted to the following colleagues for constructive criticisms of earlier versions of this report: S. Budman, R. R. Dies, K. R. MacKenzie, W. E. Piper, M. A. Richardsen, and I. D. Yalom.     

A simple computational algorithm of model-based choice preference

A broadly used computational framework posits that two learning systems operate in parallel during the learning of choice preferences—namely, the model-free and model-based reinforcement-learning systems. In this study, we examined another possibility, through which model-free learning is the basic system and model-based information is its modulator. Accordingly, we proposed several modified versions of a temporal-difference learning model to explain the choice-learning process. Using the two-stage decision task developed by Daw, Gershman, Seymour, Dayan, and Dolan (2011), we compared their original computational model, which assumes a parallel learning process, and our proposed models, which assume a sequential learning process. Choice data from 23 participants showed a better fit with the proposed models. More specifically, the proposed eligibility adjustment model, which assumes that the environmental model can weight the degree of the eligibility trace, can explain choices better under both model-free and model-based controls and has a simpler computational algorithm than the original model. In addition, the forgetting learning model and its variation, which assume changes in the values of unchosen actions, substantially improved the fits to the data. Overall, we show that a hybrid computational model best fits the data. The parameters used in this model succeed in capturing individual tendencies with respect to both model use in learning and exploration behavior. This computational model provides novel insights into learning with interacting model-free and model-based components.     

A demonstration of intransitivity in natural categories

Two experiment are reported which demonstrated intransitivity in category judgments, thus challenging a widely held assumption that the relation between categorized sets is one of class inclusion. Subjects consistently accepted the truth of certain category statements, in spite of being aware of the existence of counterexamples. Implications for semantic memory theory are discussed.     

The expectancy model in the analysis of occupational preference and occupational choice

The differences between the factors related to occupational preference and occupational choice were explored for upper-division college students from business and psychology classes. The results strongly supported a valance-instrumentality-expectancy model in the context of occupational choice. Occupational preference was primarily a function of occupational valence, the multiplicative relation between the outcome valences and the instrumentalities of the occupations for these outcomes. Occupational choice was primarily a function of the force to choose an occupation, the multiplicative relaitonship between occupational valence and the expectancy of attaining an occupation and the expected costs of attaining an occupation.     

A probabilistic multidimensional model of location information

Summary A probabilistic multidimensional model of location discrimination is developed and applied to data from an experiment in which subjects are required to determine whether a briefly presented horizontal and vertical bar are touching. The proposed gap-detection model assumes that errors in perception are due to variability in the perceived location and/or in the perceived length of the bars. A series of gap-detection models that allow variability only in perceived location were rejected on the basis of likelihood-ratio tests of overall goodness of fit. However, when the models were modified to account for: (a) a compression of the distance perceived between the bars (Wolford, 1975), or (b) the bisection illusion (Künnapas, 1955), excellent absolute fits to the data were obtained. A pair of models that suggests that the horizontal/vertical illusion or a response bias was operative failed. Applications of the model to more conventional object-perception experiments (e. g., the illusory-conjunction experiment) are discussed.     

A multidimensional system dynamics model of affect

A structural—functional perspective based on factor analysis is presented for the domain of affect. In this approach the affective system is viewed as a multivariate, hierarchically structured arrangement of affective dimensions functionally organized into three subsystems: introversion—extraversion, emotional stability, and emotional independence. Dynamic interactions within and between these subsystems are discussed in terms of information flow and feedback processes. The model that emerges is put forward in the context of accommodating the empirical findings of both experimental psychology and individual differences.     

A model of transitive choice

We report a formal model of transitive inference based on protocols from experiments on squirrel monkeys solving the 5-term series problem (McGonigle & Chalmers, 1977, 1992). These studies generate databases featuring transitive choice, task transfer (where at first a significant decrement is observed, and later substantial improvement without explicit training), and, finally, a Symbolic Distance Effect (SDE) based on decision-time data. Using a rule-based (production) system, we first established rule stacks at the group, then at the individual level, on the basis of triadic transfer performance first recorded in the McGonigle and Chalmers (1977) study. The models for each subject then accommodated data from the more intensive, later study with the same subjects (McGonigle & Chalmers, 1992). We found the initial model capable of dealing with all choice and reaction-time phenomena reported thus far, with only small changes in a rule search procedure. In common with an independent assay by McGonigle and Chalmers (1992), our model-based reassessment of decision times indicates that a major source of reaction time variation is item prominence in the rule stack rather than interitem (ordinal) distance per se.     

Individual differences in perceived similarity and preference for visual art: A multidimensional scaling analysis

Previous studies have shown that individuals may attend to different dimensions in making an overall judgment of similarity between complex stimuli. The present study investigated the nature of differences in the perceived similarity of reproductions of paintings by the use of multidimensional scaling techniques. Using the INDSCAL model, a group of art-trained students are shown to differ significantly from a group of nonart students in terms of their differential weighing of a set of common dimensions. The same subjects’ preferences are examined in relation to these differences by use of the PREFMAP hierarchy of models. While the simplest (vector) model was found to be appropriate for almost all subjects, large differences in vector direction and average subject ideal-point location are found. Implications for future studies of responses to art are discussed.     

Computer-based preference models for generic multidimensional problem sets

Alternative actions or objects can be characterized by differences among defining attributes or dimensions. Making choices among these alternatives is often a complex and difficult task, particularly when the number of alternatives or defining attributes is large or the time to decide is limited. This paper describes two programs that allow users to define a multidimensional problem domain and derive a multiplicative utility function that quantitatively describes their judgment policy within that domain. It also discusses potential applications for the derived preference models.     

A partial repetition effect in choice reaction time for multidimensional stimuli

Bertelson (1963), and others more recently, have observed that when the same stimulus is presented consecutively the reaction time (RT) to the second presentation is faster than the RT to the first presentation. In Experiments 1 and 2, two-dimensional stimuli are used in a discrete four-choice RT task where the stimuli can either be completely repeated, partially repeated, or completely different. In Experiment 3, three-dimensional stimuli were used in a discrete eight-choice RT task where the stimuli can either be completely repeated, partially respeated with respect to two dimensions, partially repeated with respect to one dimension, or completely different. The partial repetition of multidimensional stimuli showed a significant facilitation effect on RT with all sets of stimuli used in the present experiments. This result is discussed in respect to the various possible loci of the repetition effect.     

A model of situational preference amongst smokers

Seventy-five medium-light smokers rated their performance for smoking in 25 situations covering a range of high- and low-stress situations. Information on personality, age, sex, length of habit, number of cigarettes smoked, amount inhaled and amount of cigarettes smoked was also obtained. Six main factors were extracted by principal-factor analysis which were related respectively to situations involving preparatory activity, attentional stress, neutral relaxation conditions, unpleasant (anxiety) and pleasant (excited) emotional stress and social vs isolated activity. Smokers who scored high on Neuroticism tended to smoke more when experiencing unpleasant emotions; older smokers smoked predominantly in pleasant relaxing conditions; extraverts and low-activity situational smokers, reported inhaling more. A model of smoker motivation is developed which suggests that both pharmacological and psychological factors contribute to habit maintenance. Three basic motivational types of smokers are identified: high-stress smokers for whom smoking is a secondary activity, and who rely on smoking to form a minor distraction for their current task. Low-activity smokers who wish to act, or change their current affective state in some way and who rely on pharmacological factors. Neutral relaxation smokers for whom smoking is a pleasurable activity in its own right.     

Path model of multidimensional compliance with cancer therapy

Patients newly diagnosed with hematologic malignancies were followed for a 6-month treatment period to assess compliance with three regimen requirements for cancer therapy: anti-neoplastic medication self-administered intermittently, supportive medication self-administered daily, and monthly clinic appointments. The effect on compliance of three intervention "packages" (some combination of education, shaping of pill-taking behavior, and home restructuring) and the extent that patient satisfaction, knowledge, and uncertainty about illness-related events mediated the effects of the interventions were also examined. Blood levels of the drugs and self-report measures indicated that compliance with daily pill taking was higher for each intervention group compared to a control group. Similar results were obtained for compliance with clinic appointments. No improvement in intermittent self-medication was found. Although each intervention package increased patient knowledge and satisfaction, path analyses demonstrated that knowledge did not affect any aspect of compliance, whereas satisfaction was associated with increased appointment keeping only. Daily pill taking was influenced directly by the behavioral components of the interventions. Uncertainty did not influence compliance but was associated with depression, which was negatively correlated with intermittent self-medication.     

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 procedure for ordering object pairs consistent with the multidimensional unfolding model

A procedure for ordering object (stimulus) pairs based on individual preference ratings is described. The basic assumption is that individual responses are consistent with a nonmetric multidimensional unfolding model. The method requires data where a numerical response is independently generated for each individual-object pair. In conjunction with a nonmetric multidimensional scaling procedure, it provides a vehicle for recovering meaningful object configurations.The author wishes to thank Jack Hoadley, Larry Mayer, Sheldon Newhouse, Stuart Rabinowitz, Forrest Young, and three anonymous reviewers for their useful suggestions.