Multidimensional Fechnerian Scaling: Basics

Fechnerian scaling is a theory of how a certain (Fechnerian) metric can be computed in a continuous stimulus space of arbitrary dimensionality from the shapes of psychometric (discrimination probability) functions taken in small vicinities of stimuli at which these functions reach their minima. This theory is rigorously derived in this paper from three assumptions about psychometric functions: (1) that they are continuous and have single minima around which they increase in all directions; (2) that any two stimulus differences from these minimum points that correspond to equal rises in discrimination probabilities are comeasurable in the small (i.e., asymptotically proportional), with a continuous coefficient of proportionality; and (3) that oppositely directed stimulus differences from a minimum point that correspond to equal rises in discrimination probabilities are equal in the small. A Fechnerian metric derived from these assumptions is an internal (or generalized Finsler) metric whose indicatrices are asymptotically similar to the horizontal cross-sections of the psychometric functions made just above their minima. Copyright 2001 Academic Press.     

Multidimensional Fechnerian Scaling: Probability-Distance Hypothesis

The probability-distance hypothesis states that the probability with which one stimulus is discriminated from another is a function of some subjective distance between these stimuli. The analysis of this hypothesis within the framework of multidimensional Fechnerian scaling yields the following results. If the hypothetical subjective metric is internal (which means, roughly, that the distance between two stimuli equals the infimum of the lengths of all paths connecting them), then the underlying assumptions of Fechnerian scaling are satisfied and the metric in question coincides with the Fechnerian metric. Under the probability-distance hypothesis, the Fechnerian metric exists (i.e., the underlying assumptions of Fechnerian scaling are satisfied) if and only if the hypothetical subjective metric is internalizable, which means, roughly, that by a certain transformation it can be made to coincide in the small with an internal metric; and then this internal metric is the Fechnerian metric. The specialization of these results to unidimensional stimulus continua is closely related to the so-called Fechner problem proposed in 1960's as a substitute for Fechner's original theory.     

Multidimensional Fechnerian Scaling: Perceptual Separability

A new definition of the perceptual separability of stimulus dimensions is given in terms of discrimination probabilities. Omitting technical details, stimulus dimensions are considered separable if the following two conditions are met: (a) the probability of discriminating two sufficiently close stimuli is computable from the probabilities with which one discriminates the projections of these stimuli on the coordinate axes; (b) the psychometric differential for discriminating two sufficiently close stimuli that differ in one coordinate only does not depend on the value of their matched coordinates (the psychometric differential is the difference between the probability of discriminating a comparison stimulus from a reference stimulus and the probability with which the reference is discriminated from itself). Thus defined perceptual separability is analyzed within the framework of the regular variation version of multidimensional Fechnerian scaling. The result of this analysis is that the Fechnerian metric of a stimulus space with perceptually separable dimensions has the structure of a Minkowski power metric with respect to these dimensions. The exponent of this metric equals the psychometric order of the stimulus space, or 1, whichever is greater.     

The Consulting Process: A Multidimensional Approach

The consulting process seems to work similarly regardless of whether the consultant is internal or external or is doing individual or group consultation. Some differences may occur depending on whether the consultant is a process helper or content helper, although good content-focused consultants often possess excellent process skills and use them as a regular part of the consulting process. Several consultation processes were reviewed, and from this a six-step approach is presented.     

Multi-Family Group Therapy: A Multidimensional Approach

The procedures, dynamics, and process of multi-family group therapy are presented with case examples from the authors' practice.     

Multidimensional Fechnerian Scaling: Regular Variation Version

The underlying assumptions of Fechnerian scaling are complemented by an assumption that ensures that any psychometric differential (the rise in the value of a discrimination probability function as one moves away from its minimum in a given direction) regularly varies at the origin with a positive exponent. This is equivalent to the following intuitively plausible property: any two psychometric differentials are comeasurable in the small (i.e., asymptotically proportional at the origin), without, however, being asymptotically equal to each other unless the corresponding values of the Fechner-Finsler metric function are equal. The regular variation version of Fechnerian scaling generalizes the previously proposed power function version while retaining its computational and conceptual simplicity.     

The Dimensions Underlying Loneliness Counseling Interventions: A Multidimensional Scaling Solution

The purpose of this study was to investigate major dimensions affecting the acceptability of loneliness counseling interventions from the potential client's perspective and to compare the results with dimensions posited by theoretical models of treatment acceptability. Data were collected from 241 university students who were asked to sort loneliness counseling interventions on the basis of their similarities. Multidimensional scaling analyses revealed 4 dimensions underlying the acceptability of loneliness counseling interventions: the type of activity required by the intervention, the counselor-client relationship, the difficulty of the intervention, and the fit of the intervention to the problem.     

The Circumplexity of Leary's Interpersonal Circle: A Multidimensional Scaling Perspective

To investigate the circulant structure of the 16 scales comprising the Interpersonal Checklist (ICL; Laforge & Suczek, 1955), intercorrelation matrices for mate and female undergraduate ICL scores were subjected to multidimentional scaling analysis. Results showed the following: (a) while the 16 ICL scales fall in a rough circular array, measurement gaps exist in the friendly-dominant and hostile-submissive quadrants; (b) sixteenths A-B-P and L-M-N were misarranged; (c) stress coefficients for a three-dimensional solution were in an acceptable range. Implications of these findings for future research and anchoring of the ICL to Kiesler's (1983) "1982 Interpersonal Circle" are discussed.     

Cross-Validation of the Reactions to Faculty Incivility Measurement through a Multidimensional Scaling Approach

Incivility in the academic arena elicits a wide range of reactions: it interferes with learning, increases stress, feelings of disrespect and helplessness. Although reactions to incivility were mainly tested in workplaces, an extensive, robust framework to explain and measure responses to faculty incivility (FI) is yet to be offered. This study used Facet theory (FT) approach with a multidimensional scaling method of smallest space analysis (SSA), and confirmatory factor analysis (CFA) to confirm the theoretical structure of reactions to FI. A mapping sentence was constructed expressing the composite of three individual facets based on the theoretical framework: Facet A including four types of reactions (1)Exit (2) Voice (3) Loyalty and (4) Neglect (EVLN); Facet B reflecting the destructiveness – constructiveness dimension, and Facet C illustrating a dimension ranging from passive to active responses. Data were gathered by a scale measuring students’ reactions to FI. According to the findings, the CFA result presented four relatively interpretable factors (EVLN) while the SSA showed these factors as well as additional facets (B and C).     

Teaching and Learning Forgiveness: A Multidimensional Approach

Abstract. This essay seeks to illumine the teaching and learning of the practice of forgiveness by relating a range of theoretical perspectives (theological, psychological, and socio‐cultural) to the process of cultivating the practical wisdom needed for forgiveness. We discuss how a Trinitarian “epistemology of the cross” might lead one to a new way of perceiving life’s constraints and possibilities and relate this theological epistemology to three psychological approaches for understanding forgiveness – a narrative approach, object‐relations theory, and consciousness development theory. Our discussion of these theoretical perspectives is explicitly related to the practice of teaching and learning forgiveness, outlining learning activities we have used in a course we taught (which ranged from case studies and film to lectures and discussions based on close readings of biblical and theological texts) and reporting highlights in our students’ work.     

Group And Individual Regularities In Trait Inference: A Multidimensional Scaling Analysis

This study concerned regularities in trait inference, the manner in which limited information about another person is used to form a consistent total impression. An individual differences model for multidimensional scaling was appraised as a method for faithfully strong both broad culturally- determined standards, and differently-structured points of view. Judgments of likelihood of joint occurrence of trait pairs and measures of cognition were obtained from 139 subjects. Scaling analyses resulted in two points of view, the largest of which yielded dimensions of interpersonal affectivity, harmfulness, and charitableness. S e w a t e analyses of randomly divided groups yielded indices of dimension stability ranging up to .99. The results support the view that the multidimensional scaling model can serve well to explain and order trait inference data.     

Sensible Atoms: A Techno-aesthetic Approach to Representation

This essay argues that nano-images would be best understood with an aesthetical approach rather than with an epistemological critique. For this aim, I propose a ‘techno-aesthetical’ approach: an enquiry into the way instruments and machines transform the logic of the sensible itself and not just the way by which it represents something else. Unlike critical epistemology, which remains self-evidently grounded on a representationalist philosophy, the approach developed here presents the advantage of providing a clear-cut distinction between image-as-representation and other modes of existence of images, such as the one of ‘imaginaction’ that I draw from a comparison between far-field and near-field microscopies. Once this regime of imaginaction is distinguished from representation, I focus on nanotechnological percepts and argue that they follow a transmodal logic. I then draw the implications of this enquiry in terms of a new sensible condition that changes the way we think of non-living objects. Finally, I conclude that if techno-aesthetics dares to posit and articulate sensibility beyond the privileged sphere of subject/object relationships, it simultaneously engages us to consider the political character of our responsibilities towards the design of nano-engineered sensorial spaces.     

A Counseling Practitioner's Primer to the Use of Multidimensional Scaling

Applications of multidimensional scaling (MDS) to counseling practice are discussed, and the author's use of MDS as a tool in a counseling training group is described.     

Multidimensional Scaling in Counseling Research and Practice

The authors briefly describe the use of multidimensional scaling (MDS) in counseling. They discuss types of data that have been analyzed using MDS, kinds of MDS analyses, and the interpretation of MDS results and review MDS research relevant to vocational, family, group, and individual counseling. They also briefly discuss potential applications in counseling practice.     

Probabilistic Multidimensional Scaling Using a City-Block Metric

Using a probabilistic model, exact and approximate probability density functions (PDFs) for city-block distances and distance ratios are developed. The model assumes that stimuli can be represented by random vectors having multivariate normal distributions. Comparisons with the more common Euclidean PDFs are presented. The potential ability of the proposed model to correctly detect Euclidean and city-block metrics is briefly investigated. These results are then contrasted to those obtained using a deterministic, nonmetric model. Copyright 2001 Academic Press.     

Goodness-of-Fit Assessment in Multidimensional Scaling and Unfolding

Judging goodness of fit in multidimensional scaling requires a comprehensive set of diagnostic tools instead of relying on stress rules of thumb. This article elaborates on corresponding strategies and gives practical guidelines for researchers to obtain a clear picture of the goodness of fit of a solution. Special emphasis will be placed on the use of permutation tests. The second part of the article focuses on goodness-of-fit assessment of an important variant of multidimensional scaling called unfolding, which can be applied to a broad range of psychological data settings. Two real-life data sets are presented in order to walk the reader through the entire set of diagnostic measures, tests, and plots. R code is provided as supplementary information that makes the whole goodness-of-fit assessment workflow, as presented in this article, fully reproducible.     

The Rokeach Value Survey: Underlying Structure and Multidimensional Scaling

Studies have demonstrated that the ordinal, ipsative data provided by the Rokeach Value Survey (RVS; Rokeach 1973) are not suited to factor analysis. In this study, nonmetric multidimensional scaling (MDS) was used with a sorting task to identify the underlying subset of values. American college students were the participants, and the results indicate that individualism-achievement and collectivism-affiliation are the underlying dimensions of the RVS for both the terminal and the instrumental values. Observed variation in the use of MDS space was predicted, based on participants' developmental differences as measured by the Maslowian Assessment Survey (Williams & Page, 1989). Gender differences in the use of MDS space by participants were not observed. Analysis of angular variance was used to test both hypotheses.     

Estimating Optimal Weights for Compound Scores: A Multidimensional IRT Approach

A method is proposed for constructing indices as linear functions of variables such that the reliability of the compound score is maximized. Reliability is defined in the framework of latent variable modeling [i.e., item response theory (IRT)] and optimal weights of the components of the index are found by maximizing the posterior variance relative to the total latent variable variance. Three methods for estimating the weights are proposed. The first is a likelihood-based approach, that is, marginal maximum likelihood (MML). The other two are Bayesian approaches based on Markov chain Monte Carlo (MCMC) computational methods. One is based on an augmented Gibbs sampler specifically targeted at IRT, and the other is based on a general purpose Gibbs sampler such as implemented in OpenBugs and Jags. Simulation studies are presented to demonstrate the procedure and to compare the three methods. Results are very similar, so practitioners may be suggested the use of the easily accessible latter method. A real-data set pertaining to the 28-joint Disease Activity Score is used to show how the methods can be applied in a complex measurement situation with multiple time points and mixed data formats.     

The Multidimensional Scaling Of A Set Of Artistic Drawings : Perceived Structure And Scale Correlates

Product-centered research on creativity approaches the criterion problem of what is to be the referent for creativity through the analysis of tangible products such as a r t objects, writing, or scientific achievements. The present research is concerned with the evaluation and study of artist drawings contributed by sophomore students a t the Rhode Island School of Design. Multi-dimensional scaling methods were applied to similarity judgments obtained from art experts on two separate sets of 26 drawings. Three similarity dimensions accounted for the interstimulus distances for each set of drawings. Although no statistical test was available, the dimensions from the two seta appeared to correspond. Scale values of 4 drawings common to the two sets were consistent, and the dimensions appeared to define very similar stimulus characteristics. It was concluded that multidimensional scaling procedures provided a means for differentiating among a set of complex, esthetic products. Scale values of drawings on the three dimensions also correlated differentially with cognitive and achievement measures available on the students, suggesting that product dimensions identified via similarity judgments were related to characteristics of individuals producing the products. Hypotheses were developed as to the psychological meaning of the three product dimensions.