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.     

Fechnerian metrics in unidimensional and multidimensional stimulus spaces

A new theory is proposed for subjective (Fechnerian) distances among stimuli in a continuous stimulus space of arbitrary dimensionality. Each stimulus in such a space is associated with a psychometric function that determines probabilities with which it is discriminated from other stimuli, and a certain measure of its discriminability from its infinitesimally close neighboring stimuli is computed from the shape of this psychometric function in the vicinity of its minimum. This measure of discriminability can be integrated along any path connecting any two points in the stimulus space, yielding the psychometric length of this path. The Fechnerian distance between two stimuli is defined as the infimum of the psychometric lengths of all paths connecting the two stimuli. For a broad class of models defining the dichotomy of response bias versus discriminability, the Fechnerian distances are invariant under response bias changes. In the case in which physically multidimensional stimuli are discriminated along some unidimensional subjective attribute, a systematic construction of the Fechnerian metric leads to a resolution of the long-standing controversy related to the numbers of just-noticeable differences between isosensitivity curves. It is argued that for unidimensional stimulus continua, the proposed theory is close to the intended meaning of Fechner's original theory.     

Multidimensional Fechnerian Scaling: Pairwise Comparisons, Regular Minimality, and Nonconstant Self-Similarity

Stimuli presented pairwise for same-different judgments belong to two distinct observation areas (different time intervals and/or locations). To reflect this fact the underlying assumptions of multidimensional Fechnerian scaling (MDFS) have to be modified, the most important modification being the inclusion of the requirement that the discrimination probability functions possess regular minima. This means that the probability with which a fixed stimulus in one observation area (a reference) is discriminated from stimuli belonging to another observation area reaches its minimum when the two stimuli are identical (following, if necessary, an appropriate transformation of the stimulus measurements in one of the two observation areas). The remaining modifications of the underlying assumptions are rather straightforward, their main outcome being that each of the two observation areas has its own Fechnerian metric induced by a metric function obtained in accordance with the regular variation version of MDFS. It turns out that the regular minimality requirement, when combined with the empirical fact of nonconstant self-similarity (which means that the minimum level of the discrimination probability function for a fixed reference stimulus is generally different for different reference stimuli), imposes rigid constraints on the interdependence between discrimination probabilities and metric functions within each of the observation areas and on the interdependence between Fechnerian metrics and metric functions belonging to different observation areas. In particular, it turns out that the psychometric order of the stimulus space cannot exceed 1.     

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.     

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.     

Dual scaling of comparison and reference stimuli in multi-dimensional psychological space

Dzhafarov and Colonius (Psychol. Bull. Rev. 6 (1999)239; J. Math. Psychol. 45(2001)670) proposed a theory of Fechnerian scaling of the stimulus space based on the psychometric (discrimination probability) function of a human subject in a same-different comparison task. Here, we investigate a related but different paradigm, namely, referent-probe comparison task, in which the pair of stimuli (x and y) under comparison assumes substantively different psychological status, one serving as a referent and the other as a probe. The duality between a pair of psychometric functions, arising from assigning either x or y to be the fixed reference stimulus and the other to be the varying comparison stimulus, and the 1-to-1 mapping between the two stimulus spaces X and Y under either assignment are analyzed. Following Dzhafarov and Colonius, we investigate two properties characteristic of a referent-probe comparison task, namely, (i) Regular cross-minimality—for the pair of stimulus values involved in referent-probe comparison, each minimizes a discrimination probability function where the other is treated as the fixed reference stimulus; (ii) Nonconstant self-similarity—the value of the discrimination probability function at its minima is a nonconstant function of the reference stimulus value. For the particular form of psychometric functions investigated, it is shown that imposing the condition of regular cross-minimality on the pair of psychometric functions forces a consistent (but otherwise still arbitrary) mapping between X and Y, such that it is independent of the assignment of reference/comparison status to x and to y. The resulting psychometric differentials under both assignments are equal, and take an asymmetric, dualistic form reminiscent of the so-called divergence measure that appeared in the context of differential geometry of the probability manifold with dually flat connections (Differential Geometric Methods in Statistics, Lecture Notes in Statistics, Vol. 28, Springer, New York, 1985). The pair of divergence functions on X and on Y, respectively, induce a Riemannian metric in the small, with psychometric order (defined in Dzhafarov & Colonius, 1999) equal to 2. The difference between the Finsler-Riemann geometric approach to the stimulus space (Dzhafarov & Colonius, 1999) and this dually-affine Riemannian geometric approach to the dual scaling of the comparison and the reference stimuli is discussed.     

Psychophysics without physics: extension of Fechnerian scaling from continuous to discrete and discrete-continuous stimulus spaces

The computation of subjective (Fechnerian) distances from discrimination probabilities involves cumulation of appropriately transformed psychometric increments along smooth arcs (in continuous stimulus spaces) or chains of stimuli (in discrete spaces). In a space where any two stimuli that are each other's points of subjective equality are given identical physical labels, psychometric increments are positive differences ψ(x,y)-ψ(x,x) and ψ(y,x)-ψ(x,x), where x≠y and ψ is the probability of judging two stimuli different. In continuous stimulus spaces the appropriate monotone transformation of these increments (called overall psychometric transformation) is determined uniquely in the vicinity of zero, and its extension to larger values of its argument is immaterial. In discrete stimulus spaces, however, Fechnerian distances critically depend on this extension. We show that if overall psychometric transformation is assumed (A) to be the same for a sufficiently rich class of discrete stimulus spaces, (B) to ensure the validity of the Second Main Theorem of Fechnerian Scaling in this class of spaces, and (C) to agree in the vicinity of zero with one of the possible transformations in continuous spaces, then this transformation can only be identity. This result is generalized to the broad class of “discrete-continuous” stimulus spaces, of which continuous and discrete spaces are proper subclasses.     

Dissimilarity cumulation theory in smoothly connected spaces

This is the third paper in the series introducing the Dissimilarity Cumulation theory and its main psychological application, Universal Fechnerian Scaling. The previously developed dissimilarity-based theory of path length is used to construct the notion of a smooth path, defined by the property that the ratio of the dissimilarity between its points to the length of the subtended fragment of the path tends to unity as the points get closer to each other. We consider a class of stimulus spaces in which for every path there is a series of piecewise smooth paths converging to it pointwise and in length; and a subclass of such spaces where any two sufficiently close points can be connected by a smooth “geodesic in the small”. These notions are used to construct a broadly understood Finslerian geometry of stimulus spaces representable by regions of Euclidean n-spaces. With an additional assumption of comeasurability in the small between the canonical psychometric increments of the first and second kind, this establishes a link between Universal Fechnerian Scaling and Multidimensional Fechnerian Scaling in Euclidean n-spaces. The latter was a starting point for our theoretical program generalizing Fechner’s idea that sensation magnitudes can be computed by integration of a local discriminability measure.     

Is an equal interval scale an equal discriminability scale?

Stevens and Galanter’s (1957) iterative procedure for minimizing bias in category scaling was used for the scaling of loudness of white noise. The spacing obtained deviated systematically from a spacing constructed in accordance with an equal discriminability scale from a previous experiment (Eisler & Montgomery, 1972). For the stimulus spacing yielding a “pure” category scale, a magnitude scale was constructed too. Since the category scale could be predicted accurately by Fechnerian integration of this magnitude scale, it was concluded that the “pure” category scale is a pure discrimination scale. The discrepancy between the equal discriminability scale and the “pure” category scale was interpreted as a bias in the former scale due to greater recognizability of stimuli located at the extremes of the stimulus range.     

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.     

Sensation magnitude of vibrotactile stimuli

A systematic investigation of the subjective magnitude of vibrotaction was undertaken to: (1) determine the growth of sensation as a function of stimulus intensity; (2) establish contours of equal subjective magnitude; and (3) compare over a wide range of frequency and intensity the psychophysical methods of direct scaling and intensity matching. The results show that the data obtained by direct scaling are comparable to the data obtained by interfrequency matching. The subjective magnitude function is a power function with a slope of about 0.89 for frequencies up to 350 Hz. Near threshold the growth of sensation is proportional to the physical intensity. Contours of equal subjective magnitude for vibration across 10 frequencies and at 11 levels of intensity are given.     

Investigation of the psychometric properties of the Perceived Parental Autonomy Support Scale in the Southeastern European context

The Perceived Parental Autonomy Support Scale (P-PASS) is an instrument designed to measure parental autonomy-support and control of late adolescents and emerging adults. The present paper examines the process of adapting the P-PASS to the Romanian culture. Four studies were conducted, investigating: 1) the adequacy of the translation, using a multidimensional scaling of expert ratings; 2) construct validity, through exploratory approaches; 3) various psychometric properties, such as reliability and construct validity, through confirmatory approaches; convergent validity through comparisons with other measures of parental autonomy support (College-Student Scale of the Perceptions of Parents Scales) and control (Psychological Control Scale–Youth Self-Report), and predictive validity in relation with general self-efficacy; 4) test-retest reliability. The results show that the Romanian version of the P-PASS has sound psychometric properties. Confirmatory Factor Analysis indicates that a structure with two second order factors fits the data best and that the measure is equivalent with the original Canadian version. Also, it shows adequate test-retest reliability at 6 months and one year between administrations, good convergent validity, and a good prediction of general self-efficacy.

     

Adaptation et validation de la version française de l’échelle d’évaluation de la relation

IntroductionRelationship satisfaction is one of the most studied constructs in the field of relationship evaluation because of its impact on various aspects of daily life. It is therefore important to have an instrument in French.ObjectivesThis study aims to adapt the Relationship Assessment Scale (RAS) and validate its psychometric properties in French from its original version in English.MethodTwo studies were carried out. In the first study, 200 participants responded to the French version of the Relationship Assessment Scale (EER), the Dyadic Adjustment Scale (DAS), and the Hospital Anxiety Depression Scale (HADS) to assess the factor structure and psychometric properties of the French version (reliability, convergent validity, incremental validity). In the second study confirmatory factor analysis was used to validate the factor structure and to examine the gender invariance of the EER through a multi-factorial analysis in a population of 114 adults.ResultsThe results show that the psychometric properties of the EER are acceptable and comparable to the original version of the instrument. The EER presents a one dimensional factor structure. The positive correlations between the EER and the different scales tested support good external validity. The multi-group analysis showed that both women and men similarly understand the items and attribute the same meaning to the questions, confirming gender invariance of the EER.ConclusionThe French version of the EER is a valid and reliable assessment instrument of relationship satisfaction. The clinical and research implications of this scale are discussed.     

The Multigroup Ethnic Identity Measure – Revised (MEIM-R): Psychometric evaluation with adolescents from diverse ethnocultural groups in Italy

The Multigroup Ethnic Identity Measure – Revised (MEIM-R) is an extensively used questionnaire assessing ethnic identity. However, studies on its measurement characteristics in the European context are lacking. The current study addressed this gap by investigating the MEIM-R psychometric proprieties across multiple ethnocultural groups in Italy. Participants were 1445 adolescents (13–18 years) of Italian, East European, and North African origin. Results showed that the MEIM-R has good internal consistency. Multigroup confirmatory factor analyses revealed configural and metric invariance, i.e., an equal, correlated two-factor structure (ethnic identity exploration and commitment) and equal factor loadings across groups. Scalar invariance, i.e., equal item intercepts, was found only for the commitment scores that showed no group differences in latent factor mean levels. Partial structural invariance was evidenced, with the factor covariances varying across groups. These findings suggest that the MEIM-R is a valuable tool to assess the correlates of ethnic identity, although further research is needed.     

A monte carlo investigation of the statistical significance of Kruskal's nonmetric scaling procedure

Recent advances in computer based psychometric techniques have yielded a collection of powerful tools for analyzing nonmetric data. These tools, although particularly well suited to the behavioral sciences, have several potential pitfalls. Among other things, there is no statistical test for evaluating the significance of the results. This paper provides estimates of the statistical significance of results yielded by Kruskal's nonmetric multidimensional scaling. The estimates, obtained from attempts to scale many randomly generated sets of data, reveal the relative frequency with which apparent structure is erroneously found in unstructured data. For a small number of points (i.e., six or seven) it is very likely that a good fit will be obtained in two or more dimensions when in fact the data are generated by a random process. The estimates presented here can be used as a bench mark against which to evaluate the significance of the results obtained from empirically based nonmetric multidimensional scaling.A preliminary version of this paper was presented at the International Federation for Information Processing Congress 68 in Edinburgh, Scotland, August 5–10, 1968.     

The Fechnerian idea

From the principle that subjective dissimilarity between 2 stimuli is determined by their ratio, Fechner derives his logarithmic law in 2 ways. In one derivation, ignored and forgotten in modern accounts of Fechner's theory, he formulates the principle in question as a functional equation and reduces it to one with a known solution. In the other derivation, well known and often criticized, he solves the same functional equation by differentiation. Both derivations are mathematically valid (the much-derided "expedient principle" mentioned by Fechner can be viewed as merely an inept way of pointing at a certain property of the differentiation he uses). Neither derivation uses the notion of just-noticeable differences. But if Weber's law is accepted in addition to the principle in question, then the dissimilarity between 2 stimuli is approximately proportional to the number of just-noticeable differences that fit between these stimuli: The smaller Weber's fraction the better the approximation, and Weber's fraction can always be made arbitrarily small by an appropriate convention. We argue, however, that neither the 2 derivations of Fechner's law nor the relation of this law to thresholds constitutes the essence of Fechner's approach. We see this essence in the idea of additive cumulation of sensitivity values. Fechner's work contains a surprisingly modern definition of sensitivity at a given stimulus: the rate of growth of the probability-of-greater function, with this stimulus serving as a standard. The idea of additive cumulation of sensitivity values lends itself to sweeping generalizations of Fechnerian scaling.     

A Measure of the Stability of Activities in a Family Environment

The Stability of Activities in the Family Environment (SAFE) measure was developed to assess aspects of the construct of family stability. An earlier version of the SAFE was revised to improve its psychometric properties and better elaborate the construct that underlies the measure. College students completed the revised SAFE and other measures of family functioning with respect to their families of origin and measures of current adjustment. Findings support the reliability, internal consistency, and factor structure of the SAFE. It is suggested that the results of the factor analyses and findings that different aspects of family stability demonstrated different patterns of relationships to indices of family functioning, are consistent with the presumption guiding the development of the SAFE that families achieve stability in diverse ways. The validity of the SAFE was further supported by its relationship to measures of current self-esteem and depression.