A technique for the nonmetric analysis of paired comparisons of psychological intervals

A nonmetric coordinate adjustment technique is developed which determines scale values for objects whose interobject intervals (differences in subjective value) have been directly compared. In Monte Carlo simulations, the degree of metric determinancy of the scale values is shown to be quite high even when the amount of error is relatively high. This robustness under high-error conditions permitted the analysis of individual subject data in experiments on the direct comparison of loudness differences and loudness ratios where only one judgment per interval comparison was obtained per subject.This research was supported by a grant from the National Research Council of Canada.     

A method for finding the inverse of a matrix

The problem of solving simultaneous linear equations is one that frequently confronts the scientist. Whenever it is desired to state the parameters as expressions of the constant terms of these equations, the problem resolves itself into the discovery of the inverse of the matrix of coefficients. The method here described gives a routine for the solution of this problem.     

A note on the nonmetric analysis of psychological intervals for an unknown ordering of stimuli

A modified version of a coordinate adjustment technique which permits the analysis of comparisons of psychological intervals for an unknown ordering of stimuli is described and compared to the original version and to TORSCA. For configurations with a large number of points, knowledge of the rank order of the stimuli does not improve the solution. For configurations with a small number of points, the performance of the new algorithm with an unknown ordering is equivalent to TORSCA.This research was supported by a grant from the Natural Sciences and Engineering Research Council of Canada.     

Monotone mapping of similarities into a general metric space

A method of “maximum variance nondimensional scaling” is described and tested that transforms similarity measures into distances that meet just three conditions: (C1) they exactly satisfy the metric axioms, (C2) they are, as nearly as possible, monotonically related to the similarity measures, (C3) they have maximum variance possible under the two preceding conditions. By achieving an appropriate balance between the last two conditions, one can determine the true underlying distances and the form of the unknown monotone function relating the similarity measures to those distances without assuming that the underlying space has any particular Euclidean, Minkowskian, or even dimensional strucutre. The method appears to have potential applications, e.g., to studies of stimulus generalization and the structure and processing of semantic information.     

Imaginary hide and seek - A technique for opening up a psychic space in child psychotherapy

Working with concrete states of mind confronts the analyst with major countertransference problems: faced with the emptiness of the patient's psychic functioning, the analyst is threatened with the deadening of his own psychic capacities and the crumbling of the analytic process. This paper examines two cases of children who presented in very different ways, one featuring deeprooted depression and empty behaviour, the other featuring mastery and destructive omnipotence. It proposes a specific technique to enable the reconstruction of representational functioning through play. In the context of the transference/countertransference relationship and the analytic process, the introduction of a game of imaginary hide and seek, whose rules are defined and analysed, may represent an instrument of choice for opening up an undeveloped psychic space in these pathologies.     

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.     

Postural configuration affects the perception of earth-based space during pitch tilt

This study investigates the relative contribution of body parts in the elaboration of a whole-body egocentric attraction phenomenon previously observed during earth-based judgments. This was addressed through a particular earth-based task requiring estimating the possibility of passing under a projected line, imagining a forward horizontal displacement. Different postural configurations were tested, involving whole-body tilt, trunk tilt alone or head tilt alone. Two legs positions relative to the trunk were manipulated. Results showed systematic deviations of the subjective “passability” toward the tilt, linearly related to the tilt magnitude. For each postural configuration, the egocentric influence appeared to be highly dependent on the position of trunk and head axes, whereas the legs position appeared not relevant. When compared to the whole-body tilt condition, tilting the trunk alone consistently reduced the amount of the deviation toward the tilt, whereas tilting the head alone consistently increased it. Our results suggest that several specific effects from multiple body parts can account for the global deviation of the estimates observed during whole-body tilt. Most importantly, we support that the relative contribution of the body segments could mainly depend on a reweighting process, probably based on the reliability of sensory information available for a particular postural set.     

A procedure for the incremental construction of a knowledge space

Knowledge spaces are structures for the efficient assessment of the knowledge state of a student in a given field of knowledge. Existing procedures for constructing a knowledge space by querying an expert assume that the domain of questions is known in advance, and that it is fixed during the whole query process. The outcome of these procedures is a knowledge space on the questions in that domain. If the original domain is extended with new questions, a new knowledge space on the extended domain can be produced by expert query. Since in this case a knowledge space for the original domain already exists, the available information can be used to extend the existing space in an efficient way, thus avoiding to apply expert query from scratch. Existing procedures do not provide an explicit way to use such information. Although these procedures can be adapted to this purpose, in this paper a new query algorithm that is specifically tailored for the problem above mentioned is presented.     

Further considerations of a general d′ in multidimensional space

Thomas (Psychonom. Bull. Rev. 6 (1999) 224) proposed a generalization of d′,d_g′ for multidimensional distributions and demonstrated that it is not equivalent to Euclidean distance as had been assumed in some previous studies. In this note, it is further shown not to be a metric in the general sense as it fails the triangle inequality. Also, a rigorous proof is offered of the claim (found in Thomas, Psychonom. Bull. Rev. 6 (1999) 224) that in order for the definition of d_g′ to correspond to the quantity that is estimated from data in the traditional way (using hits and false alarms) one must assume a distance classifier (proved in the case of equal covariance matrices). If one assumes optimal responding instead, then the estimated d′ corresponds to (the square root of) Mahalanobis distance. This latter observation clears up an apparent paradox between the fact that d_g′ is not a metric and Ashby and Perrin's (Psychol. Rev. 95 (1988) 124) statement relating the weighted Euclidean scaling model and a signal detection model of similarity that would yield a distance classifier.     

Tests for linear trend in the smallest eigenvalues of the correlation matrix

A test for linear trend among a set of eigenvalues of a correlation matrix is developed. As a technical implementation of Cattell's scree test, this is a generalization of Anderson's test for the equality of eigenvalues, and extends Bentler and Yuan's work on linear trends in eigenvalues of a covariance matrix. The power of minimumx ² and maximum likelihood ratio tests are compared. Examples show that the linear trend hypothesis is more realistic than the standard hypothesis of equality of eigenvalues, and that the hypothesis is compatible with standard decisions on the number of factors or components to retain in data analysis.This work was supported by National Institute on Drug Abuse Grants DA01070 and DA00017. The assistance of Maia Berkane and several anonymous reviewers is gratefully acknowledged.     

Assessing sensitivity in a multidimensional space: some problems and a definition of a general d'

This article provides a formal definition for a sensivity measure,d′ _g , between two multivariate stimuli. In recent attempts to assess perceptual representations using qualitative tests on response probabilities, the concept of ad′ between two multidimensional stimuli has played a central role. For example, Kadlec and Townsend (1992a, 1992b) proposed several tests based on multidimensional signal detection theory that allow conclusions concerning the perceptual and/or decisional interactions of stimulus dimensions. One proposition, referred to as thediagonal d′test, relies on specific stimulus subsets of a feature-complete factorial identification task to infer perceptual separability. Also, Ashby and Townsend (1986), in a similar manner, attempted to relate perceptual independence to dimensional orthogonality in Tanner’s (1956) model, which also involvesd′ between two multivariate signals. An analysis of the proposedd′ _g reveals shortcomings in the diagonald′ test and also demonstrates that the assumptions behind equating perceptual independence to dimensional orthogonality are too weak. Thisd′ _g can be related to a common measure of statistical distance, Mahalanobis distance, in the special case of equal covariance matrices.     

On finding the bidimension of a relation

A method is presented for evaluating the bidimension of a finite binary relation, i.e., the number of biorders (Guttman relations) needed to yield the relation as their intersection. In case the relation is induced by a binary data matrix, the bidimension equals the minimal number of dimensions needed for a representation of the data matrix according to the conjunctive model of C. H. Coombs and R. C. Kao (Nonmetric factor analysis, Engineering Research Bulletin No. 38, Univ. of Michigan Press, Ann Arbor, 1955). Central to the evaluation of the bidimension is its characterization, provided by J.-P. Doignon, A. Ducamp, and J.-C. Falmagne (Journal of Mathematical Psychology, 28, 73–109, 1984), as the chromatic number of some associated hypergraph. A procedure is described to reduce hypergraphs of this kind to subhypergraphs with the same chromatic number. This reduction can be used throughout in applying a recurrence relation that expresses the chromatic number of a hypergraph in terms of the chromatic numbers of some of its subhypergraphs.     

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.     

A place within the heart: finding a home with parental objects

For adopted and Looked After children with compromised early experiences, there can be troubling phantasies and anxieties associated with parental objects. These internal object relations can seriously restrict the development of secure intimate relationships with new parental figures. Adoptive parents and foster carers bear the brunt of the associated difficulties. Clinical work with the child, parents/carers, family, or network, may help processes of containment, differentiation, working through, mourning, and integration. This paper is based on thinking derived from clinical work with children who are adopted or Looked After, and with adoptive parental couples, carers, and networks. The primary focus is on how the child needs to find a ‘psychic home’ in the minds of others for their damaged internal parental objects, and the emotional capacities required in adoptive parents to provide this. I describe some of the complexities for adoptive parents and others in providing what is needed to enable the child to experience their internal parental objects in a fuller way. This process can allow the child to introject, identify with, and re-imagine their internal parental objects, and through this develop a more realistic relationship to their adoptive parents and to their birth parents. It promotes trust, freedom, and emotional depth in the relationship with parental figures.     

A general model for the analysis of multilevel data

A general model is developed for the analysis of multivariate multilevel data structures. Special cases of the model include repeated measures designs, multiple matrix samples, multilevel latent variable models, multiple time series, and variance and covariance component models.We would like to acknowledge the helpful comments of Ruth Silver. We also wish to thank the referees for helping to clarify the paper. This work was partly carried out with research funds provided by the Economic and Social Research Council (U.K.).