An algorithm for clustering relational data with applications to social network analysis and comparison with multidimensional scaling

A method of hierarchical clustering for relational data is presented, which begins by forming a new square matrix of product-moment correlations between the columns (or rows) of the original data (represented as an n × m matrix). Iterative application of this simple procedure will in general converge to a matrix that may be permuted into the blocked form [_?1¹₁^?1]. This convergence property may be used as the basis of an algorithm (CONCOR) for hierarchical clustering. The CONCOR procedure is applied to several illustrative sets of social network data and is found to give results that are highly compatible with analyses and interpretations of the same data using the blockmodel approach of White (White, Boorman & Breiger, 1976). The results using CONCOR are then compared with results obtained using alternative methods of clustering and scaling (MDSCAL, INDSCAL, HICLUS, ADCLUS) on the same data sets.     

Generalized procrustes analysis

SupposeP _i ⁽ⁱ⁾ (i = 1, 2, ...,m, j = 1, 2, ...,n) give the locations ofmn points inp-dimensional space. Collectively these may be regarded asm configurations, or scalings, each ofn points inp-dimensions. The problem is investigated of translating, rotating, reflecting and scaling them configurations to minimize the goodness-of-fit criterion Σ _i=1 ^m Σ _i=1 ⁿ Δ²(P _j ⁽ⁱ⁾ G _i ), whereG _i is the centroid of them pointsP _i ⁽ⁱ⁾ (i = 1, 2, ...,m). The rotated positions of each configuration may be regarded as individual analyses with the centroid configuration representing a consensus, and this relationship with individual scaling analysis is discussed. A computational technique is given, the results of which can be summarized in analysis of variance form. The special casem = 2 corresponds to Classical Procrustes analysis but the choice of criterion that fits each configuration to the common centroid configuration avoids difficulties that arise when one set is fitted to the other, regarded as fixed.     

An alternative scaling method for priorities in hierarchical structures

The early contributions of Saaty have spawned a multitude of applications of principal right (PR) eigenvector “scaling” of a dominance matrix [R]. Prior to Saaty's work (1977–1984) scaling of dominance matrices received little attention in multidimensional scaling, e.g., see Shepard (1972, pp. 26–27). This eigenvector method (EM) of scaling [R] yields u_i scores (weights) popularly used at each branching of the Analytic Hierarchy Process (AHP) technique that has been increasingly applied in multiple criterion analysis of utility, preference, probability, and performance. In this paper, it is proposed that an alternate least squares method (LSM) scaling technique yielding least squares optimal scores (weights) provides $\text{w}{}_{\mathrm{i}}{}^1$ values having a number of important advantages over u_i scores popularly utilized to date.     

The effects of delayed reinforcement on free-operant responding

In previous studies of delayed reinforcement, response rate has been found to vary inversely with the response-reinforcer interval. However, in all of these studies the independent variable, response-reinforcer time, was confounded with the number of reinforcers presented in a fixed period of time (reinforcer frequency). In the present study, the frequency of available reinforcers was held constant, while temporal separation between response and reinforcer was independently manipulated. A repeating time cycle, T, was divided into two alternating time periods, t^D and t^Δ. The first response in t^D was reinforced at the end of the prevailing T cycle and extinction prevailed in t^Δ. Two placements for t^D were defined, an early t^D placement in which t^D precedes t^Δ and a late t^D placement in which t^D follows t^Δ. The duration of the early and late t^D was systematically decreased from 30 seconds (i.e., t^D = T) to 0.1 second. Manipulation of t^D placement and duration controlled the temporal separation between response and reinforcement, but it did not affect the frequency of programmed reinforcers, which was 1/T. The results show that early and late t^D placements of equal duration have similar overall effects upon response rate, reinforcer frequency, responses per reinforcer, and obtained response-reinforcer temporal separation. A stepwise regression analysis using log response rate as the dependent variable showed that the obtained delay was a significant first-step variable for six of eight subjects, with obtained reinforcer frequency significant for the remaining two subjects.     

A probabilistic multidimensional scaling with unique axes1

Abstract: A probabilistic multidimensional scaling model is proposed. The model assumes that the coordinates of each stimulus are normally distributed with variance Σ_i = diag(σ²₁, … σ²_Ri). The advantage of this model is that axes are determined uniquely. The distribution of the distance between two stimuli is obtained by polar coordinates transformation. The method of maximum likelihood estimation for means and variances using the EM algorithm is discussed. Further, simulated annealing is suggested as a means of obtaining initial values in order to avoid local maxima. A simulation study shows that the estimates are accurate, and a numerical example concerning the location of Japanese cities shows that natural axes can be obtained without introducing individual parameters.     

Separate and joint scaling of perceived odor intensity ofn-butanol and hydrogen sulfide

To determine which combination of scaling method and experimental procedure is optimal for measuring perceived odor intensity, the intensity perception of one odorous substance was scaled both separately and together with a second odorous substance in the same session. The methods used were magnitude estimation with and without a standard reference and cross-modal matching with finger span. The results show that separate and joint scalings ofn-butanol and H₂S give identical scales, regardless of scaling method. Different results are obtained in magnitude estimations with a homoquality, a heteroquality, and no standard. From these results, one would not expect to find a single true power function independent of method. Cross-modal matching with finger span may be the best choice for odor intensity scaling, since it results in the widest response range, thereby giving the best resolution.     

The statistical analysis of a thurstonian model for rating chess players

This paper formalizes and provides static and dynamic estimators for a scaling model for rating chess players. The model was suggested by the work of Arpad Elo, the inventor of the chess rating system in current use by both the United States and international chess federations. The model can be viewed as a Thurstone Case V model that permits draws (ties). A related model based on a linear approximation is also analyzed. In the chess application, possibly changing ability parameters are estimated sequentially from sparse data structures that often involve many fewer than $\text{M(M ? 1)}\text{2}$ observations on the M players to the rated. In contrast, psychological applications of paired-comparison scaling generally use models with no draw provision to estimate static parameters from a systematically obtained data structure such as a replicated “round robin” involving all M entities to be scaled. In the paper, both static and sequential estimators are provided and evaluated for a number of different data structures. Sampling theory for the estimators is developed. The application of rating systems to track temporally changing ability parameters may prove useful in many areas of psychology.     

Difference measurement spaces

An absolute-difference measurement space is a pair (X, e) where the real-valued function e on X² satisfies conditions which are shown in the paper to be necessary and sufficient for its representability by the absolute distance on the real line. A positive-difference measurement space is a pair (X, l), where the real-valued function l on X² satisfies conditions necessary and sufficient for its representability by positive distances on the real line. The conditions imposed on e and l make these functions extensive measurements of proximity and dominance, the two basic predicates of social enquiry. Another way of treating these conditions is to translate them to the formal language of multivalued logic. The translation is easy and the sentences obtained have plausible intuitive meanings such as reflexivity, symmetry, and transitivity. The two sets of conditions thus become formal theories of proximity and dominance. Our difference measurement spaces are relational structures for the multi-valued logic and models of the two formal theories. Thus proximity and dominance are considered dichotomous in principle and the multiple truth-values represent degrees of error. We suggest adopting multivalued logic as a framework within which the problem of measurement error can be treated together with the formal axiomatization of social and phychological theories.     

On the theory of biased bisection operations and their inverses

A theory of fundamental measurement involving multiple operations is presented. The operations are bisection, and two inverses of the bisection operation called right and left displacement, analogous to “displacement” of an interval “up” or “down” a scale. Following Pfanzagl, “biased” operations are permitted, and linear, reflexive representations are presented for the biased operations. Results are obtained on the constructability of unbiased scaling operations, and several methods of constructing standard sequences of equally spaced stimuli using biased operations, are described. A striking property of biased linear representation models is derived, showing that if bias is present, indefinite iterations of one or the other of the two displacement operations will converge to asymptote. Preliminary data support this surprising convergence property.     

Defining reward value by cross-modal scaling

Researchers in comparative psychology often use different food rewards in their studies, with food values defined by a pre-experimental preference test. While this technique rank orders food values, it provides limited information about value differences because preferences may reflect not only value differences, but also the degree to which one good may “substitute” for another (e.g., one food may substitute well for another food, but neither substitutes well for water). We propose scaling the value of food pairs by a third food that is less substitutable for either food offered in preference tests (cross-modal scaling). Here, Cebus monkeys chose between four pairwise alternatives: fruits A versus B; cereal amount X versus fruit A and cereal amount Y versus fruit B where X and Y were adjusted to produce indifference between each cereal amount and each fruit; and cereal amounts X versus Y. When choice was between perfect substitutes (different cereal amounts), preferences were nearly absolute; so too when choice was between close substitutes (fruits); however, when choice was between fruits and cereal amounts, preferences were more modest and less likely due to substitutability. These results suggest that scaling between-good value differences in terms of a third, less-substitutable good may be better than simple preference tests in defining between-good value differences.     

Contextual-Hierarchical Reconstructions of the Strengthened Liar Problem

In this paper we shall introduce two types of contextual-hierarchical (from now on abbreviated by ‘ch’) approaches to the strengthened liar problem. These approaches, which we call the ‘standard’ and the ‘alternative’ ch-reconstructions of the strengthened liar problem, differ in their philosophical view regarding the nature of truth and the relation between the truth predicates T r ⁿ and T r ⁿ⁺¹ of different hierarchy-levels. The basic idea of the standard ch-reconstruction is that the T r ⁿ⁺¹-schema should hold for all sentences of \(\mathcal {L}^{n}\). In contrast, the alternative ch-reconstruction, for which we shall argue in section four, is motivated by the idea that T r ⁿ and T r ⁿ⁺¹ are coherent in the sense that the same sentences of \(\mathcal {L}^{n}\) should be true according to T r ⁿ and T r ⁿ⁺¹. We show that instances of the standard ch-reconstruction can be obtained by iterating Kripke’s strong Kleene jump operator. Furthermore, we will demonstrate how instances of the alternative ch-reconstruction can be obtained by a slight modification of the iterated axiom system KF and of the iterated strong Kleene jump operator.     

Agency and Virtue: Dimensions Underlying Subgroups of Women

Previous research indicates that the stereotype of women can be considered to have 3 subgroups: housewife, career woman, and sex object. In 2 samples (N = 19 and 35), we found evidence that these subgroups can be reliably distinguished in terms of 2 dimensions: agency and virtue. Participants sorted 27 feminine traits and then rated these traits in terms of their agency and virtue. Cluster analysis and multidimensional scaling with property fitting were used to identify subgroups, to fit virtue and agency dimensions to the subgroups, and to test for differences among the subgroups in terms of virtue and agency. Across both samples, agency and virtue fit the subgroups well (average R ² = .75), produced many significant differences among the subgroups, and are consistent with a system-justification perspective of sexism (Glick & Fiske, 2001) in which a belief in women's virtue and lack of agency reflects and maintains status differences between men and women.     

Multidimensional scaling of rorschach inkblots: Relationships with structured self-report

This study attempted a replication of previous multidimensional scaling (MDS) results concerning the most salient perceptual features of the Rorschach inkblots. A second purpose was the investigation of the relationships between individual S's relative reliance on these salient features, and personality characteristics as assessed by the Jackson Personality Inventory (JPI). The two-dimension solution obtained in previous research, involving inkblot color and twomess of form, was closely replicated. Only very low correlations between Ss' use of these dimensions, and personality characteristics as measured by the JPI, were obtained. The MDS-Rorschach approach may prove useful in delimiting peculiarities of attention and perception in clinical states. The usefulness of the approach in furthering study of personality constructs assessed through traditional Rorschach administration is currently in question.     

ConPar: a method for identifying groups of concordant subject proximity matrices for subsequent multidimensional scaling analyses

A common representation of data within the context of multidimensional scaling (MDS) is a collection of symmetric proximity (similarity or dissimilarity) matrices for each of M subjects. There are a number of possible alternatives for analyzing these data, which include: (a) conducting an MDS analysis on a single matrix obtained by pooling (averaging) the M subject matrices, (b) fitting a separate MDS structure for each of the M matrices, or (c) employing an individual differences MDS model. We discuss each of these approaches, and subsequently propose a straightforward new method (CONcordance PARtitioning—ConPar), which can be used to identify groups of individual-subject matrices with concordant proximity structures. This method collapses the three-way data into a subject×subject dissimilarity matrix, which is subsequently clustered using a branch-and-bound algorithm that minimizes partition diameter. Extensive Monte Carlo testing revealed that, when compared to K-means clustering of the proximity data, ConPar generally provided better recovery of the true subject cluster memberships. A demonstration using empirical three-way data is also provided to illustrate the efficacy of the proposed method.     

Documenting changes in adult speakers' locus of causality during stuttering treatment using Origin and Pawn scaling

The concepts of locus of control and locus of causality are similar and refer to the degree to which a person perceives daily occurrences to be a consequence of his or her own behavior. Locus of control is considered to be a unidimensional construct indicating an inverse relationship between the polls of internality and externality. The locus of control is generally determined by using questionnaires with a limited number of items. Locus of causality is considered to be a two-dimensional construct where Origin and Pawn values, which are similar to internality and externality, respectively, are not necessarily inversely related. Locus of causality is determined by content analysis of freely spoken or written narratives. In the current study Origin and Pawn scores were obtained from twenty adults prior to and following a three-week intensive stuttering treatment program. Brief narratives written by the participants were analyzed to obtain Origin and Pawn values. These scores were compared with traditional measures of therapeutic outcome (Locus of Control, OASES, PSI, percentage of syllables stuttered). Results indicated statistically significant increases in pre- to post-treatment Origin scores (p = .001; Cohen's d = 1.44) and statistically significant decreases in pre- to post-treatment Pawn scores (p = .003; Cohen's d = 1.11). Origin and Pawn scores showed significant relationships with other measures of stuttering, indicating concurrent and construct validity. Origin and Pawn scaling procedures appear to provide a valid, sensitive, and nonreactive indicator of the speaker's locus of causality and ability to develop an autonomous and agentic lifestyle.Educational objectives: After reading this article, the readers will be able to: (1) distinguish between the concepts of locus of control and locus of causality, (2) describe the characteristics of individuals behaving as an Origin and a Pawn, (3) differentiate patterns of change for Origin and Pawn scores prior to and following treatment, and (4) describe the clinical advantages using Origin and Pawn scaling procedures for individuals who stutter.     

Anisotropic perception of visual angle: Implications for the horizontal-vertical illusion,overconstancy of size,and the moon illusion

Three experiments investigated anisotropic perception of visual angle outdoors. In Experiment 1, scales for vertical and horizontal visual angles ranging from 20° to 80° were constructed with the method of angle production (in which the subject reproduced a visual angle with a protractor) and the method of distance production (in which the subject produced a visual angle by adjusting viewing distance). In Experiment 2, scales for vertical and horizontal visual angles of 5°–30° were constructed with the method of angle production and were compared with scales for orientation in the frontal plane. In Experiment 3, vertical and horizontal visual angles of 3°-80° were judged with the method of verbal estimation. The main results of the experiments were as follows: (1) The obtained angles for visual angle are described by a quadratic equation, θ′=a+bθ+cθ² (where θ is the visual angle; θ′, the obtained angle;a, b, andc, constants). (2) The linear coefficientb is larger than unity and is steeper for vertical direction than for horizontal direction. (3) The quadratic coefficientc is generally smaller than zero and is negatively larger for vertical direction than for horizontal direction. And (4) the obtained angle for visual angle is larger than that for orientation. From these results, it was possible to predict the horizontal-vertical illusion, over-constancy of size, and the moon illusion.     

On the robustness of maximum likelihood scaling for violations of the error model

A Monte Carlo study was conducted to investigate the robustness of the assumed error distribution in maximum likelihood estimation models for multidimensional scaling. Data sets generated according to the lognormal, the normal, and the rectangular distribution were analysed with the log-normal error model in Ramsay's MULTISCALE program package. The results show that violations of the assumed error distribution have virtually no effect on the estimated distance parameters. In a comparison among several dimensionality tests, the corrected version of thex ² test, as proposed by Ramsay, yielded the best results, and turned out to be quite robust against violations of the error model.     

Biased extensive measurement: The homogeneous case

In the homogeneous case of one type of objects, we prove the existence of an additive scale unique up to a positive scaling transformation without transitivity of indifference and with a property of homothetic invariance weaker than monotonicity. The representation, which is a particular case of a semiorder representation, reveals a unique positive factor α?1 that biases extensive structures and explains departures from these standard axioms of extensive measurement (α=1). We interpret α as characterizing the qualitative influence of the underlying measurement process and we show that it induces a proportional indifference threshold.     

A statistical approach to Saaty's scaling method for priorities

This paper investigates the logarithmic least squares (LLSM) approach to Saaty's (Journal of Mathematical Psychology, 1977, 5, 234–281) scaling method for priorities in hierarchical structures. It is argued that statistical criteria are important in deciding the scaling method controversy. It is shown that LLSM is statistically optimal under a number of realistic and practical models. Variances and covariances of parameter estimates are derived. The covariance matrix associated with overall priority differences is also developed. These results allow for a significance analysis of apparent priority differences.     

Cluster differences scaling with a within-clusters loss component and a fuzzy successive approximation strategy to avoid local minima

Cluster differences scaling is a method for partitioning a set of objects into classes and simultaneously finding a low-dimensional spatial representation ofK cluster points, to model a given square table of dissimilarities amongn stimuli or objects. The least squares loss function of cluster differences scaling, originally defined only on the residuals of pairs of objects that are allocated to different clusters, is extended with a loss component for pairs that are allocated to the same cluster. It is shown that this extension makes the method equivalent to multidimensional scaling with cluster constraints on the coordinates. A decomposition of the sum of squared dissimilarities into contributions from several sources of variation is described, including the appropriate degrees of freedom for each source. After developing a convergent algorithm for fitting the cluster differences model, it is argued that the individual objects and the cluster locations can be jointly displayed in a configuration obtained as a by-product of the optimization. Finally, the paper introduces a fuzzy version of the loss function, which can be used in a successive approximation strategy for avoiding local minima. A simulation study demonstrates that this strategy significantly outperforms two other well-known initialization strategies, and that it has a success rate of 92 out of 100 in attaining the global minimum.