The relation between hierarchical and euclidean models for psychological distances

In one well-known model for psychological distances, objects such as stimuli are placed in a hierarchy of clusters like a phylogenetic tree; in another common model, objects are represented as points in a multidimensional Euclidean space. These models are shown theoretically to be mutually exclusive and exhaustive in the following sense. The distances among a set ofn objects will be strictly monotonically related either to the distances in a hierarchical clustering system, or else to the distances in a Euclidean space of less thann — 1 dimensions, but not to both. Consequently, a lower bound on the number of Euclidean dimensions necessary to represent a set of objects is one less than the size of the largest subset of objects whose distances satisfy the ultrametric inequality, which characterizes the hierarchical model.This work was supported in part by Grant GB-13588X from the National Science Foundation. I would like to thank L. M. Kelly and A. A. J. Marley for their helpful comments and suggestions.     

Multiple rectilinear prediction and the resolution into components

It is assumed that a battery ofn tests has been resolved into components in a common factor space ofr dimensions and a unique factor space of at mostn dimensions, wherer is much less thann. Simplified formulas for ordinary multiple and partial correlation of tests are derived directly in terms of the components. The best (in the sense of least squares) linear regression equations for predicting factor scores from test scores are derived also in terms of the components. Spearman's single factor prediction formulas emerge as special cases. The last part of the paper shows how the communality is an upper bound for multiple correlation. A necessary and sufficient condition is established for the square of the multiple correlation coefficient of testj on the remainingn—1 tests to approach the communality of testj as a limit asn increases indefinitely whiler remains constant. Limits are established for partial correlation and regression coefficients and for the prediction of factor scores.I am indebted to Professor Dunham Jackson for helpful criticism of most of this paper.     

A geometrical analysis of the unfolding model: General solutions

Given the complete setR of rank orders obtained from any configuration ofn stimulus points inr dimensions in accordance with the unfolding model, a configuration from which just these orders may be derived will be described as a solution forR. The space is assumed to be Euclidean. Necessary and sufficient conditions are derived for a configuration to be a solution forR. The geometrical constraints which are necessary and sufficient to determine the subset of pairs of orders and opposites contained inR are also identified and constitute the constraint system for the ordinal vector model. The relationship between the two models is discussed.     

A geometrical analysis of the unfolding model: Nondegenerate solutions

Given the complete setR of rank orders derived from some configuration ofn stimulus points inr dimensions in accordance with the unfolding model, a stimulus configuration which generates just these orders will be described as a solution forR. The space is assumed to be Euclidean. Necessary and sufficient conditions are determined for a nondegenerate configuration to be a solution forR. The geometrical conditions which are necessary and sufficient to determine the subset of pairs of opposite orders are also identified and constitute the constraint system for the ordinal vector model.The present work was completed while the writer was in receipt of an Australian Commonwealth Postgraduate Scholarship under the supervision of Professor J. A. Keats.     

Estimation ofJ scales for unidimensional unfolding

In Coombs' unidimensional unfolding theory each individual ranks all stimuli, often in terms of preference judgments. These ranks, calledI scales, are used to infer the latent continuum called aJ scale which is presumed to have generated theI scale rankings. A major problem concerns the inference of theJ scale given a set ofI scales because anI scale is not unique to any oneJ scale. This paper presents a procedure for estimatingJ scale probabilities given a set ofI scales. Models for three and four or more stimuli are presented. Results of computer tests are reported.     

Relations between multidimensional scaling and three-mode factor analysis

A combination is achieved of two lines of psychometric interest: a) multidimensional scaling and b) factor analysis. This is accomplished with the use of three-mode factor analysis of scalar product matrices, one for each subject. Two of the modes are the groups of objects scaled and the third mode is the sample of subjects. Results are an object space, a person space, and a system for changing weights given to dimensions and of angles between dimensions in the object space for individuals located at different places in the person space. The development is illustrated with data from an adjective similarity study.Supported in part by the Personnel and Training Branch of the Office of Naval Research under Contract Number 00014-67-A-0305-0003. A number of very helpful comments were made by an anonymous editorial reviewer forPsychometrika.During 1970–71 at the L. L. Thurstone Psychometric Laboratory, University of North Carolina.     

A monte carlo evaluation of interactive multidimensional scaling

Interactive Scaling with Individual Subjects (ISIS) developed by Young & Cliff [1972], is a method involving interaction between subject and computer in real time to determine which judgments made by the subject are critical to the definition of a dimensional structure. The procedure is based on the mathematical fact that it is possible to define a space ofR dimensions in terms of only the interpoint distances between all stimuli being scaled and a subset of (R+1) of these stimuli. For errorless judgments, any subset of (R + 1) stimuli is appropriate. However, fallible data require that the subset consist of stimuli that are maximally dissimilar, and the ISIS procedure is designed to obtain such an optimum subset (a basis).This research evaluates a modified version of ISIS with respect to (a) a metric MDS analysis based on all possible pairs of the stimuli, and (b) a metric MDS analysis based on a subset of one-third of the possible pairs, or about the same number as that required by ISIS. Results show that the ISIS method achieves better fit than (b) at low error levels, and may also achieve better fit than (b) at higher error levels if the size of the basis is increased. The more stimuli in the basis the more indices of fit approach those of (a).A new method of introducing error in MDS studies is used in the evaluation.This research is based in part on the doctoral dissertation of the first author, and was supported by research grant MH-16474 from the National Institute of Mental Health, Public Health Service.     

Constrained multidimensional scaling inN spaces

A gradient method is used to obtain least squares estimates of parameters of them-dimensional euclidean model simultaneously inN spaces, given the observation of all pairwise distances ofn stimuli for each space. The procedure can estimate an additive constant as well as stimulus projections and the metric of the reference axes of the configuration in each space. Each parameter in the model can be fixed to equal some a priori value, constrained to be equal to any other parameter, or free to take on any value in the parameter space. Two applications of the procedure are described.     

Ratings of Emotional Response to Pictorial Stimuli: Positive and Negative Affect Dimensions

The purpose of this study was to investigate the structure of emotional responses to pictorial stimuli within the affective space defined by orthogonal dimensions of positive affect (PA) and negative affect (NA). College student participants (N = 151) viewed 60 color photographic slides from the International Affective Picture System (Lang, Bradley, & Cuthbert, 1995) and rated their emotional responses to each slide using the Positive and Negative Affect Schedule (Watson, Clark, & Tellegen, 1988). Consistent with prediction, affective reactions to pleasant slides were characterized as high in PA, and low in NA. Unpleasant slides produced elevations on the NA scale, but contrary to expectation, also produced elevated ratings of PA, with the magnitude of PA scale scores increasing systematically with NA scale scores. The results suggest an interesting divergence between the structure of pictorially elicited affective reactions and that of ad hoc mood states. Directions for future research are discussed in relation to the unique properties of the slide viewing paradigm.     

Preferences of infants for regular and distorted natural speech stimuli

The schema hypothesis proposed by Kagan (Science, 1970, 170, 826–832) andLewis (Developmental Psychology, 1969, 1, 75–86) was used to make predictions concerning the preferences of infants 3 to 14 months old for speech stimuli. An operant response method was used in determining the infants' preferences for inflected, monotone, and scrambled natural speech stimuli. Although the infants' preferences did not change with age as predicted, the infants produced clear preference orderings for the three stimuli. The speech preferences were interpreted as being based on stimulus variables (e.g., word order, inflection, and speech rate) in addition to the realism variables assumed by the schema hypothesis.     

Norms for two types of manipulability (graspability and functional usage), familiarity,and age of acquisition for 320 photographs of objects

There is increasing interest in the role that manipulability plays in processing objects. To date, Magnié, Besson, Poncet, and Dolisi’s (2003) manipulability ratings, based on the degree to which objects can be uniquely pantomimed, have been the reference point for many studies. However, these ratings do not fully capture some relevant dimensions of manipulability, including whether an object is graspable and the extent to which functional motor associations above and beyond graspability are present. To address this, we collected ratings of these dimensions, in addition to ratings of familiarity and age of acquisition (AoA), for a set of 320 black-and-white photographs of objects. Familiarity and AoA ratings were highly correlated with previously reported ratings of the same dimensions (r =.853, p<.001, and r=.771, p<.001, respectively), validating the present norms. Grasping and functional use ratings, in contrast, were more moderately correlated with Magnié et al.’s pantomime manipulability ratings (r=.507, p<.001). These results were taken as evidence that the new manipulability ratings collected in this research capture distinct aspects of object manipulability. The complete stimuli and norms from this study may be downloaded from http://brm.psychonomic-journals.org/content/supplemental.     

The wayte Method for Investigating self-perceptions

The Who-Are-You Technique of investigating self-perceptions was extended into past, future, and ideal dimensions by additionally asking subjects Who have you been, Who will you be, and Who would you like to be. Who-Are-You-Time-Extension (WAYTE) comparisons of 232 prisoners, 55 Army Neuropsychiatric specialists, 77 college men, and 67 college women were made for twelve scoring categories and for contentment across time. Patterns of differences were noted and the WAYTE method was viewed as having potential for further self-concept research.     

The effects of random error and subsampling of dimensions on recovery of configurations by non-metric multidimensional scaling

Similarity judgments of three-dimensional stimuli were simulated, with the hypothetical subject attending to only some dimensions of stimulus variation (i.e., subsampling) on each trial. Recovery of the stimulus configuration by non-metric multidimensional scaling was investigated as a function of subsampling, the amount of random error in the judgments, and the number of stimuli being scaled.It was found that: (1) dimensions to which the subject often attends were well recovered even when dimensions seldom attended to were not, and (2) measures of recovery based on interpoint distances were inadequate. Several previous Monte Carlo studies were evaluated in light of the results.This report is based on a thesis submitted in partial fulfillment of the degree of Master of Arts at the University of Illinois. The thesis is an outgrowth of earlier work presented in Cohen, H. S., Wing, P. L., & Jones, L. E. The effects of error and subsampling of dimensions on multidimensional scaling solutions. Mathematical Psychology Meetings, Princeton, September, 1971.     

The Problem of the Condition of Arousal of the Pure Sensation of Cutaneous Pain

In order to develop scales representing curiosity-arousing components of pictures (slides) of either common objects or random shapes, a total of 56 college student Ss rated both sets of material on nine dimensions. Factor analyses indicated that the nine dimensions could be reduced to two factors. In order to test the ability of these factors to predict memorial performance and exploratory behavior, an additional 80 undergraduates were then tested for recognition or free looking time (FLT) of either object or random shape slides. Separate regression equations, using FLT and recognition performance as criterion measures, were computed. The present results not only indicate stability in visual exploratory scaling plus redundancy among various attributes of visual stimuli, but also suggest that behavior can be considered as elicited by two separate attributes of visual stimuli.     

Some boundary conditions for a monotone analysis of symmetric matrices

This paper gives a rigorous and greatly simplified proof of Guttman's theorem for the least upper-bound dimensionality of arbitrary real symmetric matricesS, where the points embedded in a real Euclidean space subtend distances which are strictly monotone with the off-diagonal elements ofS. A comparable and more easily proven theorem for the vector model is also introduced. At mostn-2 dimensions are required to reproduce the order information for both the distance and vector models and this is true for any choice of real indices, whether they define a metric space or not. If ties exist in the matrices to be analyzed, then greatest lower bounds are specifiable when degenerate solutions are to be avoided. These theorems have relevance to current developments in nonmetric techniques for the monotone analysis of data matrices.This research in nonmetric methods is supported in part by a grant from the National Science Foundation (GS-929 & -2850).The very helpful comments and encouragement of Louis Guttman and J. Douglas Carroll are greatly appreciated. Finally, to that unknown, but not unsung, reviewer who helped in the clarification of the argument, I express my thanks.     

Some necessary conditions for common-factor analysis

LetR be any correlation matrix of ordern, with unity as each main diagonal element. Common-factor analysis, in the Spearman-Thurstone sense, seeks a diagonal matrixU ² such thatG = R – U ² is Gramian and of minimum rankr. Lets ₁ be the number of latent roots ofR which are greater than or equal to unity. Then it is proved here thatr s ₁. Two further lower bounds tor are also established that are better thans ₁. Simple computing procedures are shown for all three lower bounds that avoid any calculations of latent roots. It is proved further that there are many cases where the rank of all diagonal-free submatrices inR is small, but the minimum rankr for a GramianG is nevertheless very large compared withn. Heuristic criteria are given for testing the hypothesis that a finiter exists for the infinite universe of content from which the sample ofn observed variables is selected; in many cases, the Spearman-Thurstone type of multiple common-factor structure cannot hold.This research was made possible in part by an uncommitted grant-in-aid from the Behavioral Sciences Division of the Ford Foundation.     

A method of successive cumulations for the scaling of pair-comparison preference judgments

The present model treats the scaling of pair-comparison preference judgments among a unidimensional set of stimuli across a population of individuals. Given a setS ofn stimuli,S = {S ₁,S ₂, ,S _n }, the model yields a partially ordered metric on the interstimulus distances which may be used to construct an interval scale of values forS. Obtained also are a set of predictionsP = {P ₁,P ₂, ,P _n } whereP _i is the proportion of individuals in the population whose first choice among the elements ofS isS _i . A numerical illustration is offered and comparisons are drawn with Coombs' unfolding technique.This work was supported in part by Grant GB 2345 from the National Science Foundation. An earlier version of this paper was prepared while the author was a consultant to Proctor & Gamble Co. during the summer of 1964.Now with Proctor & Gamble Co.     

Some properties of the communality in multiple factor theory

Several theorems concerning properties of the communaltiy of a test in the Thurstone multiple factor theory are established. The following theorems are applicable to a battery ofn tests which are describable in terms ofr common factors, with orthogonal reference vectors.1. The communality of a testj is equal to the square of the multiple correlation of testj with ther reference vectors.2. The communality of a testj is equal to the square of the multiple correlation of testj with ther reference vectors and then—1 remaining tests. Corollary: The square of the multiple correlation of a testj with then—1 remaining tests is equal to or less than the communality of testj. It cannot exceed the communality.3. The square of the multiple correlation of a testj with then—1 remaining tests equals the communality of testj if the group of tests containsr statistically independent ests teach with a communality of unity.4. With correlation coefficients corrected for attenuation, when the number of tests increases indefinitely while the rank of the correlational matrix remains unchanged, the communality of a testj equals the square of the multiple correlation of testj with then—1 remaining tests.5. With raw correlation coefficients, it is shown in a special case that the square of the multiple correlation of a testj with then—1 remaining tests approaches the communality of testj as a limit when the number of tests increases indefinitely while the rank of correlational matrix remains the same. This has not yet been proved for the general case.The author wishes to express his appreciation of the encouragement and assistance given him by Dr. L. L. Thurstone.     

Faces are special for newly hatched chicks: evidence for inborn domain‐specific mechanisms underlying spontaneous preferences for face‐like stimuli

It is currently being debated whether human newborns’ preference for faces is due to an unlearned, domain‐specific and configural representation of the appearance of a face, or to general mechanisms, such as an up‐down bias (favouring top‐heavy stimuli, which have more elements in their upper part). Here we show that 2‐day‐old domestic chicks, visually naïve for the arrangement of inner facial features, spontaneously prefer face‐like, schematic, stimuli. This preference is maintained when the up‐down bias is controlled for (Experiment1) or when put in direct conflict with facedness (Experiment 4). In contrast, we found no evidence for the presence of an up‐down bias in chicks (Experiment 2). Moreover, our results indicate that the eye region of stimuli is crucial in determining the expression of spontaneous preferences for faces (Experiments 3 and 4).     

Problems with bootstrapping pearson correlations in very small bivariate samples

Efron'sMonte Carlo bootstrap algorithm is shown to cause degeneracies in Pearson'sr for sufficiently small samples. Two ways of preventing this problem when programming the bootstrap ofr are considered.