The joint analysis of direct ratings,pairwise preferences,and dissimilarities

In studies involving judgments of similarity or dissimilarity, a variety of other variables may also be measured. Examples might be direct ratings of the stimuli, pairwise preference judgments, and physical measurements of the stimuli with respect to various properties. In such cases, there are important advantages to joint analyses of the dissimilarity and collateral variables. A variety of models are described for relating these and algorithms described for fitting these to data. A number of hypothesis tests are developed and an example offered.The research reported here was supported by grant number APA 320 to the author by the National Research Council of Canada.     

Dissimilarity, Quasimetric, Metric

Dissimilarity is a function that assigns to every pair of stimuli a nonnegative number vanishing if and only if two stimuli are identical, and that satisfies the following two conditions called the intrinsic uniform continuity and the chain property, respectively: it is uniformly continuous with respect to the uniformity it induces, and, given a set of stimulus chains (finite sequences of stimuli), the dissimilarity between their initial and terminal elements converges to zero if the chains’ length (the sum of the dissimilarities between their successive elements) converges to zero. The four properties axiomatizing this notion are shown to be mutually independent. Any conventional, symmetric metric is a dissimilarity function. A quasimetric (satisfying all metric axioms except for symmetry) is a dissimilarity function if and only if it is symmetric in the small. It is proposed to reserve the term metric (not necessarily symmetric) for such quasimetrics. A real-valued binary function satisfies the chain property if and only if whenever its value is sufficiently small it majorates some quasimetric and converges to zero whenever this quasimetric does. The function is a dissimilarity function if, in addition, this quasimetric is a metric with respect to which the function is uniformly continuous.     

Multidimensional successive categories scaling: A maximum likelihood method

A single-step maximum likelihood estimation procedure is developed for multidimensional scaling of dissimilarity data measured on rating scales. The procedure can fit the euclidian distance model to the data under various assumptions about category widths and under two distributional assumptions. The scoring algorithm for parameter estimation has been developed and implemented in the form of a computer program. Practical uses of the method are demonstrated with an emphasis on various advantages of the method as a statistical procedure.The research reported here was partly supported by Grant A6394 to the author by Natural Sciences and Engineering Research Council of Canada. Portions of this research were presented at the Psychometric Society meeting in Uppsala, Sweden, in June, 1978. MAXSCAL-2.1, a program to perform the computations discussed in this paper may be obtained from the author. Thanks are due to Jim Ramsay for his helpful comments.     

A Newton procedure for conditionally linear mixed-effects models

This article reviews Newton procedures for the analysis of mean and covariance structures that may be functions of parameters that enter a model nonlinearly. The kind of model considered is a mixed-effects model that is conditionally linear with regard to its parameters. This means parameters entering the model nonlinearly must be fixed, whereas those entering linearly may vary across individuals. This framework encompasses several models, including hierarchical linear models, linear and nonlinear factor analysis models, and nonlinear latent curve models. A full maximum-likelihood estimation procedure is described. Mx, a statistical software package often used to estimate structural equation models, is considered for estimation of these models. An example with Mx syntax is provided.     

Dissimilarity cumulation as a procedure correcting for violations of triangle inequality

Dissimilarity on a finite set is a function that assigns to every pair of points (stimuli) a nonnengative number vanishing if and only if the two points are identical. For any two points in a finite set, the minimum of the length of all finite chains connecting these points is a quasimetric (i.e., it satisfies all metric axioms except for symmetry). The computation of the quasimetric from a dissimilarity can be viewed as a data-analytic procedure of “correcting” dissimilarities into (quasi)distances, an alternative to nonmetric Multidimensional Scaling: the procedure simply replaces the dissimilarity between any two points with the shortest length across all chains of points connecting them. It is shown in this paper that this procedure can be equivalently described as a series of recursive corrections for violations of the triangle inequality across all triples of points considered in an arbitrary order.     

Perceptions As Reality: The Influence of Actual and Perceived Demographic Dissimilarity

Two studies were conducted to examine the effects of demographic dissimilarity from others in the workgroup. Results from Study 1 (n = 87 coaches) indicate that actual demographic differences were reliably related to perceptions of such differences. Study 2 investigated the effects of perceived demographic dissimilarity. Data were gathered from an additional 175 coaches. Structural equation modeling indicated that perceived demographic differences were associated with perceived deep-level dissimilarity (i.e., differences in personality, values, and attitudes). In turn, perceived deep-level dissimilarity was negatively related to coworker satisfaction and held a positive association with organizational turnover intentions. Results are discussed in terms of theoretical contributions to the relational demography perspective.     

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.     

Two routes for the processing of verbal numbers: evidence from the SNARC effect

The functional locus of the semantic system is an important issue in number processing. In the present article, the necessity of addressing a central semantic magnitude system in the processing of printed verbal number words is evaluated by looking at the presence of a spatial-numerical association of response codes or SNARC effect. This effect consists of an association of number magnitude and response-preference (preferred responses to small numbers with the left hand and to large numbers with the right hand) and reflects semantic access. Two experiments were run. In Experiment 1, participants performed a parity judgment task which requires access to number semantics. A SNARC effect was observed. In Experiment 2 a phoneme monitoring task was used, which can, in principle, be performed through direct asemantic transcoding. No SNARC effect occurred. Apparently, written number words access the semantic system only if this is necessary for correct task completion. Hence, a semantic and an asemantic route can be postulated for the processing of word numerals. These observations contrast with the processing of Arabic numerals for which semantic effects are omnipresent. Implications of this explicit demonstration of a dissimilarity between the processing of digits and of number words are discussed. Received: 25 October 2000 / Accepted: 21 May 2001     

Surface‐ and deep‐level dissimilarity effects on social integration and individual effectiveness related outcomes in work groups: A meta‐analytic integration

Prior research linking demographic (e.g., age, ethnicity/race, gender, and tenure) and underlying psychological (e.g., personality, attitudes, and values) dissimilarity variables to individual group member's work‐related outcomes produced mixed and contradictory results. To account for these findings, this study develops a contingency framework and tests it using meta‐analytic and structural equation modelling techniques. In line with this framework, results showed different effects of surface‐level (i.e., demographic) dissimilarity and deep‐level (i.e., underlying psychological) dissimilarity on social integration, and ultimately on individual effectiveness related outcomes (i.e., turnover, task, and contextual performance). Specifically, surface‐level dissimilarity had a negative effect on social integration under low but not under high team interdependence. In return, social integration fully mediated the negative relationship between surface‐level dissimilarity and individual effectiveness related outcomes under low interdependence. In contrast, deep‐level dissimilarity had a negative effect on social integration, which was stronger under high and weaker under low team interdependence. Contrary to our predictions, social integration did not mediate the negative relationship between deep‐level dissimilarity and individual effectiveness related outcomes but suppressed positive direct effects of deep‐level dissimilarity on individual effectiveness related outcomes. Possible explanations for these counterintuitive findings are discussed.     

Principal component analysis of three-mode data by means of alternating least squares algorithms

A new method to estimate the parameters of Tucker's three-mode principal component model is discussed, and the convergence properties of the alternating least squares algorithm to solve the estimation problem are considered. A special case of the general Tucker model, in which the principal component analysis is only performed over two of the three modes is briefly outlined as well. The Miller & Nicely data on the confusion of English consonants are used to illustrate the programs TUCKALS3 and TUCKALS2 which incorporate the algorithms for the two models described.     

Do perceived differences in religion matter at work?

Over 90% of the United States population affiliates with an organised religion. There has been, however, very little research regarding religion in work organisations. This study focuses on the effects of individual perceptions of religious dissimilarity, as a characteristic of deep-level diversity, on perceived cohesion. A scale is developed to measure individual perceptions of the dissimilarity of religion. Two different studies are used to establish reliability and validity of this scale. In addition, some implications of the negative relationship between individual perceptions of religious dissimilarity and perceived cohesion for organisations are discussed.     

Structural modeling and psychometrika: An historical perspective on growth and achievements

The field of linear structural equation modeling with continuous variables is reviewed. Trends in psychometric theory and data analysis across the five decades of publication ofPsychometrika are discussed, especially the clarification of concepts of population and sample, explication of the parametric structure of models, delineation of concepts of exploratory and confirmatory data analysis, expansion of statistical theory in psychometrics, estimation via optimization of an explicit objective function, and implementation of general function minimization methods. Developments in the ideas of factor analysis, latent variables, as well as structural and causal modeling are noted. Some major conceptual achievements involving general covariance structure representations, multiple population models, and moment structures are reviewed. The major statistical achievements of normal theory generalized least squares estimation, elliptical and distribution-free estimation, and higher-moment estimation are discussed. Computer programs that implement some of the theoretical developments are described.This review was supported in part by USPHS grants DA00017 and DA01070.     

An information theory analysis of visual complexity and dissimilarity

The subjective complexity of a computer-generated bitmap image can be measured by magnitude estimation scaling, and its objective complexity can be measured by its compressed file size. There is a high correlation between these measures of subjective and objective complexity over a large set of marine electronic chart and radar images. The subjective dissimilarity of a pair of bitmap images can be predicted from subjective and objective measures of the complexity of each image, and from the subjective and objective complexity of the image produced by overlaying the two simple images. In addition, the subjective complexity of the image produced by overlaying two simple images can be predicted from the subjective complexity of the simple images and the subjective dissimilarity of the image pair. The results of the experiments that generated these complexity and dissimilarity judgments are consistent with a theory, outlined here, that treats objective and subjective measures of image complexity and dissimilarity as vectors in Euclidean space.     

Contributions to Estimation of Polychoric Correlations

This research concerns the estimation of polychoric correlations in the context of fitting structural equation models to observed ordinal variables by multistage estimation. The first main contribution of this research is to propose and evaluate a Monte Carlo estimator for the asymptotic covariance matrix (ACM) of the polychoric correlation estimates. In multistage estimation, the ACM plays a prominent role, as overall test statistics, derived fit indices, and parameter standard errors all depend on this quantity. The ACM, however, must itself be estimated. Established approaches to estimating the ACM use a sample-based version, which can yield poor estimates with small samples. A simulation study demonstrates that the proposed Monte Carlo estimator can be more efficient than its sample-based counterpart. This leads to better calibration for established test statistics, in particular with small samples. The second main contribution of this research is a further exploration of the consequences of violating the normality assumption for the underlying response variables. We show the consequences depend on the type of nonnormality, and the number and location of thresholds. The simulation study also demonstrates that overall test statistics have little power to detect the studied forms of nonnormality, regardless of the ACM estimator.     

Importance of friendship potential in reducing the negative effects of dissimilarity

The purpose of this study was to examine the influence of friendship potential on perceptions of dissimilarity among persons in diverse groups. The author gathered data at the beginning and end of a 15-week semester from college students enrolled in physical activity classes. Structural equation modeling indicated that friendship potential was negatively related to perceptions of deep-level dissimilarity at the end of the semester. This relation was moderated by perceptions of such dissimilarity at the beginning of the semester. The author found the perceived dissimilarity to be negatively associated with affective reactions to the class. The author discusses findings in terms of contributions and implications for diverse groups.     

An extended class of matched pairs tests based on powers of ranks

A large class of rank tests, which includes the familiar sign test and the Wilcoxon signed-ranks test, is described and discussed. This class of distribution-free tests provides a flexible basis for testing research hypotheses of various forms. Exact small sample and approximate large sample procedures are considered. Applications of these procedures are presented, including simple numerical examples.The authors wish to acknowledge the constructive comments by the reviewers.     

On measures of explained variance in nonrecursive structural equation models

Whereas measures of explained variance in a regression and an equation of a recursive structural equation model can be simply summarized by a standard R2 measure, this is not possible in nonrecursive models in which there are reciprocal interdependencies among variables. This article provides a general approach to defining variance explained in latent dependent variables of nonrecursive linear structural equation models. A new method of its estimation, easily implemented in EQS or LISREL and available in EQS 6, is described and illustrated.     

Comparison of Methods for Collecting and Modeling Dissimilarity Data: Applications to Complex Sound Stimuli

Sorting procedures are frequently adopted as an alternative to dissimilarity ratings to measure the dissimilarity of large sets of stimuli in a comparatively short time. However, systematic empirical research on the consequences of this experiment-design choice is lacking. We carried out a behavioral experiment to assess the extent to which sorting procedures compare to dissimilarity ratings in terms of efficiency, reliability, and accuracy, and the extent to which data from different data-collection methods are redundant and are better fit by different distance models. Participants estimated the dissimilarity of either semantically charged environmental sounds or semantically neutral synthetic sounds. We considered free and hierarchical sorting and derived indications concerning the properties of constrained and truncated hierarchical sorting methods from hierarchical sorting data. Results show that the higher efficiency of sorting methods comes at a considerable cost in terms of data reliability and accuracy. This loss appears to be minimized with truncated hierarchical sorting methods that start from a relatively low number of groups of stimuli. Finally, variations in data-collection method differentially affect the fit of various distance models at the group-average and individual levels. On the basis of these results, we suggest adopting sorting as an alternative to dissimilarity-rating methods only when strictly necessary. We also suggest analyzing the raw behavioral dissimilarities, and avoiding modeling them with one single distance model.     

多维测验项目参数的估计：基于SEM与MIRT方法的比较

作者简要回顾了SEM框架下分类数据因素分析(CCFA)模型和MIRT框架下测验题目和潜在能力的关系模型, 对两种框架下的主要参数估计方法进行了总结。通过模拟研究, 比较了SEM框架下WLSc和WLSMV估计方法与MIRT框架下MLR和MCMC估计方法的差异。研究结果表明：(1) WLSc得到参数估计的偏差最大, 且存在参数收敛的问题; (2)随着样本量增大, 各种项目参数估计的精度均提高, WLSMV方法与MLR方法得到的参数估计精度差异很小, 大多数情况下不比MCMC方法差; (3)除WLSc方法外, 随着每个维度测验题目的增多参数估计的精度逐渐增高; (4)测验维度对区分度参数和难度参数的影响较大, 而测验维度对项目因素载荷和阈值的影响相对较小; (5)项目参数的估计精度受项目测量维度数的影响, 只测量一个维度的项目参数估计精度较高。另外文章还对两种方法在实际应用中应该注意的问题提供了一些建议。