A multiple group method of factoring the correlation matrix

There are a number of methods of factoring the correlation matrix which require the calculation of a table of residual correlations after each factor has been extracted. This is perhaps the most laborious part of factoring. The method to be described here avoids the computation of residuals after each factor has been computed. Since the method turns on the selection of a set of constellations or clusters of test vectors, it will be calleda multiple group method of factoring. The method can be used for extracting one factor at a time if that is desired but it will be considered here for the more interesting case in which a number of constellations are selected from the correlation matrix at the start. The result of this method of factoring is a factor matrixF which satisfies the fundamental relationFF'=R.This study is one of a series of investigations in the development of multiple factor analysis and application to the study of primary mental abilities. We wish to acknowledge the financial assistance from the Social Science Research Committee of The University of Chicago which has made possible the work of the Psychometric Laboratory.     

基于假设检验的项目相合性指标研究

在认知诊断评估中,评价认知模型与作答数据的拟合非常重要。已有的层级相合性指标(HCI)仅能用于评价连接规则下模型与数据的拟合情况,有必要研究分离规则下相合性指标。HCI假设某项目上正确作答,便推断其子项目上的错误作答为失拟。由于作答反应的随机性,提出基于假设检验的项目相合性指标。该指标可用于区分连接规则和分离规则的作答数据、评价Q矩阵质量和衡量作答数据中的噪音、还可为评价认知模型和选择认知诊断模型提供参考。     

Psychometric latent response models

In this paper, some psychometric models will be presented that belong to the larger class oflatent response models (LRMs). First, LRMs are introduced by means of an application in the field ofcomponential item response theory (Embretson, 1980, 1984). Second, a general definition of LRMs (not specific for the psychometric subclass) is given. Third, some more psychometric LRMs, and examples of how they can be applied, are presented. Fourth, a method for obtaining maximum likelihood (ML) and some maximum a posteriori (MAP) estimates of the parameters of LRMs is presented. This method is then applied to theconjunctive Rasch model. Fifth and last, an application of the conjunctive Rasch model is presented. This model was applied to responses to typical verbal ability items (open synonym items).This paper presents theoretical and empirical results of a research project supported by the Research Council [Onderzoeksraad] of the University of Leuven (grant number 89-9) to Paul De Boeck and Luc Delbeke.     

Discovery algorithms for hierarchical relations

Two algorithms based on a latent class model are presented for discovering hierarchical relations that exist among a set ofK dichotomous items. The two algorithms, stepwise forward selection and backward elimination, incorporate statistical criteria for selecting (or deleting) 0–1 response pattern vectors to form the subset of the total possible 2 ^k vectors that uniquely describe the hierarchy. The performances of the algorithms are compared, using computer-constructed data, with those of three competing deterministic approaches based on ordering theory and the calculation of Phi/Phi-max coefficients. The discovery algorithms are also demonstrated on real data sets investigated in the literature.     

Cognitive appraisal,vulnerability and coping: An integrative analysis of appraisal and coping mechanisms

Research on stress has generally concluded that stress occurs within the person as a result of their cognitive appraisal of experience. However although a number of cognitive theories of emotion exist, an integrative analysis of cognitive appraisal and vulnerability to stress has not been produced. This report contains two studies. Study 1 draws together the major variables identified in the literature in an analysis of their power to differentiate between a clinical sample (n = 48), a self-help group (n = 54), and a control sample (n = 509). The analysis shows that achievement motivation, attributional style, problem-solving style, emotional reactivity; hopelessness, perceived control and perceived social support are significant discriminators. In a second study the relationship between these variables and aspects of life stress, work stress, life satisfaction, depression, anxiety, and positive affect were investigated. Again the data support an explanatory role for these variables. The suggestion is that these variables can be utilized in an integrative model of cognitive appraisal in the stress process to guide intervention.     

Validating Clusters with the Lower Bound for Sum-of-Squares Error

Given that a minor condition holds (e.g., the number of variables is greater than the number of clusters), a nontrivial lower bound for the sum-of-squares error criterion in K-means clustering is derived. By calculating the lower bound for several different situations, a method is developed to determine the adequacy of cluster solution based on the observed sum-of-squares error as compared to the minimum sum-of-squares error. The author was partially supported by the Office of Naval Research Grant #N00014-06-0106.     

Choice model analysis of the “pick any/n” type of binary data

In the “pick any/n” method, subjects are asked to choose any number of items from a list of n items according to some criterion. This kind of data can be analyzed as a special case of either multiple-choice data or successive categories data where the number of response categories is limited to two. An item response model was proposed for the latter case, which is a combination of an unfolding model and a choice model. The marginal maximum-likelihood estimation method was developed for parameter estimation to avoid incidental parameters, and an expectation-maximization algorithm used for numerical optimization. Two examples of analysis are given to illustrate the proposed method, which we call MAXSC.     

A Model of Knower-Level Behavior in Number Concept Development

Abstract We develop and evaluate a model of behavior on the Give- N task, a commonly used measure of young children's number knowledge. Our model uses the knower-level theory of how children represent numbers. To produce behavior on the Give- N task, the model assumes that children start out with a base rate that makes some answers more likely a priori than others but is updated on each experimental trial in a way that depends on the interaction between the experimenter's request and the child's knower level. We formalize this process as a generative graphical model, so that the parameters—including the base rate distribution and each child's knower level—can be inferred from data using Bayesian methods. Using this approach, we evaluate the model on previously published data from 82 children spanning the whole developmental range. The model provides an excellent fit to these data, and the inferences about the base rate and knower levels are interpretable and insightful. We discuss how our modeling approach can be extended to other developmental tasks and can be used to help evaluate alternative theories of number representation against the knower-level theory.     

A study of the classification capabilities of neural networks using unsupervised learning: A comparison withK-means clustering

Several neural networks have been proposed in the general literature for pattern recognition and clustering, but little empirical comparison with traditional methods has been done. The results reported here compare neural networks using Kohonen learning with a traditional clustering method (K-means) in an experimental design using simulated data with known cluster solutions. Two types of neural networks were examined, both of which used unsupervised learning to perform the clustering. One used Kohonen learning with a conscience and the other used Kohonen learning without a conscience mechanism. The performance of these nets was examined with respect to changes in the number of attributes, the number of clusters, and the amount of error in the data. Generally, theK-means procedure had fewer points misclassified while the classification accuracy of neural networks worsened as the number of clusters in the data increased from two to five.Acknowledgements: Sara Dickson, Vidya Nair, and Beth Means assisted with the neural network analyses.     

Polytomous IRT models and monotone likelihood ratio of the total score

In a broad class of item response theory (IRT) models for dichotomous items the unweighted total score has monotone likelihood ratio (MLR) in the latent trait. In this study, it is shown that for polytomous items MLR holds for the partial credit model and a trivial generalization of this model. MLR does not necessarily hold if the slopes of the item step response functions vary over items, item steps, or both. MLR holds neither for Samejima's graded response model, nor for nonparametric versions of these three polytomous models. These results are surprising in the context of Grayson's and Huynh's results on MLR for nonparametric dichotomous IRT models, and suggest that establishing stochastic ordering properties for nonparametric polytomous IRT models will be much harder.Hemker's research was supported by the Netherlands Research Council, Grant 575-67-034. Junker's research was supported in part by the National Institutes of Health, Grant CA54852, and by the National Science Foundation, Grant DMS-94.04438.     

Number-knower levels in young children: insights from Bayesian modeling

Lee and Sarnecka (2010) developed a Bayesian model of young children’s behavior on the Give-N test of number knowledge. This paper presents two new extensions of the model, and applies the model to new data. In the first extension, the model is used to evaluate competing theories about the conceptual knowledge underlying children’s behavior. One, the knower-levels theory, is basically a “stage” theory involving real conceptual change. The other, the approximate-meanings theory, assumes that the child’s conceptual knowledge is relatively constant, although performance improves over time. In the second extension, the model is used to ask whether the same latent psychological variable (a child’s number-knower level) can simultaneously account for behavior on two tasks (the Give-N task and the Fast-Cards task) with different performance demands. Together, these two demonstrations show the potential of the Bayesian modeling approach to improve our understanding of the development of human cognition.     

CHull as an alternative to AIC and BIC in the context of mixtures of factor analyzers

Mixture analysis is commonly used for clustering objects on the basis of multivariate data. When the data contain a large number of variables, regular mixture analysis may become problematic, because a large number of parameters need to be estimated for each cluster. To tackle this problem, the mixtures-of-factor-analyzers (MFA) model was proposed, which combines clustering with exploratory factor analysis. MFA model selection is rather intricate, as both the number of clusters and the number of underlying factors have to be determined. To this end, the Akaike (AIC) and Bayesian (BIC) information criteria are often used. AIC and BIC try to identify a model that optimally balances model fit and model complexity. In this article, the CHull (Ceulemans & Kiers, 2006) method, which also balances model fit and complexity, is presented as an interesting alternative model selection strategy for MFA. In an extensive simulation study, the performances of AIC, BIC, and CHull were compared. AIC performs poorly and systematically selects overly complex models, whereas BIC performs slightly better than CHull when considering the best model only. However, when taking model selection uncertainty into account by looking at the first three models retained, CHull outperforms BIC. This especially holds in more complex, and thus more realistic, situations (e.g., more clusters, factors, noise in the data, and overlap among clusters).     

Hierarchical number estimation

We investigated number estimation using dot patterns grouped by proximity into larger clusters. Participants estimated the number of dots and clusters in separate trials. Estimation was most accurate when the numbers of elements on both scales were the same. When the number of elements on the unattended scale was higher, overestimation occurred. Conversely, when the number of elements on the unattended scale was lower, underestimation occurred. In Experiment 2, response cues were blocked to reduce any tendency toward attending the irrelevant level. The results were essentially unchanged, indicating response confusion alone cannot account for the effect. The data support the existence of an opposite scale effect in which the number of elements at the unattended level influence the processing of number.The preliminary results of the two experiments were presented at the Annual Meeting of The Association for Research in Vision and Ophthalmology, April 29–May 4, 2001, in Fort Lauderdale, Florida, USA     

The Role of Semantic Clustering in Optimal Memory Foraging

Recent studies of semantic memory have investigated two theories of optimal search adopted from the animal foraging literature: Lévy flights and marginal value theorem. Each theory makes different simplifying assumptions and addresses different findings in search behaviors. In this study, an experiment is conducted to test whether clustering in semantic memory may play a role in evidence for both theories. Labeled magnets and a whiteboard were used to elicit spatial representations of semantic knowledge about animals. Category recall sequences from a separate experiment were used to trace search paths over the spatial representations of animal knowledge. Results showed that spatial distances between animal names arranged on the whiteboard were correlated with inter‐response intervals (IRIs) during category recall, and distributions of both dependent measures approximated inverse power laws associated with Lévy flights. In addition, IRIs were relatively shorter when paths first entered animal clusters, and longer when they exited clusters, which is consistent with marginal value theorem. In conclusion, area‐restricted searches over clustered semantic spaces may account for two different patterns of results interpreted as supporting two different theories of optimal memory foraging.     

A Typology of Ideological Attitudes Towards Social Solidarity and Social Control

Research on ideological attitudes has identified two main dimensions that refer to two fundamental features of group organization: social solidarity and social control. In response to prior research that has studied their relationship mainly from a correlational perspective, this paper introduces a social reality model based on psychological functionality of ideological attitudes. Social position variables (education, income and material vulnerability) and insecurity variables (fear of crime and distrust) are used to predict the interplay between ideological attitudes towards social solidarity and social control. Using K‐means cluster analysis, a typology with four patterns of support for solidarity and control (‘socials’, ‘repressives’, ‘minimalists’ and ‘social‐repressives’) was created, on the basis of representative survey data for the UK, France and Germany (N = 7034). Results from logistic regression analyses show that the proposed social reality model explains membership in typology categories, with similar results across the three countries. Overall, the model underscores the social origins of ideological attitudes as functional responses to perceived social reality. The paper illustrates how the social psychological study of ideological attitudes may be enriched by a typological approach that examines patterns of attitudes rather than single dimensions. Copyright © 2014 John Wiley & Sons, Ltd.     

Fitting multilevel models with ordinal outcomes: performance of alternative specifications and methods of estimation

Previous research has compared methods of estimation for fitting multilevel models to binary data, but there are reasons to believe that the results will not always generalize to the ordinal case. This article thus evaluates (a) whether and when fitting multilevel linear models to ordinal outcome data is justified and (b) which estimator to employ when instead fitting multilevel cumulative logit models to ordinal data, maximum likelihood (ML), or penalized quasi-likelihood (PQL). ML and PQL are compared across variations in sample size, magnitude of variance components, number of outcome categories, and distribution shape. Fitting a multilevel linear model to ordinal outcomes is shown to be inferior in virtually all circumstances. PQL performance improves markedly with the number of ordinal categories, regardless of distribution shape. In contrast to binary data, PQL often performs as well as ML when used with ordinal data. Further, the performance of PQL is typically superior to ML when the data include a small to moderate number of clusters (i.e., ≤ 50 clusters).     

A multidimensional item response model: Constrained latent class analysis using the gibbs sampler and posterior predictive checks

In this paper it will be shown that a certain class of constrained latent class models may be interpreted as a special case of nonparametric multidimensional item response models. The parameters of this latent class model will be estimated using an application of the Gibbs sampler. It will be illustrated that the Gibbs sampler is an excellent tool if inequality constraints have to be taken into consideration when making inferences. Model fit will be investigated using posterior predictive checks. Checks for manifest monotonicity, the agreement between the observed and expected conditional association structure, marginal local homogeneity, and the number of latent classes will be presented.This paper is supported by grant S40-645 of the Dutch Organization for Scientific Research (NWO).