Exploratory second-order analyses for components and factors

Abstract: Exploratory methods using second‐order components and second‐order common factors were proposed. The second‐order components were obtained from the resolution of the correlation matrix of obliquely rotated first‐order principal components. The standard errors of the estimates of the second‐order component loadings were derived from an augmented information matrix with restrictions for the loadings and associated parameters. The second‐order factor analysis proposed was similar to the classical method in that the factor correlations among the first‐order factors were further resolved by the exploratory method of factor analysis. However, in this paper the second‐order factor loadings were estimated by the generalized least squares using the asymptotic variance‐covariance matrix for the first‐order factor correlations. The asymptotic standard errors for the estimates of the second‐order factor loadings were also derived. A numerical example was presented with simulated results.     

Oblique factors and components with independent clusters

Relationships between the results of factor analysis and component analysis are derived when oblique factors have independent clusters with equal variances of unique factors. The factor loadings are analytically shown to be smaller than the corresponding component loadings while the factor correlations are shown to be greater than the corresponding component correlations. The condition for the inequality of the factor/component contributions is derived in the case with different variances for unique factors. Further, the asymptotic standard errors of parameter estimates are obtained for a simplified model with the assumption of multivariate normality, which shows that the component loading estimate is more stable than the corresponding factor loading estimate.     

Concise formulas for the standard errors of component loading estimates

Concise formulas for the asymptotic standard errors of component loading estimates were derived. The formulas cover the cases of principal component analysis for unstandardized and standardized variables with orthogonal and oblique rotations. The formulas can be used under any distributions for observed variables as long as the asymptotic covariance matrix for sample covariances/correlations is available. The estimated standard errors in numerical examples were shown to be equivalent to those by the methods using information matrices.The author is indebted to anonymous reviewers for the corrections and suggestions on this study, which have led to improvements of earlier versions of this article.     

Standard errors for the class of orthomax-rotated factor loadings: Some matrix results

The partial derivative matrices of the class of orthomax-rotated factor loadings with respect to the unrotated maximum likelihood factor loadings are derived. The reported results are useful for obtaining standard errors of the orthomax-rotated factor loadings, with or without row normalization (standardization) of the initial factor loading matrix for rotation. Using a numerical example, we verify our analytic formulas by comparing the obtained standard error estimates with that from some existing methods. Some advantages of the current approach are discussed.Authorship is determined by alphabetical order. The authors contributed equally to the research. Kentaro Hayashi is now at the Department of Mathematics, Bucknell University, Lewisburg, PA 17837 (email: khayashi@Bucknell.edu). Yiu-Fai Yung is now at the SAS Institute, Inc., SAS Campus Drive, Cary, NC 27513 (email: yiyung@wnt.sas.com).Part of the research was completed while Yiu-Fai Yung was a visiting scholar at the Department of Psychology, the Ohio State University. The visit was supported in part by grant N4856118101 from the NIMH and the Mason and Linda Stephenson Travel Award from the Department of Psychology, University of North Carolina at Chapel Hill. The authors are grateful to Michael Browne who suggested some relevant references and provided valuable comments on the research, and to Robert Cudeck who provided the FAS program for the numerical comparison. The expert comments by the reviewers are deeply appreciated.     

Some standard errors in item response theory

The mathematics required to calculate the asymptotic standard errors of the parameters of three commonly used logistic item response models is described and used to generate values for some common situations. It is shown that the maximum likelihood estimation of a lower asymptote can wreak havoc with the accuracy of estimation of a location parameter, indicating that if one needs to have accurate estimates of location parameters (say for purposes of test linking/equating or computerized adaptive testing) the sample sizes required for acceptable accuracy may be unattainable in most applications. It is suggested that other estimation methods be used if the three parameter model is applied in these situations.The research reported here was supported, in part, by contract #F41689-81-6-0012 from the Air Force Human Resources Laboratory to McFann-Gray & Associates, Benjamin A. Fairbank, Jr., Principal Investigator. Further support of Wainer's effort was supplied by the Educational Testing Service, Program Statistics Research Project.     

A look,by simulation,at the validity of some asymptotic distribution results for rotated loadings

In the last few years, a number of asymptotic results for the distribution of unrotated and rotated factor loadings have been given. This paper investigates the validity of some of these results based on simulation techniques. In particular, it looks at principal component extraction and quartimax rotation on a problem with 13 variables. The indication is that the asymptotic results are quite good.     

The relationship between external variables and common factors

A theorem is presented which gives the range of possible correlations between a common factor and an external variable (i.e., a variable not included in the test battery factor analyzed). Analogous expressions for component (and regression component) theory are also derived. Some situations involving external correlations are then discussed which dramatize the theoretical differences between components and common factors.Support by the National Research Council of Canada (NRC Grant No. A4640) and the University of British Columbia (UBC Humanities and Social Sciences Grant 26-9718) is gratefully acknowledged.This paper is based in part on the author's Ph.D. dissertation. I am particularly grateful to my dissertation advisor, Dr. Peter H. Schönemann. Thanks also to the editor and the anonymous reviewers, who contributed many helpful comments and suggestions.     

On the estimation of polychoric correlations and their asymptotic covariance matrix

A general theory for parametric inference in contingency tables is outlined. Estimation of polychoric correlations is seen as a special case of this theory. The asymptotic covariance matrix of the estimated polychoric correlations is derived for the case when the thresholds are estimated from the univariate marginals and the polychoric correlations are estimated from the bivariate marginals for given thresholds. Computational aspects are also discussed.The research was supported by the Swedish Council for Research in the Humanities and Social Sciences (HSFR) under the programMultivariate Statistical Analysis. The author thanks a reviewer for pointing out an error in the original version of the paper.     

Standard errors for procrustes solutions

The asymptotic standard errors for the procrustes solutions are derived for orthogonal rotation, direct oblique rotation and indirect oblique rotation. The standard errors for the first two rotations are obtained using the augmented information matrices. For the indirect oblique solution, the standard errors of rotated parameters are derived from the information matrix of unrotated loadings using the chain rule for information matrices. For all three types of rotation, the standard errors of rotated parameters are presented for unstandardized and standardized manifest variables. Numerical examples show the similarity of theoretical and simulated values.     

Standard errors of fit indices using residuals in structural equation modeling

The asymptotic standard errors of the correlation residuals and Bentler's standardized residuals in covariance structures are derived based on the asymptotic covariance matrix of raw covariance residuals. Using these results, approximations of the asymptotic standard errors of the root mean square residuals for unstandardized or standardized residuals are derived by the delta method. Further, in mean structures, approximations of the asymptotic standard errors of residuals, standardized residuals and their summary statistics are derived in a similar manner. Simulations are carried out, which show that the asymptotic standard errors of the various types of residuals and the root mean square residuals in covariance, correlation and mean structures are close to actual ones.The author is indebted to the reviewers for their comments and suggestions which have led to an improvement of this work.     

Some relationships between behavioral and traditional assessment

Psychometric principles and procedures were developed within the context of traditional approaches to assessment. In order to illustrate the frequent, if not uniform, applicability of these principles to behavior assessment, we first discuss the differences and similarities between traditional and behavioral assessment. We then point out ways in which, given the nature of behavior assessment, attention to certain psychometric principles and procedures would expedite the development of more adequate behavior assessment devices and procedures.     

The rise of the nones and the changing relationships between identity,belief, and behavior

ABSTRACT

One of the more remarkable trends of the past 30 years is the dramatic rise of individuals who do not identify with any religious tradition. While this trend has been well documented, some of the underlying dynamics and consequences have not been fully appreciated or explicated. We examine the General Social Survey in the period from 1972 to 2014 to examine how the increase in the ‘nones’ is tied to changes in the strength of religious identity among US adults and, in turn, how the rise of the nones has affected the relationships between religious identity, religious belief, and religious behavior. In particular, we show that, as the percentage of US adults who do not identify with a religion has grown, the correlations between religious identification, belief, and behavior have increased. In short, the rise of the nones has led to more congruence between measures of religion.     

Some contributions to efficient statistics in structural models: Specification and estimation of moment structures

Current practice in structural modeling of observed continuous random variables is limited to representation systems for first and second moments (e.g., means and covariances), and to distribution theory based on multivariate normality. In psychometrics the multinormality assumption is often incorrect, so that statistical tests on parameters, or model goodness of fit, will frequently be incorrect as well. It is shown that higher order product moments yield important structural information when the distribution of variables is arbitrary. Structural representations are developed for generalizations of the Bentler-Weeks, Jöreskog-Keesling-Wiley, and factor analytic models. Some asymptotically distribution-free efficient estimators for such arbitrary structural models are developed. Limited information estimators are obtained as well. The special case of elliptical distributions that allow nonzero but equal kurtoses for variables is discussed in some detail. The argument is made that multivariate normal theory for covariance structure models should be abandoned in favor of elliptical theory, which is only slightly more difficult to apply in practice but specializes to the traditional case when normality holds. Many open research areas are described.     

Multiple factors of cognitive failure and their relationships with stress vulnerability

Use of the Cognitive Failures Questionnaire (CFQ) in clinical research assumes that the questionnaire measures a single factor. A factor analysis of the CFQ was performed, on a sample of 475 students, using accurate procedures for determining the number of factors. These procedures gave mutually inconsistent results: at least two alternative factor structures can be extracted from the CFQ. An interpretable seven factor solution was found, but the CFQ probably has insufficient items to measure more than two strongly defined factors. The capacity of CFQ multiple factors to predict measures related to stress vulnerability was tested in a further study (N=57). Correlations between CFQ total score and neuroticism and use of coping strategies appeared to depend mainly on a single CFQ multiple factor, related to failures of concentration. Some of the other CFQ factors predicted other criteria. Further research on multiple factors of cognitive failure might improve the validity of the CFQ as a predictor of stress vulnerability and of cognitive performance.     

Individual differences in cognition: Some relationships between personality and memory

Individual differences in cognition were studied in the form of the hypothesis that arousal, as indexed by personality measures of extraversion and neuroticism, affects the way in which verbal material is organized in memory. Subjects pretested on measures of these personality variables participated in either a paired-associates learning or a free-recall experiment. On the paired-associates task, subjects who were thought to be high on arousal made fewer errors when response terms were semantically similar than low arousal subjects. On the other hand, subjects thought low on arousal made fewer errors when response words were phonetically similar than high arousal subjects. On the free-recall task, low arousal subjects were found to cluster words together on the basis of semantic category at a higher rate than high arousal subjects. These results were taken to support the view that high arousal (as indexed by personality measures) leads to a focus on the physical aspects of verbal material, whereas low arousal leads to a memory organized around semantic aspects. The implications of these findings for other views of memory are discussed.     

Exploring the relationships between maternal and child metacognitions and child anxiety

Research has shown that anxiety may be transmitted through verbal information pathways, for example, when parents share their anxious cognitions with their child. Less is known about the influence of parental metacognitions, that is, beliefs regarding thoughts, on child anxiety. We explored the relations between metacognitions in mothers and their children and anxiety in the children. Our study included 111 non‐clinical children aged 8 to 12 years and their mothers. Children rated their metacognitions, worry and anxiety, and mothers rated their metacognitions. Results indicated agreement between maternal and child metacognitions. Maternal metacognitions were positively associated with children's anxiety symptoms and worry, and this relation was mediated by the children's metacognitions. Our results warrant further examination of the role of parental metacognitions in child anxiety.     

General and supplementary factors of personality in genetic and environmental correlation matrices

Genetic and environmental correlation matrices among 7 personality scales were estimated, using samples of adult and young adult twins from Australia (2081 and 1178 pairs, respectively). A general factor of personality and two supplemental factors were obtained in each. The supplemental factors were tentatively identified as Social conformity and Other-dependence. Factors from the genetic and environmental correlations replicated well across samples, and they were similar in both the genetic and environmental covariation. It was concluded that the structure of personality is inherent in the evolved phenotype, and is not the immediate consequence of either genetic or environmental organizing factors.     

Some relationships between adaptation and electromyographic activity at laryngeal and masseter sites in stutterers

Cumulative integrated electromyographic (EMG) activity was recorded from laryngeal and masseter sites from ten stutterers during a five trial oral adaptation sequence. Results showed more activity at the laryngeal site than the masseter site. While significant reductions in EMG activity were not found, patterns were shown to be related to the subjects' dysfluency patterns over trials. Results were interpreted as suggesting that reduced muscle activity and adaptation (when they occur) are artifacts of other behavioral modifications resulting from overt motor practice.     

Beyond principal component analysis: A trilinear decomposition model and least squares estimation

The paper derives sufficient conditions for the consistency and asymptotic normality of the least squares estimator of a trilinear decomposition model for multiway data analysis.