Minimum rank and minimum trace of covariance matrices

This paper considers some mathematical aspects of minimum trace factor analysis (MTFA). The uniqueness of an optimal point of MTFA is proved and necessary and sufficient conditions for a point x to be optimal are established. Finally, some results about the connection between MTFA and the classical minimum rank factor analysis will be presented.     

Image and anti-image covariance matrices from a correlation matrix that may be singular

Following the work of Tucker, Cooper, and Meredith, image and anti-image covariance matrices from a correlation matrix that may be singular are derived.     

A numerical approach to the approximate and the exact minimum rank of a covariance matrix

A concept of approximate minimum rank for a covariance matrix is defined, which contains the (exact) minimum rank as a special case. A computational procedure to evaluate the approximate minimum rank is offered. The procedure yields those proper communalities for which the unexplained common variance, ignored in low-rank factor analysis, is minimized. The procedure also permits a numerical determination of the exact minimum rank of a covariance matrix, within limits of computational accuracy. A set of 180 covariance matrices with known or bounded minimum rank was analyzed. The procedure was successful throughout in recovering the desired rank.The authors are obliged to Paul Bekker for stimulating and helpful comments.     

Structural analysis of covariance and correlation matrices

A general approach to the analysis of covariance structures is considered, in which the variances and covariances or correlations of the observed variables are directly expressed in terms of the parameters of interest. The statistical problems of identification, estimation and testing of such covariance or correlation structures are discussed.Several different types of covariance structures are considered as special cases of the general model. These include models for sets of congeneric tests, models for confirmatory and exploratory factor analysis, models for estimation of variance and covariance components, regression models with measurement errors, path analysis models, simplex and circumplex models. Many of the different types of covariance structures are illustrated by means of real data.1978 Psychometric Society Presidential Address.This research has been supported by the Bank of Sweden Tercentenary Foundation under the project entitledStructural Equation Models in the Social Sciences, Karl G. Jöreskog, project director.     

The robustness of estimates of total indirect effects in covariance structure models estimated by maximum

The large sample distribution of total indirect effects in covariance structure models in well known. Using Monte Carlo methods, this study examines the applicability of the large sample theory to maximum likelihood estimates oftotal indirect effects in sample sizes of 50, 100, 200, 400, and 800. Two models are studied. Model 1 is a recursive model with observable variables and Model 2 is a nonrecursive model with latent variables. For the large sample theory to apply, the results suggest that sample szes of 200 or more and 400 or more are required for models such as Model 1 and Model 2, respectively.For helpful comments on a previous draft of this paper, we are grateful to Gerhard Arminger, Clifford C. Clogg, and several anonymous reviewers.     

Pseudo maximum likelihood estimation and a test for misspecification in mean and covariance structure models

Using the theory of pseudo maximum likelihood estimation the asymptotic covariance matrix of maximum likelihood estimates for mean and covariance structure models is given for the case where the variables are not multivariate normal. This asymptotic covariance matrix is consistently estimated without the computation of the empirical fourth order moment matrix. Using quasi-maximum likelihood theory a Hausman misspecification test is developed. This test is sensitive to misspecification caused by errors that are correlated with the independent variables. This misspecification cannot be detected by the test statistics currently used in covariance structure analysis.For helpful comments on a previous draft of the paper we are indebted to Kenneth A. Bollen, Ulrich L. Küsters, Michael E. Sobel and the anonymous reviewers of Psychometrika. For partial research support, the first author wishes to thank the Department of Sociology at the University of Arizona, where he was a visiting professor during the fall semester 1987.     

Power of the likelihood ratio test in covariance structure analysis

A procedure for computing the power of the likelihood ratio test used in the context of covariance structure analysis is derived. The procedure uses statistics associated with the standard output of the computer programs commonly used and assumes that a specific alternative value of the parameter vector is specified. Using the noncentral Chi-square distribution, the power of the test is approximated by the asymptotic one for a sequence of local alternatives. The procedure is illustrated by an example. A Monte Carlo experiment also shows how good the approximation is for a specific case.This research was made possible by a grant from the Dutch Organization for Advancement of Pure Research (ZWO). The authors also like to acknowledge the helpful comments and suggestions from the editor and anonymous reviewers.     

Estimation of covariance structure models with parameters subject to functional restraints

This paper demonstrates the feasibility of using the penalty function method to estimate parameters that are subject to a set of functional constraints in covariance structure analysis. Both types of inequality and equality constraints are studied. The approaches of maximum likelihood and generalized least squares estimation are considered. A modified Scoring algorithm and a modified Gauss-Newton algorithm are implemented to produce the appropriate constrained estimates. The methodology is illustrated by its applications to Heywood cases in confirmatory factor analysis, quasi-Weiner simplex model, and multitrait-multimethod matrix analysis.The author is indebted to several anonymous reviewers for creative suggestions for improvement of this paper. Computer funding is provided by the Computer Services Centre, The Chinese University of Hong Kong.     

Improper solutions in the analysis of covariance structures: Their interpretability and a comparison of alternate respecifications

A Monte Carlo approach was employed to investigate the interpretability of improper solutions caused by sampling error in maximum likelihood confirmatory factor analysis. Four models were studied with two sample sizes. Of the overall goodness-of-fit indices provided by the LISREL VI program significant differences between improper and proper solutions were found only for the root mean square residual. As expected, indicators of the factor on which the negative uniqueness estimate occurred had biased loadings, and the correlations of its factor with other factors were also biased. In contrast, the loadings of indicators on other factors and those factor intercorrelations did not have any bias of practical significance. For initial solutions with one negative uniqueness estimate, three respecifications were studied: Fix the uniqueness at .00, fix it at .20, or constrain the domain of the solution to be proper. For alternate, respecified solutions that were converged and proper, the constrained solutions and uniqueness fixed at .00 solutions were equivalent. The mean goodness-of-fit and pattern coefficient values for the original improper solutions were not meaningfully different from those obtained under the constrained and uniqueness fixed at .00 respecifications.This investigation was supported in part by a grant from the Baylor University Research Committee (#018-F83-URC). The authors gratefully acknowledge the comments and suggestions of Claes Fornell and Roger E. Kirk, and the assistance of Timothy J. Vance with the analysis.     

Equivalent models in covariance structure analysis

Defining equivalent models as those that reproduce the same set of covariance matrices, necessary and sufficient conditions are stated for the local equivalence of two expanded identified modelsM ₁ andM ₂ when fitting the more restricted modelM ₀. Assuming several regularity conditions, the rank deficiency of the Jacobian matrix, composed of derivatives of the covariance elements with respect to the union of the free parameters ofM ₁ andM ₂ (which characterizes modelM ₁₂), is a necessary and sufficient condition for the local equivalence ofM ₁ andM ₂. This condition is satisfied, in practice, when the analysis dealing with the fitting ofM ₀, predicts that the decreases in the chi-square goodness-of-fit statistic for the fitting ofM ₁ orM ₂, orM ₁₂ are all equal for any set of sample data, except on differences due to rounding errors.This research was supported by the Foundation of Social-Cultural Sciences which is subsidized by the Dutch Scientific Organization (N.W.O.) under project number 500-278-003. The author wishes to thank Anne Boomsma, Ivo Molenaar, Albert Satorra, and Tom Snijders for their stimulating and crucial comments during the research, and the Editor, Paul Bekker, Henk Broer, and anonymous reviewers for their helpful suggestions.     

因子分析元分析的三阶段整合模型

因子分析元分析(meta-analysis of factor analyses, MAFA)整合了因子分析和元分析两种方法学传统。开展MAFA方法学研究可结束过去几十年MAFA研究缺乏根据的状况, 服务心理学等领域内的知识生产。三阶段整合模型认为MAFA包括数据准备、数据合成和数据分析三个主要阶段。模型及数据合成技术的有效性将用心理测量学方法, 包括蒙特卡洛模拟研究进行检验。研究结果将阐明正确运用MAFA的一般规则、操作步骤和注意事项, 进而增进我们对MAFA的认识, 丰富应用心理等领域的方法学体系, 为MAFA方法的推广应用奠定基础。     

A study of algorithms for covariance structure analysis with specific comparisons using factor analysis

Several algorithms for covariance structure analysis are considered in addition to the Fletcher-Powell algorithm. These include the Gauss-Newton, Newton-Raphson, Fisher Scoring, and Fletcher-Reeves algorithms. Two methods of estimation are considered, maximum likelihood and weighted least squares. It is shown that the Gauss-Newton algorithm which in standard form produces weighted least squares estimates can, in iteratively reweighted form, produce maximum likelihood estimates as well. Previously unavailable standard error estimates to be used in conjunction with the Fletcher-Reeves algorithm are derived. Finally all the algorithms are applied to a number of maximum likelihood and weighted least squares factor analysis problems to compare the estimates and the standard errors produced. The algorithms appear to give satisfactory estimates but there are serious discrepancies in the standard errors. Because it is robust to poor starting values, converges rapidly and conveniently produces consistent standard errors for both maximum likelihood and weighted least squares problems, the Gauss-Newton algorithm represents an attractive alternative for at least some covariance structure analyses.Work by the first author has been supported in part by Grant No. Da01070 from the U. S. Public Health Service. Work by the second author has been supported in part by Grant No. MCS 77-02121 from the National Science Foundation.     

A mixture model for responses and response times with a higher-order ability structure to detect rapid guessing behaviour

Many educational and psychological assessments focus on multidimensional latent traits that often have a hierarchical structure to provide both overall-level information and fine-grained diagnostic information. A test will usually have either separate time limits for each subtest or an overall time limit for administrative convenience and test fairness. In order to complete the items within the allocated time, examinees frequently adopt different test-taking behaviours during the test, such as solution behaviour and rapid guessing behaviour. In this paper we propose a new mixture model for responses and response times with a hierarchical ability structure, which incorporates auxiliary information from other subtests and the correlation structure of the abilities to detect rapid guessing behaviour. A Markov chain Monte Carlo method is proposed for model estimation. Simulation studies reveal that all model parameters could be recovered well, and the parameter estimates had smaller absolute bias and mean squared error than the mixture unidimensional item response theory (UIRT) model. Moreover, the true positive rate of detecting rapid guessing behaviour is also higher than when using the mixture UIRT model separately for each subscale, whereas the false detection rate is much lower than the mixture UIRT model. The deviance information criterion and the logarithm of the pseudo-marginal likelihood are employed to evaluate the model fit. Finally, a real data analysis is presented to demonstrate the practical value of the proposed model.     

Selecting the number of classes under latent class regression: a factor analytic analogue

Recently, the regression extension of latent class analysis (RLCA) model has received much attention in the field of medical research. The basic RLCA model summarizes shared features of measured multiple indicators as an underlying categorical variable and incorporates the covariate information in modeling both latent class membership and multiple indicators themselves. To reduce complexity and enhance interpretability, one usually fixes the number of classes in a given RLCA. Often, goodness of fit methods comparing various estimated models are used as a criterion to select the number of classes. In this paper, we propose a new method that is based on an analogous method used in factor analysis and does not require repeated fitting. Two ideas with application to many settings other than ours are synthesized in deriving the method: a connection between latent class models and factor analysis, and techniques of covariate marginalization and elimination. A Monte Carlo simulation study is presented to evaluate the behavior of the selection procedure and compare to alternative approaches. Data from a study of how measured visual impairments affect older persons’ functioning are used for illustration.This work was supported by National Institute on Aging (NIA) Program Project P01-AG-10184-03. The author wishes to thank Dr. Karen Bandeen-Roche for her stimulating comments and helpful discussions, and Drs. Gary Rubin and Sheila West for kindly making the Salisbury Eye Evaluation data available.     

The direct product model for the mtmm matrix parameterized as a second order factor analysis model

The composite direct product model for the multitrait-multimethod matrix is reparameterized as a second-order factor analysis model. This facilitates the use of widely available computer programs such as LISREL and LISCOMP for fitting the model.Bruce Bloxom. Paul Horst and Karl Jöreskog contributed helpful comments to an earlier version of this paper. Their suggestions are gratefully acknowledged.     

Bayesian Factor Analysis as a Variable-Selection Problem: Alternative Priors and Consequences

Factor analysis is a popular statistical technique for multivariate data analysis. Developments in the structural equation modeling framework have enabled the use of hybrid confirmatory/exploratory approaches in which factor-loading structures can be explored relatively flexibly within a confirmatory factor analysis (CFA) framework. Recently, Muthén & Asparouhov proposed a Bayesian structural equation modeling (BSEM) approach to explore the presence of cross loadings in CFA models. We show that the issue of determining factor-loading patterns may be formulated as a Bayesian variable selection problem in which Muthén and Asparouhov's approach can be regarded as a BSEM approach with ridge regression prior (BSEM-RP). We propose another Bayesian approach, denoted herein as the Bayesian structural equation modeling with spike-and-slab prior (BSEM-SSP), which serves as a one-stage alternative to the BSEM-RP. We review the theoretical advantages and disadvantages of both approaches and compare their empirical performance relative to two modification indices-based approaches and exploratory factor analysis with target rotation. A teacher stress scale data set is used to demonstrate our approach.     

MCMC estimation and some model-fit analysis of multidimensional IRT models

A Bayesian procedure to estimate the three-parameter normal ogive model and a generalization of the procedure to a model with multidimensional ability parameters are presented. The procedure is a generalization of a procedure by Albert (1992) for estimating the two-parameter normal ogive model. The procedure supports analyzing data from multiple populations and incomplete designs. It is shown that restrictions can be imposed on the factor matrix for testing specific hypotheses about the ability structure. The technique is illustrated using simulated and real data. The authors would like to thank Norman Verhelst for his valuable comments and ACT, CITO group and SweSAT for the use of their data.     

运用均数与协方差结构模型侦查项目功能差异

阐释了运用多组均数与协方差结构(MACS)模型侦查多级反应项目的一致性与非一致性项目功能差异(DIF)的原理与程序, 以道德自我概念量表DIF的侦查进行示例, 并对该方法进行了评价。与项目反应理论比照, MACS采用系统的、迭代的方式利用修正指数来侦查DIF, 并提供多个拟合指数协同评价模型拟合;与标准验证性因素分析相较, MACS不仅能侦查非一致性DIF, 而且能侦查一致性DIF。运用MACS侦查DIF是一种值得推荐的方法。     

中文读写能力及其相关因素的结构模型

运用问卷调查法考查了中文儿童读写能力及其相关因素之间的关系.通过验证性因素分析对量表的信度和效度指标进行了分析,发现量表具有较好的项目信度和结构效度.结构模型分析发现阅读听写能力影响书面语意义理解和书写技能,而书面语意义理解和书写技能对书面表达有显著影响.家庭阅读背景影响儿童的口语能力和阅读听写能力,口语能力影响意义理解和阅读听写能力,基本知觉加工影响口语能力和阅读听写,动作技能影响书写技能.上述结果表明中文字词识别、书写技能和意义理解水平都影响书面表达,即段落产生能力.而口语能力、家庭阅读背景、基本知觉加工和动作技能分别对读写能力的不同侧面产生影响.本研究还分析了模型的年龄和性别特点.     

Model Fit and Item Factor Analysis: Overfactoring,Underfactoring, and a Program to Guide Interpretation

In exploratory item factor analysis (IFA), researchers may use model fit statistics and commonly invoked fit thresholds to help determine the dimensionality of an assessment. However, these indices and thresholds may mislead as they were developed in a confirmatory framework for models with continuous, not categorical, indicators. The present study used Monte Carlo simulation methods to investigate the ability of popular model fit statistics (chi-square, root mean square error of approximation, the comparative fit index, and the Tucker–Lewis index) and their standard cutoff values to detect the optimal number of latent dimensions underlying sets of dichotomous items. Models were fit to data generated from three-factor population structures that varied in factor loading magnitude, factor intercorrelation magnitude, number of indicators, and whether cross loadings or minor factors were included. The effectiveness of the thresholds varied across fit statistics, and was conditional on many features of the underlying model. Together, results suggest that conventional fit thresholds offer questionable utility in the context of IFA.