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Peter M Bentler - One of the best experts on this subject based on the ideXlab platform.
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mean and Covariance Structure analysis theoretical and practical improvements
Department of Statistics UCLA, 2011Co-Authors: Kehai Yuan, Peter M BentlerAbstract:UCLA Statistical Series Report No. 194 Mean and Covariance Structure Analysis: Theoretical and Practical Improvements Ke-Hai Yuan and Peter M. Bentler June 21, 1995 Ke-Hai Yuan is Statistician, and Peter M. Bentler is Professor, Department of Psychology and Center for Statistics, University of California, Los Angeles, Box 951563, Los Angeles, CA 90095-1563. This work was supported by National Institute on Drug Abuse Grants DA01070 and DA00017.
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Normal theory likelihood ratio statistic for mean and Covariance Structure analysis under alternative hypotheses
Journal of Multivariate Analysis, 2007Co-Authors: Kehai Yuan, Kentaro Hayashi, Peter M BentlerAbstract:The normal distribution based likelihood ratio (LR) statistic is widely used in structural equation modeling. Under a sequence of local alternative hypotheses, this statistic has been shown to asymptotically follow a noncentral chi-square distribution. In practice, the population mean vector and Covariance matrix as well as the model and sample size are always fixed. It is hard to justify the validity of the noncentral chi-square distribution for the resulting LR statistic even when data are normally distributed and sample size is large. By extending results in the literature, this paper develops normal distributions to describe the behavior of the LR statistic for mean and Covariance Structure analysis. A sequence of local alternative hypotheses is not necessary for the proposed distributions to be asymptotically valid. When the effect size is medium and above or when the model is not trivially misspecified, empirical results indicate that a refined normal distribution describes the behavior of the LR statistic better than the commonly used noncentral chi-square distribution, as measured by the Kolmogorov-Smirnov distance. Quantile-quantile plots are also provided to better understand the different distributions.
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the effect of skewness and kurtosis on mean and Covariance Structure analysis the univariate case and its multivariate implication
Sociological Methods & Research, 2005Co-Authors: Kehai Yuan, Peter M Bentler, Wei ZhangAbstract:The maximum likelihood (ML) method, based on the normal distribution assumption, is widely used in mean and Covariance Structure analysis. With typical nonnormal data, the ML method will lead to biased statistics and inappropriate scientific conclusions. This article develops a simple but informative case to show how ML results are influenced by skewness and kurtosis. Specifically, the authors discuss how skewness and kurtosis in a univariate distribution affect the standard errors of the ML estimators, the Covariances between the estimators, and the likelihood ratio test of hypotheses on mean and variance parameters. They also describe corrections that have been developed to allow appropriate inference. Enough details are provided so that this material can be used in graduate instruction. For each result, the corresponding results in the higher dimensional case are pointed out, and references are provided.
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on chi square difference and z tests in mean and Covariance Structure analysis when the base model is misspecified
Educational and Psychological Measurement, 2004Co-Authors: Kehai Yuan, Peter M BentlerAbstract:In mean and Covariance Structure analysis, the chi-square difference test is often applied to evaluate the number of factors, cross-group constraints, and other nested model comparisons. Let model Ma be the base model within which model Mb is nested. In practice, this test is commonly used to justify Mb even when Ma is misspecified. The authors study the behavior of the chi-square difference test in such a circumstance. Monte Carlo results indicate that a nonsignificant chi-square difference cannot be used to justify the constraints in Mb. They also show that when the base model is misspecified, the z test for the statistical significance of a parameter estimate can also be misleading. For specific models, the analysis further shows that the intercept and slope parameters in growth curve models can be estimated consistently even when the Covariance Structure is misspecified, but only in linear growth models. Similarly, with misspecified Covariance Structures, the mean parameters in multiple group models can be estimated consistently under null conditions.
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three likelihood based methods for mean and Covariance Structure analysis with nonnormal missing data
Sociological Methodology, 2000Co-Authors: Kehai Yuan, Peter M BentlerAbstract:Survey and longitudinal studies in the social and behavioral sciences generally contain missing data. Mean and Covariance Structure models play an important role in analyzing such data. Two promising methods for dealing with missing data are a direct maximum-likelihood and a two-stage approach based on the unStructured mean and Covariance estimates obtained by the EM-algorithm. Typical assumptions under these two methods are ignorable nonresponse and normality of data. However, data sets in social and behavioral sciences are seldom normal, and experience with these procedures indicates that normal theory based methods for nonnormal data very often lead to incorrect model evaluations. By dropping the normal distribution assumption, we develop more accurate procedures for model inference. Based on the theory of generalized estimating equations, a way to obtain consistent standard errors of the two-stage estimates is given. The asymptotic efficiencies of different estimators are compared under various assump...
Robert C Maccallum - One of the best experts on this subject based on the ideXlab platform.
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testing differences between nested Covariance Structure models power analysis and null hypotheses
Psychological Methods, 2006Co-Authors: Robert C Maccallum, Michael W Browne, Li CaiAbstract:For comparing nested Covariance Structure models, the standard procedure is the likelihood ratio test of the difference in fit, where the null hypothesis is that the models fit identically in the population. A procedure for determining statistical power of this test is presented where effect size is based on a specified difference in overall fit of the models. A modification of the standard null hypothesis of zero difference in fit is proposed allowing for testing an interval hypothesis that the difference in fit between models is small, rather than zero. These developments are combined yielding a procedure for estimating power of a test of a null hypothesis of small difference in fit versus an alternative hypothesis of larger difference.
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power analysis in Covariance Structure modeling using gfi and agfi
Multivariate Behavioral Research, 1997Co-Authors: Robert C Maccallum, Sehee HongAbstract:Extending earlier work by MacCallum, Browne, and Sugawara (1996), procedures are shown for conducting power analysis of tests of overall fit of Covariance Structure models when null and alternative levels of model fit are specified in terms of values of the GFI or AGFl fit indexes. Results show that for GFI-based power analyses, holding null and alternative values of GFI fixed, power decreases as degrees of freedom increase, which is a counter-intuitive and undesirable phenomenon indicating lower power for detecting false null hypotheses about simpler models. For AGFI-based analyses, power increases as degrees of freedom increase. However, for both indexes it is shown that it is problematic to establish consistently appropriate values for null and alternative hypotheses about model fit. Because of these problems, it is recommended that the RMSEA index is preferable as a basis for power analysis and model evaluation.
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power analysis and determination of sample size for Covariance Structure modeling
Psychological Methods, 1996Co-Authors: Robert C Maccallum, Michael W Browne, Hazuki M SugawaraAbstract:A framework for hypothesis testing and power analysis in the assessment of fit of Covariance Structure models is presented. We emphasize the value of confidence intervals for fit indices, and we stress the relationship of confidence intervals to a framework for hypothesis testing. The approach allows for testing null hypotheses of not-good fit, reversing the role of the null hypothesis in conventional tests of model fit, so that a significant result provides strong support for good fit. The approach also allows for direct estimation of power, where effect size is defined in terms of a null and alternative value of the root-mean-square error of approximation fit index proposed by J. H. Steiger and J. M. Lind (1980). It is also feasible to determine minimum sample size required to achieve a given level of power for any test of fit in this framework. Computer programs and examples are provided for power analyses and calculation of minimum sample sizes.
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alternative strategies for cross validation of Covariance Structure models
Multivariate Behavioral Research, 1994Co-Authors: Robert C Maccallum, Mary Roznowski, Corinne M Mar, Janet ReithAbstract:Alternative strategies for two-sample cross-validation of Covariance Structure models are described and investigated. The strategies vary according to whether all (tight strategy) or some (partial strategy) of the model parameters are held constant when a calibration sample solution is re-fit to a validation sample Covariance matrix. Justification is provided for three partial strategies. Conventional and alternative strategies for cross-validation are discussed as methods for evaluating overall discrepancy of a model fit to a particular sample, where overall discrepancy arises from the combined influences of discrepancy of approximation and discrepancy of estimation (Cudeck & Henly, 1991). Results of a sampling study using empirical data show that for tighter strategies simpler models are preferred in smaller samples. However, when partial cross-validation is employed, a more complex model may be supported even in a small sample. Implications for model comparison and evaluation, as well as the issues of model complexity and sample size are discussed.
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effect of estimation method on incremental fit indexes for Covariance Structure models
Applied Psychological Measurement, 1993Co-Authors: Hazuki M Sugawara, Robert C MaccallumAbstract:In a typical study involving Covariance Structure modeling, fit of a model or a set of alternative models is evaluated using several indicators of fit under one estimation method, usually maximum likelihood. This study examined the stability across estimation methods of incremental and non incremental fit measures that use the information about the fit of the most restricted (null) model as a reference point in assessing the fit of a more substantive model to the data. A set of alternative models for a large empirical dataset was analyzed by asymptotically distribution-free, generalized least squares, maximum likelihood, and ordinary least squares estimation methods. Four incremental and four nonincremental fit indexes were com pared. Incremental indexes were quite unstable across estimation methods—maximum likelihood and ordinary least squares solutions indicated better fit of a given model than asymptotically distribution-free and generalized least squares solu tions. The cause of this phenomenon is exp...
Kehai Yuan - One of the best experts on this subject based on the ideXlab platform.
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mean and Covariance Structure analysis theoretical and practical improvements
Department of Statistics UCLA, 2011Co-Authors: Kehai Yuan, Peter M BentlerAbstract:UCLA Statistical Series Report No. 194 Mean and Covariance Structure Analysis: Theoretical and Practical Improvements Ke-Hai Yuan and Peter M. Bentler June 21, 1995 Ke-Hai Yuan is Statistician, and Peter M. Bentler is Professor, Department of Psychology and Center for Statistics, University of California, Los Angeles, Box 951563, Los Angeles, CA 90095-1563. This work was supported by National Institute on Drug Abuse Grants DA01070 and DA00017.
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Normal theory likelihood ratio statistic for mean and Covariance Structure analysis under alternative hypotheses
Journal of Multivariate Analysis, 2007Co-Authors: Kehai Yuan, Kentaro Hayashi, Peter M BentlerAbstract:The normal distribution based likelihood ratio (LR) statistic is widely used in structural equation modeling. Under a sequence of local alternative hypotheses, this statistic has been shown to asymptotically follow a noncentral chi-square distribution. In practice, the population mean vector and Covariance matrix as well as the model and sample size are always fixed. It is hard to justify the validity of the noncentral chi-square distribution for the resulting LR statistic even when data are normally distributed and sample size is large. By extending results in the literature, this paper develops normal distributions to describe the behavior of the LR statistic for mean and Covariance Structure analysis. A sequence of local alternative hypotheses is not necessary for the proposed distributions to be asymptotically valid. When the effect size is medium and above or when the model is not trivially misspecified, empirical results indicate that a refined normal distribution describes the behavior of the LR statistic better than the commonly used noncentral chi-square distribution, as measured by the Kolmogorov-Smirnov distance. Quantile-quantile plots are also provided to better understand the different distributions.
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the effect of skewness and kurtosis on mean and Covariance Structure analysis the univariate case and its multivariate implication
Sociological Methods & Research, 2005Co-Authors: Kehai Yuan, Peter M Bentler, Wei ZhangAbstract:The maximum likelihood (ML) method, based on the normal distribution assumption, is widely used in mean and Covariance Structure analysis. With typical nonnormal data, the ML method will lead to biased statistics and inappropriate scientific conclusions. This article develops a simple but informative case to show how ML results are influenced by skewness and kurtosis. Specifically, the authors discuss how skewness and kurtosis in a univariate distribution affect the standard errors of the ML estimators, the Covariances between the estimators, and the likelihood ratio test of hypotheses on mean and variance parameters. They also describe corrections that have been developed to allow appropriate inference. Enough details are provided so that this material can be used in graduate instruction. For each result, the corresponding results in the higher dimensional case are pointed out, and references are provided.
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on chi square difference and z tests in mean and Covariance Structure analysis when the base model is misspecified
Educational and Psychological Measurement, 2004Co-Authors: Kehai Yuan, Peter M BentlerAbstract:In mean and Covariance Structure analysis, the chi-square difference test is often applied to evaluate the number of factors, cross-group constraints, and other nested model comparisons. Let model Ma be the base model within which model Mb is nested. In practice, this test is commonly used to justify Mb even when Ma is misspecified. The authors study the behavior of the chi-square difference test in such a circumstance. Monte Carlo results indicate that a nonsignificant chi-square difference cannot be used to justify the constraints in Mb. They also show that when the base model is misspecified, the z test for the statistical significance of a parameter estimate can also be misleading. For specific models, the analysis further shows that the intercept and slope parameters in growth curve models can be estimated consistently even when the Covariance Structure is misspecified, but only in linear growth models. Similarly, with misspecified Covariance Structures, the mean parameters in multiple group models can be estimated consistently under null conditions.
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three likelihood based methods for mean and Covariance Structure analysis with nonnormal missing data
Sociological Methodology, 2000Co-Authors: Kehai Yuan, Peter M BentlerAbstract:Survey and longitudinal studies in the social and behavioral sciences generally contain missing data. Mean and Covariance Structure models play an important role in analyzing such data. Two promising methods for dealing with missing data are a direct maximum-likelihood and a two-stage approach based on the unStructured mean and Covariance estimates obtained by the EM-algorithm. Typical assumptions under these two methods are ignorable nonresponse and normality of data. However, data sets in social and behavioral sciences are seldom normal, and experience with these procedures indicates that normal theory based methods for nonnormal data very often lead to incorrect model evaluations. By dropping the normal distribution assumption, we develop more accurate procedures for model inference. Based on the theory of generalized estimating equations, a way to obtain consistent standard errors of the two-stage estimates is given. The asymptotic efficiencies of different estimators are compared under various assump...
Michael D Coovert - One of the best experts on this subject based on the ideXlab platform.
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the tripartite influence model of body image and eating disturbance a Covariance Structure modeling investigation testing the mediational role of appearance comparison
Journal of Psychosomatic Research, 2002Co-Authors: Patricia Van Den Berg, Kevin J Thompson, Karen Obremskibrandon, Michael D CoovertAbstract:Recent theoretical approaches to the etiology of eating disorders and body image disturbances have begun to focus on multifactorial models. In the current study, the Tripartite Influence model was examined in a large sample of college females (ages 18–22). This model proposes that three primary core sources of influence— parents, peers and media—contribute to the development of body imageandeatingdisturbances.Additionally,themodelsuggests that at least two factors mediate the relationship between influences and disturbance—appearance comparison and internalization of media information. In this study, appearance comparison was examined as a mediational link between peer, family and media influence variables and the outcome disturbance measures of eating dysfunction and body image dissatisfaction. Covariance Structure modeling (CSM)wasusedtotesttheproposedpathways.Theresultsindicated thatappearancecomparisonmediatedtheeffectsoffamilyandmedia influences on body dissatisfaction, which in turn influenced restrictive and bulimic behaviors. In addition, peer influences had a direct influence on restriction. Perfectionism was hypothesized to relate to body dissatisfaction, but was in fact found to influence appearance comparison. The findings were limited by the necessity of several modifications to the originally proposed models, yet offer replication and extension of previous work with appearance comparison and support for further testing of the Tripartite Influence model. D 2002 Elsevier Science Inc. All rights reserved.
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the tripartite influence model of body image and eating disturbance a Covariance Structure modeling investigation testing the mediational role of appearance comparison
Journal of Psychosomatic Research, 2002Co-Authors: Patricia Van Den Berg, Karen Obremskibrandon, Kevin J Thompson, Michael D CoovertAbstract:Recent theoretical approaches to the etiology of eating disorders and body image disturbances have begun to focus on multifactorial models. In the current study, the Tripartite Influence model was examined in a large sample of college females (ages 18-22). This model proposes that three primary core sources of influence--parents, peers and media--contribute to the development of body image and eating disturbances. Additionally, the model suggests that at least two factors mediate the relationship between influences and disturbance-appearance comparison and internalization of media information. In this study, appearance comparison was examined as a mediational link between peer, family and media influence variables and the outcome disturbance measures of eating dysfunction and body image dissatisfaction. Covariance Structure modeling (CSM) was used to test the proposed pathways. The results indicated that appearance comparison mediated the effects of family and media influences on body dissatisfaction, which in turn influenced restrictive and bulimic behaviors. In addition, peer influences had a direct influence on restriction. Perfectionism was hypothesized to relate to body dissatisfaction, but was in fact found to influence appearance comparison. The findings were limited by the necessity of several modifications to the originally proposed models, yet offer replication and extension of previous work with appearance comparison and support for further testing of the Tripartite Influence model.
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body image social comparison and eating disturbance a Covariance Structure modeling investigation
International Journal of Eating Disorders, 1999Co-Authors: Kevin J Thompson, Michael D Coovert, Susan M StormerAbstract:Objective To evaluate the role of appearance-based social comparison processes as a possible mediational link between developmental factors (negative social feedback, i.e., teasing about appearance; biological status, i.e.,early physical maturation) and levels of body dissatisfaction, eating disturbance, and global psychological functioning. Method In this study of 173 female undergraduates, Covariance Structure modeling (CSM) was used to examine direct and mediational relationships among these variables. Results Appearance-based social comparison mediated the effect of appearance-related teasing on body image and eating disturbance; body image mediated the effect of teasing on eating disturbance; eating disturbance had a direct effect on overall psychological functioning. Discussion Implications for devising and testing other theoretical models are discussed. © 1999 John Wiley & Sons, Inc. Int J Eat Disord 26: 43–51, 1999.
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development of body image eating disturbance and general psychological functioning in female adolescents Covariance Structure modeling and longitudinal investigations
International Journal of Eating Disorders, 1995Co-Authors: Kevin J Thompson, Michael D Coovert, Kevin J Richards, Sylvia Johnson, M Jill A CattarinAbstract:Objective: Covariance Structure modeling (CSM) and longitudinal analyses were used to identify a possible causal sequence for factors that might lead to the development of body image and eating disturbance in adolescent females. Method: In Study 1, subjects were measured for level of obesity, perceived weight status, maturational timing, history of being teased about weight/size, body image, eating disturbance, and global psychological functioning. Study 2 was a systematic replication of Study 1 using different indices of body image and a more comprehensive assessment of eating disturbance. In Study 3, subjects from Study 1 were reassessed at a 3-year follow-up. Results: The findings from Study 1 indicated that level of obesity and perceived weight, but not maturational timing, had a directional influence on the other variables. In addition, teasing history was significantly related to the development of body image and eating disturbance. CSM and path analyses with Studies 2 and 3 data generally replicated and extended the results from Study 1. Discussion: These findings offer important insights into possible causal sequences for the development of body image and eating disturbance. © 1995 by John Wiley & Sons, Inc.
Peter Guttorp - One of the best experts on this subject based on the ideXlab platform.
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considering covariates in the Covariance Structure of spatial processes
Environmetrics, 2011Co-Authors: Alexandra M Schmidt, Peter Guttorp, Anthony OhaganAbstract:In spatial statistics one usually assumes that observations are partial realizations of a stochastic process , where commonly C = 2, and the components of the location vector x are geographical coordinates. Frequently, it is assumed that follows a Gaussian process (GP) with stationary Covariance Structure. In this setting the usual aim is to make spatial interpolation to unobserved locations of interest, based on observed values at monitored locations. This interpolation is heavily based on the specification of the mean and Covariance Structure of the GP. In environmental problems the assumption of stationary Covariance Structures is commonly violated due to local influences in the Covariance Structure of the process. We propose models which relax the assumption of stationary GP by accounting for covariate information in the Covariance Structure of the process. Usually at each location x, covariates related to are also observed. We initially propose the use of covariates to allow the latent space model of Sampson and Guttorp to be of dimension C > 2. Then we discuss a particular case of the latent space model by using a representation projected down from C dimensions to 2 in order to model the 2D correlation Structure better. Inference is performed under the Bayesian paradigm, and Markov chain Monte Carlo methods are used to obtain samples from the resultant posterior distributions under each model. As illustration of the proposed models, we analyse solar radiation in British Columbia, and mean temperature in Colorado. Copyright © 2011 John Wiley & Sons, Ltd.
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nonparametric estimation of nonstationary spatial Covariance Structure
Journal of the American Statistical Association, 1992Co-Authors: Paul D Sampson, Peter GuttorpAbstract:Abstract Estimation of the Covariance Structure of spatial processes is a fundamental prerequisite for problems of spatial interpolation and the design of monitoring networks. We introduce a nonparametric approach to global estimation of the spatial Covariance Structure of a random function Z(x, t) observed repeatedly at times ti (i = 1, …, T) at a finite number of sampling stations xi (i = 1, 2, …, N) in the plane. Our analyses assume temporal stationarity but do not assume spatial stationarity (or isotropy). We analyze the spatial dispersions var(Z(xi, t) − Z(xj, t)) as a natural metric for the spatial Covariance Structure and model these as a general smooth function of the geographic coordinates of station pairs (xi, xj ). The model is constructed in two steps. First, using nonmetric multidimensional scaling (MDS) we compute a two-dimensional representation of the sampling stations for which a monotone function of interpoint distances δij approximates the spatial dispersions. MDS transforms the problem...
