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16
A Scaled Difference Chisquare Test Statistic for Moment Structure Analysis
"... A family of scaling corrections aimed to improve the chisquare approximation of goodnessoffit test statistics in small samples, large models, and nonnormal data was proposed in Satorra and Bentler (1994). For structural equations models, SatorraBentler's (SB) scaling corrections are available in ..."
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Cited by 45 (0 self)
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A family of scaling corrections aimed to improve the chisquare approximation of goodnessoffit test statistics in small samples, large models, and nonnormal data was proposed in Satorra and Bentler (1994). For structural equations models, SatorraBentler's (SB) scaling corrections are available in standard computer software. Often, however, the interest is not on the overall fit of a model, but on a test of the restrictions that a null model say M 0 implies on a less restricted one M 1 .IfT 0 and T 1 denote the goodnessoffit test statistics associated to M 0 and M 1 , respectively, then typically the difference T d = T 0 ; T 1 is used as a chisquare test statistic with degrees of freedom equal to the difference on the number of independent parameters estimated under the models M 0 and M 1 . As in the case of the goodnessoffit test, it is of interest to scale the statistic T d in order to improveitschisquare approximation in realistic, i.e., nonasymptotic and nonn...
The robustness of test statistics to nonnormality and specification error in confirmatory factor analysis
 Psychological Methods
, 1996
"... Monte Carlo computer simulations were used to investigate the performance of three X 2 test statistics in confirmatory factor analysis (CFA). Normal theory maximum likelihood)~2 (ML), Browne's asymptotic distribution free X 2 (ADF), and the SatorraBentler rescaled X 2 (SB) were examined under varyi ..."
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Cited by 42 (1 self)
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Monte Carlo computer simulations were used to investigate the performance of three X 2 test statistics in confirmatory factor analysis (CFA). Normal theory maximum likelihood)~2 (ML), Browne's asymptotic distribution free X 2 (ADF), and the SatorraBentler rescaled X 2 (SB) were examined under varying conditions of sample size, model specification, and multivariate distribution. For properly specified models, ML and SB showed no evidence of bias under normal distributions across all sample sizes, whereas ADF was biased at all but the largest sample sizes. ML was increasingly overestimated with increasing nonnormality, but both SB (at all sample sizes) and ADF (only at large sample sizes) showed no evidence of bias. For misspecified models, ML was again inflated with increasing nonnormality, but both SB and ADF were underestimated with increasing nonnormality. It appears that the power of the SB and ADF test statistics to detect a model misspecification is attenuated given nonnormally distributed data. Confirmatory factor analysis (CFA) has become an increasingly popular method of investigating the structure of data sets in psychology. In contrast to traditional exploratory factor analysis that does not place strong a priori restrictions on the structure of the model being tested, CFA requires the investigator to specify both the number of factors
Principles and practice in reporting structural equation analyses
 Psychological Methods
, 2002
"... Principles for reporting analyses using structural equation modeling are reviewed, with the goal of supplying readers with complete and accurate information. It is recommended that every report give a detailed justification of the model used, along with plausible alternatives and an account of ident ..."
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Cited by 29 (0 self)
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Principles for reporting analyses using structural equation modeling are reviewed, with the goal of supplying readers with complete and accurate information. It is recommended that every report give a detailed justification of the model used, along with plausible alternatives and an account of identifiability. Nonnormality and missing data problems should also be addressed. A complete set of parameters and their standard errors is desirable, and it will often be convenient to supply the correlation matrix and discrepancies, as well as goodnessoffit indices, so that readers can exercise independent critical judgment. A survey of fairly representative studies compares recent practice with the principles of reporting recommended here. Structural equation modeling (SEM), also known as path analysis with latent variables, is now a regularly used method for representing dependency (arguably “causal”) relations in multivariate data in the behavioral and social sciences. Following the seminal work of Jöreskog (1973), a number of models for linear structural relations have been developed
Bayesian Estimation and Testing of Structural Equation Models
 Psychometrika
, 1999
"... The Gibbs sampler can be used to obtain samples of arbitrary size from the posterior distribution over the parameters of a structural equation model (SEM) given covariance data and a prior distribution over the parameters. Point estimates, standard deviations and interval estimates for the parameter ..."
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Cited by 27 (8 self)
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The Gibbs sampler can be used to obtain samples of arbitrary size from the posterior distribution over the parameters of a structural equation model (SEM) given covariance data and a prior distribution over the parameters. Point estimates, standard deviations and interval estimates for the parameters can be computed from these samples. If the prior distribution over the parameters is uninformative, the posterior is proportional to the likelihood, and asymptotically the inferences based on the Gibbs sample are the same as those based on the maximum likelihood solution, e.g., output from LISREL or EQS. In small samples, however, the likelihood surface is not Gaussian and in some cases contains local maxima. Nevertheless, the Gibbs sample comes from the correct posterior distribution over the parameters regardless of the sample size and the shape of the likelihood surface. With an informative prior distribution over the parameters, the posterior can be used to make inferences about the parameters of underidentified models, as we illustrate on a simple errorsinvariables model.
Mean and Covariance Structure Analysis: Theoretical and Practical Improvements
, 1995
"... The most widely used multivariate statistical models in the social and behavioral sciences involve linear structural relations among observed and latent variables. In practice, these variables are generally nonnormally distributed, and hence classical multivariate analysis, based on multinormal erro ..."
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Cited by 12 (4 self)
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The most widely used multivariate statistical models in the social and behavioral sciences involve linear structural relations among observed and latent variables. In practice, these variables are generally nonnormally distributed, and hence classical multivariate analysis, based on multinormal errorfree variables having no simultaneous interrelations, is not adequate to deal with such data. Since structural relations among variables imply a structure for the multivariate product moments of the variables, general methods for the analysis of mean and covariance structures have been proposed to estimate and test particular model structures. Unfortunately, extant statistical tests, such as the likelihood ratio test (LRT) and a test based on asymptotically distribution free (ADF) covariance structure analysis, have been found to be virtually useless in practical model evaluation at finite sample sizes with nonnormal data. For example, in one condition of a simulation on confirmatory facto...
The Robustness of LISREL Modeling Revisited
 Structural equation modeling: Present and future: A Festschrift in honor of Karl Jöreskog (pp. 139–168). Chicago: Scientific Software International
, 2001
"... Somer obustness questions in str uctur al equation modeling (SEM) ar intr duced. Factor that a#ect the occuruv ce of nonconver gence and impr: er solutions arr/7 ewed in detail. ..."
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Cited by 6 (2 self)
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Somer obustness questions in str uctur al equation modeling (SEM) ar intr duced. Factor that a#ect the occuruv ce of nonconver gence and impr: er solutions arr/7 ewed in detail.
Corrections to test statistics in principal Hessian directions
, 1999
"... Li’s pHd method uses an asymptotic chisquared test statistic to evaluate a hypothesized dimensionality of a reduceddimension space in a largely nonparametric setting. This statistic is based on an assumed normal distribution of the predictors. When the distributional assumption is violated, a mixtu ..."
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Cited by 3 (0 self)
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Li’s pHd method uses an asymptotic chisquared test statistic to evaluate a hypothesized dimensionality of a reduceddimension space in a largely nonparametric setting. This statistic is based on an assumed normal distribution of the predictors. When the distributional assumption is violated, a mixture chisquared test proposed by Cook is theoretically more appropriate. However, both tests may not perform well with small or intermediate sized nonnormal samples. We propose two corrections to Li’s statistic to enable the chisquared approximation to be more accurate in such samples. The corrections are based on the mean and variance of the statistic of Cook’s mixture distribution. The performance of Li’s, Cook’s, and the two new statistics are compared in some small simulation studies. Results show that one of the new tests performs about as well as Cook’s, while the other performs better than the previously proposed tests.
Structure learning in causal cyclic networks
 In JMLR Workshop and Conference Proceedings
, 2010
"... Cyclic graphical models are unnecessary for accurate representation of joint probability distributions, but are often indispensable when a causal representation of variable relationships is desired. For variables with a cyclic causal dependence structure, DAGs are guaranteed not to recover the corre ..."
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Cyclic graphical models are unnecessary for accurate representation of joint probability distributions, but are often indispensable when a causal representation of variable relationships is desired. For variables with a cyclic causal dependence structure, DAGs are guaranteed not to recover the correct causal structure, and therefore may yield false predictions about the outcomes of perturbations (and even inference.) In this paper, we introduce an approach to generalize Bayesian Network structure learning to structures with cyclic dependence. We introduce a structure learning algorithm, prove its performance given reasonable assumptions, and use simulated data to compare its results to the results of standard Bayesian network structure learning. We then propose a modified, heuristic algorithm with more modest data requirements, and test its performance on a reallife dataset from molecular biology, containing causal, cyclic dependencies. c○2010 S. Itani and M. OhannessianITANI OHANNESSIAN SACHS NOLAN DAHLEH 1.
Some New Test Statistics for Mean and Covariance Structure Analysis with High Dimensional Data
"... Covariance structure analysis is often used for inference and for dimension reduction with high dimensional data. When data is not normally distributed, the asymptotic distribution free (ADF) method is often used to fit a proposed model. This approach uses a weight matrix based on the inverse of the ..."
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Covariance structure analysis is often used for inference and for dimension reduction with high dimensional data. When data is not normally distributed, the asymptotic distribution free (ADF) method is often used to fit a proposed model. This approach uses a weight matrix based on the inverse of the matrix formed by the sample fourth moments and sample covariances. The ADF test statistic is asymptotically distributed as a chisquare variate, but its empirical performance rejects the true model too often at all but impractically large sample sizes. By comparing mean and covariance structure analysis with its peer in the multivariate linear model, we propose some modified ADF test statistics as Ftests whose distributions we approximate using Fdistributions. Empirical studies show that the distributions of the new Ftests are more closely approximated by Fdistributions than are the original ADF statistics when referred to chisquare distributions. Detailed analysis indicates why the AD...
Psicol6gica ('2000.), 21, 301323.
"... This paper analyzes some of the aspects of the problem by means of simulation studies, and proposes a procedure that may be useful for dealing with the problem. The procedure is illustrated by means of an empirical example ..."
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This paper analyzes some of the aspects of the problem by means of simulation studies, and proposes a procedure that may be useful for dealing with the problem. The procedure is illustrated by means of an empirical example