An EM algorithm for fitting two-level structural equation models

Liang, Jiajuan; Bentler, Peter M.

doi:10.1007/bf02295842

Cited by 60 publications

(58 citation statements)

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“…For continuous data, maximum likelihood estimation has been proposed for unbalanced multilevel designs with missing items (Longford & Muthén, 1992), for example using an EM algorithm (Raudenbush, 1995;Lee & Tsang, 1999) or a generalization of the iterated generalized least squares algorithm (Yang, Pickles, & Taylor, 1999;Yang & Pickles, 2004). An EM algorithm for unbalanced continuous two-level data has recently been implemented in EQS (e.g., Liang & Bentler, 2003). For binary data, MCMC methods have been proposed by Ansari and Jedidi (2000) and Fox and Glas (2001).…”

Section: Discussionmentioning

confidence: 99%

Generalized multilevel structural equation modeling

2004

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A unifying framework for generalized multilevel structural equation modeling is introduced. The models in the framework, called generalized linear latent and mixed models (GLLAMM), combine features of generalized linear mixed models (GLMM) and structural equation models (SEM) and consist of a response model and a structural model for the latent variables. The response model generalizes GLMMs to incorporate factor structures in addition to random intercepts and coefficients. As in GLMMs, the data can have an arbitrary number of levels and can be highly unbalanced with different numbers of lower-level units in the higher-level units and missing data. A wide range of response processes can be modeled including ordered and unordered categorical responses, counts, and responses of mixed types. The structural model is similar to the structural part of a SEM except that it may include latent and observed variables varying at different levels. For example, unit-level latent variables (factors or random coefficients) can be regressed on cluster-level latent variables. Special cases of this framework are explored and data from the British Social Attitudes Survey are used for illustration. Maximum likelihood estimation and empirical Bayes latent score prediction within the GLLAMM framework can be performed using adaptive quadrature in gllamm, a freely available program running in Stata.

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Section: Discussionmentioning

confidence: 99%

Generalized multilevel structural equation modeling

2004

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“…The standard approach of assessing overall fit has this limitation because the sample size is usually much larger at the within-group level than at the betweengroup level. The ML fitting function for a two-level multilevel SEM is written as follows (Bentler & Liang, 2003;Liang & Bentler, 2004):…”

Section: Level-specific ML Test Statisticmentioning

confidence: 99%

Effects of skewness and kurtosis on normal-theory based maximum likelihood test statistic in multilevel structural equation modeling

Ryu

2011

Behav Res

155

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A simulation study investigated the effects of skewness and kurtosis on level-specific maximum likelihood (ML) test statistics based on normal theory in multilevel structural equation models. The levels of skewness and kurtosis at each level were manipulated in multilevel data, and the effects of skewness and kurtosis on level-specific ML test statistics were examined. When the assumption of multivariate normality was violated, the level-specific ML test statistics were inflated, resulting in Type I error rates that were higher than the nominal level for the correctly specified model. Q-Q plots of the test statistics against a theoretical chi-square distribution showed that skewness led to a thicker upper tail and kurtosis led to a longer upper tail of the observed distribution of the level-specific ML test statistic for the correctly specified model. Keywords Maximum likelihood test statistic . Violation of multivariate normality . Multilevel structural equation modelingStructural equation modeling (SEM) has become an important and widely used analysis approach in social and behavioral science over the past 3 decades (Bollen, 2002;MacCallum & Austin, 2000). SEM (Bentler, 1980;Jöreskog, 1978) is a general multivariate technique that accounts for the means, variances, and covariances of a set of variables in terms of a smaller number of parameters associated with a hypothesized model.One of the assumptions made in SEM is that the observations are independent. When data have multilevel structure such that individuals are nested within clusters, the independence assumption typically is violated, because individuals in the same cluster are likely to be more homogeneous than those from different clusters. Multilevel SEM has been developed to apply SEM to multilevel data, dealing with non independent observations by explicitly modeling the clustered nature of multilevel data (Goldstein & McDonald, 1988;Lee, 1990;Longford & Muthén, 1992;Muthén, 1990Muthén, , 1994Muthén & Satorra, 1995).The typical application of SEM and multilevel SEM has interest in both (1) assessing the goodness of fit of the model to the data and (2) estimating and testing individual parameters in the hypothesized model. The most widely used method for estimation and testing is maximum likelihood (ML) estimation based on normal theory. ML estimation finds the set of parameter estimates that maximizes the likelihood that the data will be observed, given that the data have a multivariate normal distribution in the population. ML estimation also provides the likelihood ratio test statistic, which provides a test of whether the hypothesized model fits the data. The multivariate normality assumption is important for the ML estimator to achieve its asymptotic properties and for statistical inference to be valid.The present study investigated how skewness and kurtosis affect the performance of the normal-theory ML test statistic for overall model fit in multilevel structural equation models, under violations of the multivariate normality assumpti...

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“…Some recent technical and computational developments have enabled the implementation of full information maximum likelihood (FIML) approaches to estimating disaggregated structural equation models with multilevel data (e.g., du Toit & du Toit, 2007;Lee & Poon, 1998;Liang & Bentler, 2004). In this article we use a large, nationally representative sample of law students to test substantive theory on the role of educational diversity in higher education settings and to identify some existing practical limitations of employing disaggregated multilevel structural equation models within a real-world data situation.…”

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confidence: 99%

“…A number of quantitative researchers have published expository papers demonstrating empirical examples of the use of the MSEM procedure (e.g., Kaplan & Elliott, 1997;Liang & Bentler, 2004;B. O. Muthén, 1994;Muthén, Khoo, & Gustafsson, 1997;Skrondal & Rabe-Hesketh, 2004).…”

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confidence: 99%

The Effects of Educational Diversity in a National Sample of Law Students: Fitting Multilevel Latent Variable Models in Data With Categorical Indicators

Gottfredson

Panter

Daye

et al. 2009

Multivariate Behavioral Research

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Controversy surrounding the use of race-conscious admissions can be partially resolved with improved empirical knowledge of the effects of racial diversity in educational settings. We use a national sample of law students nested in 64 law schools to test the complex and largely untested theory regarding the effects of educational diversity on student outcomes. Social scientists who study these outcomes frequently encounter both latent variables and nested data within a single analysis. Yet, until recently, an appropriate modeling technique has been computationally infeasible, and consequently few applied researchers have estimated appropriate models to test their theories, sometimes limiting the scope of their research question. Our results, based on disaggregated multilevel structural equation models, show that racial diversity is related to a reduction in prejudiced attitudes and

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An EM algorithm for fitting two-level structural equation models

Abstract: Chi-square statistic, mean and covariance structures, EM algorithm, maximum likelihood, multivariate normal distribution, two-level structural equation models,

Cited by 60 publications

References 19 publications

Generalized multilevel structural equation modeling

Generalized multilevel structural equation modeling

Effects of skewness and kurtosis on normal-theory based maximum likelihood test statistic in multilevel structural equation modeling

The Effects of Educational Diversity in a National Sample of Law Students: Fitting Multilevel Latent Variable Models in Data With Categorical Indicators

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