Bayesian estimation of single and multilevel models with latent variable interactions
TL;DR: The Bayesian estimation for single and multilevel SEM models with latent variable interactions is described and it is shown through simulation studies that the Bayesian method outperforms other methods such as the maximum-likelihood method.
Abstract: In this article, we discuss single and multilevel SEM models with latent variable interactions We describe the Bayesian estimation for these models and show through simulation studies that the Bay
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09 Oct 2008
TL;DR: In this paper, the authors show that the classical chi-square goodness-of-fit test is unable to detect the presence of interaction (non-linear) terms in the model and explain this phenomenon by exploiting results on asymptotic robustness (AR) in structural equation models.
Abstract: In this note we show that for some structural equation models (SEM), the classical chi-square goodness-of-fit test is unable to detect the presence of interaction (non-linear) terms in the model. Not only the model test has zero power against that type of misspecifications, but even the theoretical (chi-square) distribution of the test is not distorted when severe interaction term misspecification is present in the postulated model. We explain this phenomenon by exploiting results on asymptotic robustness (AR) in structural equation models. The importance of this paper is to warn against the conclusion that if a proposed linear model fits the data well according to the chi-quare goodness-of-fit test, then the underlying model is linear indeed; it will be shown that the underlying model may in fact be be severely nonlinear. In addition, the present paper shows that such insensitivity to interaction terms is only a particular instance of a more general problem, namely, the incapacity of the classical chi-square goodness-of-fit test to detect deviations from zero correlation among exogenous regressors (either being them observable, or latent) when the structural part of the model is just saturated.
42 citations
TL;DR: This article shows how factor score regression (FSR) can be used to obtain estimates of multilevel mediation, moderation, and nonlinear effects and results were generally promising, which renders FSR an attractive alternative to ML.
Abstract: In the analysis of hierarchical data, multilevel structural equation modeling (multilevel SEM) has become the standard in the social sciences. To estimate these models, maximum likelihood (ML) appr...
18 citations
Cites background from "Bayesian estimation of single and m..."
...Since version 8.3, Bayesian approaches have been available even for multilevel moderation and nonlinear effects (Asparouhov & Muthén, 2020)....
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TL;DR: Information and communication technologies facilitate connectivity to work-related matters after official working hours, and therefore more and more employees engage in technology-assisted supplementa... as discussed by the authors, which is a good idea.
Abstract: Information and communication technologies facilitate connectivity to work-related matters after official working hours. Therefore, more and more employees engage in technology-assisted supplementa...
17 citations
Cites methods from "Bayesian estimation of single and m..."
...We ran this analysis using Bayesian estimation to decrease computational complexity (Asparouhov and Muthén, 2020)....
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...Specifically, there are advantages of an alternative approach to within-level moderation analysis using latent interaction terms (Asparouhov and Muthén, 2020; Preacher et al., 2016)....
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8 citations
TL;DR: In this article , the authors provide recommendations, tools, resources, and a checklist that can be useful for scholars involved in conducting or assessing multilevel studies, in which Level-1 entities are neatly nested within Level-2 entities, and top-down effects are estimated.
Abstract: Multilevel methods allow researchers to investigate relationships that expand across levels (e.g., individuals, teams, and organizations). The popularity of these methods for studying organizational phenomena has increased in recent decades. Methodologists have examined how these methods work under different conditions, providing an empirical base for making sound decisions when using these methods. In this article, we provide recommendations, tools, resources, and a checklist that can be useful for scholars involved in conducting or assessing multilevel studies. The focus of our article is on two-level designs, in which Level-1 entities are neatly nested within Level-2 entities, and top-down effects are estimated. However, some of our recommendations are also applicable to more complex multilevel designs.
7 citations
References
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01 Jan 2016
TL;DR: The introduction to multivariate statistical analysis is universally compatible with any devices to read, and will help you to cope with some harmful bugs inside their desktop computer.
Abstract: Thank you for downloading introduction to multivariate statistical analysis. Maybe you have knowledge that, people have look hundreds times for their favorite books like this introduction to multivariate statistical analysis, but end up in infectious downloads. Rather than reading a good book with a cup of tea in the afternoon, instead they cope with some harmful bugs inside their desktop computer. introduction to multivariate statistical analysis is available in our digital library an online access to it is set as public so you can get it instantly. Our books collection hosts in multiple countries, allowing you to get the most less latency time to download any of our books like this one. Merely said, the introduction to multivariate statistical analysis is universally compatible with any devices to read.
3,304 citations
Additional excerpts
...…the mean of a product EðXj XkÞ ¼ CovðXj;XkÞ: (5) Assuming multivariate normality for four random variables Xi, Xj, Xk, Xl any third-order moment about the mean (μ) is zero (see, e.g., Anderson, 1984), EððXi μiÞðXj μjÞðXk μkÞÞ ¼ 0; (6) while the fourth-order moment about the mean is a function…...
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TL;DR: The Latent Moderated Structural Equations (LMS) as mentioned in this paper approach is a new method developed for the analysis of the general interaction model that utilizes the mixture distribution and provides a ML estimation of model parameters by adapting the EM algorithm.
Abstract: In the context of structural equation modeling, a general interaction model with multiple latent interaction effects is introduced. A stochastic analysis represents the nonnormal distribution of the joint indicator vector as a finite mixture of normal distributions. The Latent Moderated Structural Equations (LMS) approach is a new method developed for the analysis of the general interaction model that utilizes the mixture distribution and provides a ML estimation of model parameters by adapting the EM algorithm. The finite sample properties and the robustness of LMS are discussed. Finally, the applicability of the new method is illustrated by an empirical example.
1,122 citations
"Bayesian estimation of single and m..." refers methods in this paper
...For ease of interpretation, the regression Equation (1) can be re-written in the equivalent form η3 ¼ ðβ1 þ β3 η2Þ η1 þ β2 η2 þ ζ3; (2) where ðβ1 þ β3 η2Þ is a moderator function (Klein & Moosbrugger, 2000) so that the β1 strength of influence of η1 on η3 is moderated by β3 η2....
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...Interactions between two latent variables, however, do not lead to closed form expressions for the likelihood and cannot be estimated with the maximum-likelihood method without numerical integration, see Klein and Moosbrugger (2000)....
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...In the single-level context, the maximum-likelihood (ML) estimation is described in Klein and Moosbrugger (2000)....
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...Interactions between two latent variables, however, do not lead to closed form expressions for the likelihood and cannot be estimated with the maximum-likelihood method without numerical integration, see Klein and Moosbrugger (2000). In Mplus, the dimension of numerical integration corresponds to the number of latent variables used in the interaction terms....
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...The method is asymptotically equivalent to the LMS method discussed in Klein and Moosbrugger (2000)....
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TL;DR: This article proposes a new approach to factor analysis and structural equation modeling using Bayesian analysis, which replaces parameter specifications of exact zeros with approximate zeros based on informative, small-variance priors.
Abstract: This article proposes a new approach to factor analysis and structural equation modeling using Bayesian analysis. The new approach replaces parameter specifications of exact zeros with approximate zeros based on informative, small-variance priors. It is argued that this produces an analysis that better reflects substantive theories. The proposed Bayesian approach is particularly beneficial in applications where parameters are added to a conventional model such that a nonidentified model is obtained if maximum-likelihood estimation is applied. This approach is useful for measurement aspects of latent variable modeling, such as with confirmatory factor analysis, and the measurement part of structural equation modeling. Two application areas are studied, cross-loadings and residual correlations in confirmatory factor analysis. An example using a full structural equation model is also presented, showing an efficient way to find model misspecification. The approach encompasses 3 elements: model testing using posterior predictive checking, model estimation, and model modification. Monte Carlo simulations and real data are analyzed using Mplus. The real-data analyses use data from Holzinger and Swineford's (1939) classic mental abilities study, Big Five personality factor data from a British survey, and science achievement data from the National Educational Longitudinal Study of 1988.
1,045 citations
TL;DR: The traditional constrained approach performed more poorly than did 3 new approaches--unconstrained, generalized appended product indicator, and quasi-maximum-likelihood (QML); the authors' new unconstrained approach was easiest to apply.
Abstract: Interactions between (multiple indicator) latent variables are rarely used because of implementation complexity and competing strategies. Based on 4 simulation studies, the traditional constrained approach performed more poorly than did 3 new approaches--unconstrained, generalized appended product indicator, and quasi-maximum-likelihood (QML). The authors' new unconstrained approach was easiest to apply. All 4 approaches were relatively unbiased for normally distributed indicators, but the constrained and QML approaches were more biased for nonnormal data; the size and direction of the bias varied with the distribution but not with the sample size. QML had more power, but this advantage was qualified by consistently higher Type I error rates. The authors also compared general strategies for defining product indicators to represent the latent interaction factor.
891 citations
"Bayesian estimation of single and m..." refers methods in this paper
...For an overview of the ML approach and various estimators suggested in earlier work, see Marsh et al. (2004)....
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TL;DR: A new multilevel latent covariate (MLC) approach is introduced that corrects for unreliability at L2 and results in unbiased estimates of L2 constructs under appropriate conditions and suggests when researchers should most appropriately use one, the other, or a combination of both approaches.
Abstract: In multilevel modeling (MLM), group-level (L2) characteristics are often measured by aggregating individual-level (L1) characteristics within each group so as to assess contextual effects (e.g., group-average effects of socioeconomic status, achievement, climate). Most previous applications have used a multilevel manifest covariate (MMC) approach, in which the observed (manifest) group mean is assumed to be perfectly reliable. This article demonstrates mathematically and with simulation results that this MMC approach can result in substantially biased estimates of contextual effects and can substantially underestimate the associated standard errors, depending on the number of L1 individuals per group, the number of groups, the intraclass correlation, the sampling ratio (the percentage of cases within each group sampled), and the nature of the data. To address this pervasive problem, the authors introduce a new multilevel latent covariate (MLC) approach that corrects for unreliability at L2 and results in unbiased estimates of L2 constructs under appropriate conditions. However, under some circumstances when the sampling ratio approaches 100%, the MMC approach provides more accurate estimates. Based on 3 simulations and 2 real-data applications, the authors evaluate the MMC and MLC approaches and suggest when researchers should most appropriately use one, the other, or a combination of both approaches.
607 citations
"Bayesian estimation of single and m..." refers background or result in this paper
...These findings are in line with the findings in Lüdtke et al. (2008) and Asparouhov and Muthén (2019) regarding latent variable centering for predictors in multilevel modeling....
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...E(η3) = β1 0 + β2 0 + β3 E(η1 η2) + 0 (12) = β3 Cov(η1, η2)....
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...That bias is generally referred to as Lüdtke’s bias, see Lüdtke et al. (2008) and Asparouhov and Muthén (2019)....
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