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The integration of continuous and discrete latent variable models: Potential problems and promising opportunities
 Psychological Methods
, 2004
"... Structural equation mixture modeling (SEMM) integrates continuous and discrete latent variable models. Drawing on prior research on the relationships between continuous and discrete latent variable models, the authors identify 3 conditions that may lead to the estimation of spurious latent classes i ..."
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Cited by 48 (6 self)
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Structural equation mixture modeling (SEMM) integrates continuous and discrete latent variable models. Drawing on prior research on the relationships between continuous and discrete latent variable models, the authors identify 3 conditions that may lead to the estimation of spurious latent classes in SEMM: misspecification of the structural model, nonnormal continuous measures, and nonlinear relationships among observed and/or latent variables. When the objective of a SEMM analysis is the identification of latent classes, these conditions should be considered as alternative hypotheses and results should be interpreted cautiously. However, armed with greater knowledge about the estimation of SEMMs in practice, researchers can exploit the flexibility of the model to gain a fuller understanding of the phenomenon under study. In recent years, many exciting developments have taken place in structural equation modeling, but perhaps none more so than the development of structural equation models that account for unobserved popula
Have multilevel models been structural equation models all along
 Multivariate Behavioral Research
, 2003
"... A core assumption of the standard multiple regression model is independence of residuals, the violation of which results in biased standard errors and test statistics. The structural equation model (SEM) generalizes the regression model in several key ways, but the SEM also assumes independence of ..."
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Cited by 38 (2 self)
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A core assumption of the standard multiple regression model is independence of residuals, the violation of which results in biased standard errors and test statistics. The structural equation model (SEM) generalizes the regression model in several key ways, but the SEM also assumes independence of residuals. The multilevel model (MLM) was developed to extend the regression model to dependent data structures. Attempts have been made to extend the SEM in similar ways, but several complications currently limit the general application of these techniques in practice. Interestingly, it is well known that under a broad set of conditions SEM and MLM longitudinal "growth curve" models are analytically and empirically identical. This is intriguing given the clear violation of independence in growth modeling that does not detrimentally affect the standard SEM. Better understanding the source and potential implications of this isomorphism is my focus here. I begin by exploring why SEM and MLM are analytically equivalent methods in the presence of nesting due to repeated observations over time. I then capitalize on this equivalency to allow for the extension of SEMs to a general class of nested data structures. I conclude with a description of potential opportunities for multilevel SEMs and directions for future developments. The structural equation model (SEM) is a flexible and powerful analytical method that has become a mainstay in many areas of social science research. The generality of this approach is evidenced in the ability to parameterize the SEM to estimate well known members of the general linear modeling (GLM) family including the ttest, ANOVA, ANCOVA, MANOVA, MANCOVA, and the multiple regression model. However, the
Quasi Maximum Likelihood Estimation of Structural Equation Models With Multiple Interaction and Quadratic Effects
"... The development of statistically efficient and computationally practicable estimation methods for the analysis of structural equation models with multiple nonlinear effects has been called for by substantive researchers in psychology, marketing research, and sociology. But the development of efficie ..."
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Cited by 14 (0 self)
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The development of statistically efficient and computationally practicable estimation methods for the analysis of structural equation models with multiple nonlinear effects has been called for by substantive researchers in psychology, marketing research, and sociology. But the development of efficient methods is complicated by the fact that a nonlinear model structure implies specifically nonnormal multivariate distributions for the indicator variables. In this paper, nonlinear structural equation models with quadratic forms are introduced and a new QuasiMaximum Likelihood method for simultaneous estimation of model parameters is developed with the focus on statistical efficiency and computational practicability. The QuasiML method is based on an approximation of the nonnormal density function of the joint indicator vector by a product of a normal and a conditionally normal density. The results of MonteCarlo studies for the new QuasiML method indicate that the parameter estimation is almost as efficient as ML estimation, whereas ML estimation is only computationally practical for elementary models. Also, the QuasiML method outperforms other currently available methods with respect to efficiency. It is demonstrated in a MonteCarlo study that the QuasiML method permits computationally feasible and very efficient analysis of models with multiple latent nonlinear effects. Finally, the applicability of the QuasiML method is illustrated by an empirical example of an aging study in psychology. Key words: structural equation modeling, quadratic form of normal variates, latent interaction effect, moderator effect, QuasiML estimation, variance function model. 1 1.
A semiparametric approach to modeling nonlinear relations among latent variables. Structural Equation Modeling
, 2005
"... To date, finite mixtures of structural equation models (SEMMs) have been developed and applied almost exclusively for the purpose of providing modelbased cluster analyses. This type of analysis constitutes a direct application of the model wherein the estimated component distributions of the latent ..."
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Cited by 7 (3 self)
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To date, finite mixtures of structural equation models (SEMMs) have been developed and applied almost exclusively for the purpose of providing modelbased cluster analyses. This type of analysis constitutes a direct application of the model wherein the estimated component distributions of the latent classes are thought to represent the characteristics of distinct unobserved subgroups of the population. This article instead considers an indirect application of the SEMM in which the latent classes are estimated only in the service of more flexibly modeling the characteristics of the aggregate population as a whole. More specifically, the SEMM is used to semiparametrically model nonlinear latent variable regression functions. This approach is first developed analytically and then demonstrated empirically through analyses of simulated and real data. The modeling of nonlinear relations between latent variables has been a topic of longstanding interest. Within the factoranalytic tradition, early contributions to nonlinear latent variable modeling were made by Gibson (1959), McDonald (1967), and EtezadiAmoli and McDonald (1983). Whereas these approaches focused mainly on nonlinear factortoitem relations, subsequent contributions have focused specifically on modeling nonlinear effects between latent factors in structural equation models. These include the seminal paper by Kenny and Judd (1984) using products of manifest variables to model latent interactions and quadratic effects, as well as subsequent papers refining and extending this product indicant approach (see Schumacker & Marcoulides, 1998, and references therein). Problems with the product indicant approach included the tedium of properly specifying the necessary nonlinear constraints of the model and the fact that the
TwoStage Least Squares (2SLS) and Structural Equation Models (SEM). http://csusap.csu. edu.au./~eoczkows/home.htm Samuel Gebreselassie and E. Ludi (2007), Agricultural Commercialisation in Coffeegrowing Areas of Ethiopia. Paper presented at the
 Fifth International Conference on the Ethiopian
, 2003
"... These notes describe the 2SLS estimator for latent variable models developed by Bollen (1996). The technique separately estimates the measurement model and structural model of SEM. One can therefore use it either as a stand alone procedure for a full SEM or combine it with factor analysis, for examp ..."
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Cited by 3 (0 self)
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These notes describe the 2SLS estimator for latent variable models developed by Bollen (1996). The technique separately estimates the measurement model and structural model of SEM. One can therefore use it either as a stand alone procedure for a full SEM or combine it with factor analysis, for example, establish the measurement model using factor analysis and then employ 2SLS for the structural model only. The advantages of using 2SLS over the more conventional maximum likelihood (ML) method for SEM include: • It does not require any distributional assumptions for RHS independent variables, they can be nonnormal, binary, etc. • In the context of a multiequation nonrecursive SEM it isolates specification errors to single equations, see Bollen (2001). • It is computationally simple and does not require the use of numerical optimisation algorithms. • It easily caters for nonlinear and interactions effects, see Bollen and Paxton (1998). • It permits the routine use of often ignored diagnostic testing procedures for problems such as heteroscedasticity and specification error, see Pesaran and Taylor (1999). • Simulation evidence from econometrics suggests that 2SLS may perform better in small samples than ML, see Bollen (1996, pp120121). There are however some disadvantages in using 2SLS compared to ML, these include: • The ML estimator is more efficient than 2SLS given its simultaneous estimation of all relationships, hence ML will dominate 2SLS always in sufficiently large samples if all assumptions are valid and the model specification is correct. Effectively ML is more efficient (if the model is valid) as it uses much more information than 2SLS. • Unlike the ML method, the 2SLS estimator depends upon the choice of reference variable. The implication being that different 2SLS estimates result given different scaling variables. • Programs with diagram facilities such as EQS do not exist for 2SLS. One needs to logically work through the structure of the model to specify individual equations for all the relationships for the 2SLS estimator.
Detecting Latent Interaction Effects in Behavioral Data
"... Harsh discipline is a wellreplicated risk factor for aggressive, antisocial and delinquent behavior. In this paper we investigate the moderator hypothesis that parental perceptions of early child manageability problems, moderate parental discipline responses to the child’s disruptive behavior. We ..."
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Harsh discipline is a wellreplicated risk factor for aggressive, antisocial and delinquent behavior. In this paper we investigate the moderator hypothesis that parental perceptions of early child manageability problems, moderate parental discipline responses to the child’s disruptive behavior. We describe the application of an interaction model to the analysis of a behavioral data set, show how the data are initially screened for an interaction effect, and implement the model by use of the newly developed QuasiML method for analysis of latent interaction effects. The new methodology detects the hypothesized interaction effect by use of a likelihood ratio test. A quantitative interpretation of the estimated model parameters is provided, and conclusions for the analysis of synergistic effects in behavioral data by implementing appropriate interaction hypotheses are drawn.
Interactions of Latent Variables in Structural Equation Models
"... .the.':?' ' indicates a variable is adjusted by its grand mean Level 2 model for bu (see Equation 6) Level 2 model for bzj (see Equation 7) Request for maximum of 100 iterations Print ordinary least square estimates for the first 50 Level 2 units No equality constraints imposed on fix ..."
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.the.':?' ' indicates a variable is adjusted by its grand mean Level 2 model for bu (see Equation 6) Level 2 model for bzj (see Equation 7) Request for maximum of 100 iterations Print ordinary least square estimates for the first 50 Level 2 units No equality constraints imposed on fixed effects No residual file will be output Option 3, "automatic fixup, " invoked if variancecovariance matrix at Level 2 is not positively definite No optional hypothesis testing requested.00005 is used as convergence criterion Results be saved in CIGHLM2.0UT Title for the setup
Accuracy of Parameter Estimates and Confidence Intervals in Moderated Mediation Models: A Comparison of Regression and Latent Moderated Structural Equations
"... Currently, the most popular analytical method for testing moderated mediation is the regression approach, which is based on observed variables and assumes no measurement error. It is generally acknowledged that measurement errors result in biased estimates of regression coefficients. What has drawn ..."
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Currently, the most popular analytical method for testing moderated mediation is the regression approach, which is based on observed variables and assumes no measurement error. It is generally acknowledged that measurement errors result in biased estimates of regression coefficients. What has drawn relatively less attention is that the confidence intervals produced by regression are also biased when the variables are measured with errors. Therefore, we extend the latent moderated structural equations (LMS) method—which corrects for measurement errors when estimating latent interaction effects—to the study of the moderated mediation of latent variables. Simulations were conducted to compare the regression approach and the LMS approach. The results show that the LMS method produces accurate estimated effects and confidence intervals. By contrast, regression not only substantially underestimates the effects but also produces inaccurate confidence intervals. It is likely that the statistically significant moderated mediation effects that have been reported in previous studies using regression include biased estimated effects and confidence intervals that do not include the true values.
The Ontario HIV Treatment Network
"... early years of marriage is important because it is during the first 2 years of marriage that signs of disillusionment surface, putting marriages at risk (Huston, Caughlin, Houts, Smith, & George, 2001). Does idealization early in marriage set spouses up for disappointment, as Waller (1938) sugg ..."
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early years of marriage is important because it is during the first 2 years of marriage that signs of disillusionment surface, putting marriages at risk (Huston, Caughlin, Houts, Smith, & George, 2001). Does idealization early in marriage set spouses up for disappointment, as Waller (1938) suggests, or does it help protect people from becoming disillusioned? Although research on positive illusions shows that people who idealize their partner generally establish more satisfying relationships during courtship and early
Joint Statistical Meetings Business & Economic Statistics Section PSEUDO LIKELIHOOD APPROACH FOR NONLINEAR AND NONNORMAL STRUCTURAL EQUATION ANALYSIS
"... standard error estimation, deconvolution, bootstrap, latent variable modeling. Structural equation analysis is widely used in economics and social sciences. The model considered in this paper consists of two parts; a linear measurement model relating observed measurements to underlying latent variab ..."
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standard error estimation, deconvolution, bootstrap, latent variable modeling. Structural equation analysis is widely used in economics and social sciences. The model considered in this paper consists of two parts; a linear measurement model relating observed measurements to underlying latent variables, and a nonlinear structural model representing relationships among the latent variables. When the distributional form of the latent variables is unspecified, a pseudo likelihood approach based on a hypothetical normal mixture assumption is proposed. To obtain the pseudo likelihood parameter estimates, the Monte Carlo EM algorithm is developed. Standard error estimates for the estimated structural parameters are obtained combining an empirical observed information estimates and a bootstrap estimated covariance matrix for the nuisance parameters. Simulation studies are reported. 1.