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53
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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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
An Unscented Kalman Filter Approach to the Estimation of Nonlinear Dynamical Systems Models
"... In the past several decades, methodologies used to estimate nonlinear relationships among latent variables have been developed almost exclusively to fit crosssectional models. We present a relatively new estimation approach, the unscented Kalman filter (UKF), and illustrate its potential as a tool ..."
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In the past several decades, methodologies used to estimate nonlinear relationships among latent variables have been developed almost exclusively to fit crosssectional models. We present a relatively new estimation approach, the unscented Kalman filter (UKF), and illustrate its potential as a tool for fitting nonlinear dynamic models in two ways: (1) as a building block for approximating the log–likelihood of nonlinear state–space models and (2) to fit timevarying dynamic models wherein parameters are represented and estimated online as other latent variables. Furthermore, the substantive utility of the UKF is demonstrated using simulated examples of (1) the classical predatorprey model with time series and multiple–subject data, (2) the chaotic Lorenz system and (3) an empirical example of dyadic interaction. Dynamical systems are systems that change over time such that their current states are somehow dependent upon their previous states (Alligood, Sauer, & Yorke, 1996). Change concepts described in most dynamical systems models are by no means novel to psychologists. From the rather controversial difference scores (e.g., Bereiter, 1963; CronWe thank Jack McArdle, Ellen Bass, Howard Epstein, Fumiaki Hamagami and a few anonymous reviewers for their valuable comments on earlier versions of this article. This study was supported by a National
Default Priors and Efficient Posterior Computation in Bayesian Factor Analysis
"... Abstract. Factor analytic models are widely used in social sciences. These models have also proven useful for sparse modeling of the covariance structure in multidimensional data. Normal priors for factor loadings and inverse gamma priors for residual variances are a popular choice because of their ..."
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Abstract. Factor analytic models are widely used in social sciences. These models have also proven useful for sparse modeling of the covariance structure in multidimensional data. Normal priors for factor loadings and inverse gamma priors for residual variances are a popular choice because of their conditionally conjugate form. However, such priors require elicitation of many hyperparameters and tend to result in poorly behaved Gibbs samplers. In addition, one must choose an informative specification, as high variance priors face problems due to impropriety of the posterior. This article proposes a default, heavy tailed prior specification, which is induced through parameter expansion while facilitating efficient posterior computation. We also develop an approach to allow uncertainty in the number of factors. The methods are illustrated through simulated examples and epidemiology and toxicology applications.
The Role of Nonlinear FactortoIndicator Relationships in Tests of Measurement Equivalence
"... Measurement invariance is a necessary condition for the evaluation of factor mean differences over groups or time. This article considers the potential problems that can arise for tests of measurement invariance when the true factortoindicator relationship is nonlinear (quadratic) and invariant bu ..."
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Measurement invariance is a necessary condition for the evaluation of factor mean differences over groups or time. This article considers the potential problems that can arise for tests of measurement invariance when the true factortoindicator relationship is nonlinear (quadratic) and invariant but the linear factor model is nevertheless applied. The factor loadings and indicator intercepts of the linear model will diverge across groups as the factor mean difference increases. Power analyses show that even apparently small quadratic effects can result in rejection of measurement invariance at moderate sample sizes when the factor mean difference is medium to large. Recommendations include the identification of nonlinear relationships using diagnostic plots and consideration of newly developed methods for fitting nonlinear factor models.
Efficient Bayesian model averaging in factor analysis
 Duke University
, 2006
"... Summary. Although factor analytic models have proven useful for covariance structure modeling and dimensionality reduction in a wide variety of applications, a challenging problem is uncertainty in the number of latent factors. This article proposes an efficient Bayesian approach for model selectio ..."
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Summary. Although factor analytic models have proven useful for covariance structure modeling and dimensionality reduction in a wide variety of applications, a challenging problem is uncertainty in the number of latent factors. This article proposes an efficient Bayesian approach for model selection and averaging in hierarchical models having one or more factor analytic components. In particular, the approach relies on a method for embedding each of the smaller models within the largest possible model. Bayesian computation can proceed within the largest model, while moving between submodels based on posterior model probabilities. The approach represents a type of parameter expansion, as one always samples within an encompassing model, incorporating extra parameters and latent variables when a smaller model is true. This results in a highly efficient stochastic search factor selection algorithm (SSFS) for identifying good factor models and performing modelaveraged inferences. The approach is illustrated using simulated examples and a toxicology application.
Statistics in Sociology, 19502000: A Selective Review
, 2001
"... Statistical methods have had a successful halfcentury in sociology, contributing to a greatly improved standard of scientic rigor in the discipline. I identify three overlapping postwar generations of statistical methods in sociology, based on the kinds of data they address. The rst generation, whi ..."
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Statistical methods have had a successful halfcentury in sociology, contributing to a greatly improved standard of scientic rigor in the discipline. I identify three overlapping postwar generations of statistical methods in sociology, based on the kinds of data they address. The rst generation, which started in the late 1940s, deals with crosstabulations, and focuses on measures of association and loglinear models, perhaps the area of statistics to which sociology has contributed the most. The second generation, which began in the 1960s, deals with unitlevel survey data, and focuses on LISRELtype causal models and event history analysis. The third generation, starting to emerge in the late 1980s, deals with data that do not fall easily into either of these categories, either because they have a dierent form, such as texts or narratives, or because dependence is a crucial aspect, as with spatial or social network data. There are many new challenges and the area is ripe for statis...
Simple, Efficient and Distributionfree Approach to Interaction Effects in Complex Structural Equation Models
"... Abstract. Structural equation models with mean structure and nonlinear constraints are the most frequent choice for estimating interaction effects when measurement errors are present. This article proposes eliminating the mean structure and all the constraints but one, which leads to a more easily ..."
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Abstract. Structural equation models with mean structure and nonlinear constraints are the most frequent choice for estimating interaction effects when measurement errors are present. This article proposes eliminating the mean structure and all the constraints but one, which leads to a more easily handled model that is more robust to nonnormality and more general as it can accommodate endogenous interactions and thus indirect effects. Our approach is compared to other approaches found in the literature with a Monte Carlo simulation and is found to be equally efficient under normality and less biased under nonnormality. An empirical illustration is included. Key words: Interaction effects; structural equation models; indirect effects; nonlinear constraints; mean structure; normality assumption. 1.
Maximum likelihood estimation and model comparison for mixtures of structural equation models with ignorable missing data”,
 Journal of Classification,
, 2003
"... Abstract Recently, it is recognized that nonlinear relationships among latent variables in a structural equation model are important. In this article, maximum likelihood (ML) analysis of a general nonlinear structural equation model that contains ÿxed covariates in the measurement equation and the ..."
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Abstract Recently, it is recognized that nonlinear relationships among latent variables in a structural equation model are important. In this article, maximum likelihood (ML) analysis of a general nonlinear structural equation model that contains ÿxed covariates in the measurement equation and the nonlinear structural equation is investigated. A MCEM algorithm is implemented to obtain the ML estimates, in which the Estep is completed with the help of a hybrid algorithm that combines the Gibbs sampler and the MetropolisHastings algorithm whilst the Mstep is completed by conditional maximization. The importance sampling is employed to compute the observeddata likelihood in the Bayesian Information Criterion for model comparison. The methodology is illustrated with a simulation study and a real example.
Bayesian model selection for mixtures of structural equation models with an unknown number of components.
 British J. Math. Statist. Psych.
, 2003
"... This paper considers mixtures of structural equation models with an unknown number of components. A Bayesian model selection approach is developed based on the Bayes factor. A procedure for computing the Bayes factor is developed via path sampling, which has a number of nice features. The key idea ..."
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This paper considers mixtures of structural equation models with an unknown number of components. A Bayesian model selection approach is developed based on the Bayes factor. A procedure for computing the Bayes factor is developed via path sampling, which has a number of nice features. The key idea is to construct a continuous path linking the competing models; then the Bayes factor can be estimated ef ciently via grids in [0, 1] and simulated observations that are generated by the Gibbs sampler from the posterior distribution. Bayesian estimates of the structural parameters, latent variables, as well as other statistics can be produced as byproducts. The properties and merits of the proposed procedure are discussed and illustrated by means of a simulation study and a real example.