A Nonlinear Structural Equation Mixture Modeling Approach for Nonnormally Distributed Latent Predictor Variables

Structural equation models with interaction and quadratic effects have become a standard tool for testing nonlinear hypotheses in the social sciences. Most of the current approaches assume normally distributed latent predictor variables. In this article, we describe a nonlinear structural equation mixture approach that integrates the strength of parametric approaches (specification of the nonlinear functional relationship) and the flexibility of semiparametric structural equation mixture approaches for approximating the nonnormality of latent predictor variables. In a comparative simulation study, the advantages of the proposed mixture procedure over contemporary approaches [Latent Moderated Structural Equations approach (LMS) and the extended unconstrained approach] are shown for varying degrees of skewness of the latent predictor variables. Whereas the conventional approaches show either biased parameter estimates or standard errors of the nonlinear effects, the proposed mixture approach provides unbiased estimates and standard errors. We present an empirical example from educational research. Guidelines for applications of the approaches and limitations are discussed.     

A Specification Error Test That Uses Instrumental Variables to Detect Latent Quadratic and Latent Interaction Effects

The relations between the latent variables in structural equation models are typically assumed to be linear in form. This article aims to explain how a specification error test using instrumental variables (IVs) can be employed to detect unmodeled interactions between latent variables or quadratic effects of latent variables. An empirical example is presented, and the results of a simulation study are reported to evaluate the sensitivity and specificity of the test and compare it with the commonly employed chi-square model test. The results show that the proposed test can identify most unmodeled latent interactions or latent quadratic effects in moderate to large samples. Furthermore, its power is higher when the number of indicators used to define the latent variables is large. Altogether, this article shows how the IV-based test can be applied to structural equation models and that it is a valuable tool for researchers using structural equation models.     

A Bayesian Approach for Analyzing Hierarchical Data With Missing Outcomes Through Structural Equation Models

Structural equation models are widely appreciated in behavioral, social, and psychological research to model relations between latent constructs and manifest variables, and to control for measurement errors. Most applications of structural equation models are based on fully observed data that are independently distributed. However, hierarchical data with a correlated structure are common in behavioral research, and very often, missing data are encountered. In this article, we propose a 2-level structural equation model for analyzing hierarchical data with missing entries, and describe a Bayesian approach for estimation and model comparison. We show how to use WinBUGS software to get the solution conveniently. The proposed methodologies are illustrated through a simulation study, and a real application in relation to organizational and management research concerning the study of the interrelationships of the latent constructs about job satisfaction, job responsibility, and life satisfaction for citizens in 43 countries.     

Bayesian Analysis of Mixture Structural Equation Models With an Unknown Number of Components

Multivariate heterogenous data with latent variables are common in many fields such as biological, medical, behavioral, and social-psychological sciences. Mixture structural equation models are multivariate techniques used to examine heterogeneous interrelationships among latent variables. In the analysis of mixture models, determination of the number of mixture components is always an important and challenging issue. This article aims to develop a full Bayesian approach with the use of reversible jump Markov chain Monte Carlo method to analyze mixture structural equation models with an unknown number of components. The proposed procedure can simultaneously and efficiently select the number of mixture components and conduct parameter estimation. Simulation studies show the satisfactory empirical performance of the method. The proposed method is applied to study risk factors of osteoporotic fractures in older people.     

Model Comparison of Bayesian Semiparametric and Parametric Structural Equation Models

Structural equation models have wide applications. One of the most important issues in analyzing structural equation models is model comparison. This article proposes a Bayesian model comparison statistic, namely the L _ν-measure for both semiparametric and parametric structural equation models. For illustration purposes, we consider a Bayesian semiparametric approach for estimation and model comparison in the context of structural equation models with fixed covariates. A finite dimensional Dirichlet process is used to model the crucial latent variables, and a blocked Gibbs sampler is implemented for estimation. Empirical performance of the L _ν-measure is evaluated through a simulation study. Results obtained indicate that the L _ν-measure, which additionally requires very minor computational effort, gives satisfactory performance. Moreover, the methodologies are demonstrated through an example with a real data set on kidney disease. Finally, the application of the L _ν-measure to Bayesian semiparametric nonlinear structural equation models is outlined.     

Regularized Structural Equation Modeling

A new method is proposed that extends the use of regularization in both lasso and ridge regression to structural equation models. The method is termed regularized structural equation modeling (RegSEM). RegSEM penalizes specific parameters in structural equation models, with the goal of creating easier to understand and simpler models. Although regularization has gained wide adoption in regression, very little has transferred to models with latent variables. By adding penalties to specific parameters in a structural equation model, researchers have a high level of flexibility in reducing model complexity, overcoming poor fitting models, and the creation of models that are more likely to generalize to new samples. The proposed method was evaluated through a simulation study, two illustrative examples involving a measurement model, and one empirical example involving the structural part of the model to demonstrate RegSEM’s utility.     

Inference and Interval Estimation Methods for Indirect Effects With Latent Variable Models

Although much is known about the performance of recent methods for inference and interval estimation for indirect or mediated effects with observed variables, little is known about their performance in latent variable models. This article presents an extensive Monte Carlo study of 11 different leading or popular methods adapted to structural equation models with latent variables. Manipulated variables included sample size, number of indicators per latent variable, internal consistency per set of indicators, and 16 different path combinations between latent variables. Results indicate that some popular or previously recommended methods, such as the bias-corrected bootstrap and asymptotic standard errors had poorly calibrated Type I error and coverage rates in some conditions. Likelihood-based confidence intervals, the distribution of the product method, and the percentile bootstrap emerged as leading methods for both interval estimation and inference, whereas joint significance tests and the partial posterior method performed well for inference.     

Alternative approaches to structural modeling of ordinal data: A Monte Carlo study

In practice, several measures of association are used when analyzing structural equation models with ordinal variables: ordinary Pearson correlations (PE approach), polychoric and polyserial correlations (PO approach), and conditional polychoric correlations (CPO approach). In the case of structural equation models without latent variables, the literature has shown that the PE approach is outperformed by the alternatives. In this article we report a Monte Carlo study showing the comparative performance of the aforementioned alternative approaches under deviations from their respective assumptions in the case of structural equation models with latent variables when attention is restricted to point estimates of model parameters. The CPO approach is shown to be the most robust against nonnormality. It is also robust to randomness of the exogenous variables, but not to the existence of measurement errors in them. The PO approach lacks robustness against nonnormality. The PE approach lacks robustness against transformation errors but otherwise it can perform about as well as the alternative approaches.     

结构方程模型(SEM)的原理及操作

结构方程模型(SEM)是应用线性方程系统表示观测变量与潜在变量之间及潜在变量之间关系的一种统计方法。当前，SEM及相应的LISREL软件已成为心理学等社会学科中广泛应用的一种分析思想和技术。文章简要介绍了SEM的特点、原理及LISREL的操作方法。     

Structural pattern differences in course enrollment rates among community college students

This study tested a structural equation model of enrollment patterns of white and Hispanic males and females in two-year institutions and the invariance of parameter estimates among the different subgroups in the study. The model represented a multiequation model with three latent endogenous variables, high school academic preparation in mathematics and science, mathematics and science attitudes, and the dependent variable, enrollment patterns in mathematics and science courses. Exogenous variables included parents' education, levels of encouragement by others, and high school grades. Structural equation modeling was used to examine the structural and measurement coefficients of the hypothesized causal model for all subgroups in the study. In summary, an examination of the direct and total effect coefficients revealed different underlying patterns of factors for white and Hispanic females. No convergence on the model was found for white and Hispanic males. Equality constraints on all structural coefficients for both white and Hispanic females were tested and results indicated that all parameter estimates in the structural models for both subgroups were significantly different from each other.     

Using Fixed Thresholds with Grouped Data in Structural Equation Modeling

Valuable methods have been developed for incorporating ordinal variables into structural equation models using a latent response variable formulation. However, some model parameters, such as the means and variances of latent factors, can be quite difficult to interpret because the latent response variables have an arbitrary metric. This limitation can be particularly problematic in growth models, where the means and variances of the latent growth parameters typically have important substantive meaning when continuous measures are used. However, these methods are often applied to grouped data, where the ordered categories actually represent an interval-level variable that has been measured on an ordinal scale for convenience. The method illustrated in this article shows how category threshold values can be incorporated into the model so that interpretation is more meaningful, with particular emphasis given to the application of this technique with latent growth models.     

Using structural equation models with latent variables to study student growth and development

Calls for accountability, coupled with a desire to improve teaching and learning, have prompted many colleges and universities to consider ways of assessing the effects of postsecondary education on student growth and development. Despite widespread support for the concept of assessing student change, relatively few institutions have implemented this type of assessment, in part because of a concern about the best method of measuring change. This article describes the use of structural equation models with latent variables to assess the effects of education on change. Advantages of using structural equation models with latent variables include error-free measurement of change, direct tests of the assumptions underlying change research, along with the power and flexibility of maximum likelihood estimation. An analysis of data on freshman-to-senior gains provides evidence of the advantages of latent variable structural equation modeling and also suggests that the group differences identified by traditional analysis of variance and covariance techniques may be an artifact of measurement error.     

A Non-arbitrary Method of Identifying and Scaling Latent Variables in SEM and MACS Models

A non-arbitrary method for the identification and scale setting of latent variables in general structural equation modeling is introduced. This particular technique provides identical model fit as traditional methods (e.g., the marker variable method), but it allows one to estimate the latent parameters in a nonarbitrary metric that reflects the metric of the measured indicators. This technique, therefore, is particularly useful for mean and covariance structures (MACS) analyses, where the means of the indicators and latent constructs are of key interest. By introducing this alternative method of identification and scale setting, researchers are provided with an additional tool for conducting MACS analyses that provides a meaningful and nonarbitrary scale for the estimates of the latent variable parameters. Importantly, this tool can be used with single-group single-occasion models as well as with multiple-group models, multiple-occasion models, or both.     

Implementing Restricted Maximum Likelihood Estimation in Structural Equation Models

Structural equation modeling (SEM) is now a generic modeling framework for many multivariate techniques applied in the social and behavioral sciences. Many statistical models can be considered either as special cases of SEM or as part of the latent variable modeling framework. One popular extension is the use of SEM to conduct linear mixed-effects modeling (LMM) such as cross-sectional multilevel modeling and latent growth modeling. It is well known that LMM can be formulated as structural equation models. However, one main difference between the implementations in SEM and LMM is that maximum likelihood (ML) estimation is usually used in SEM, whereas restricted (or residual) maximum likelihood (REML) estimation is the default method in most LMM packages. This article shows how REML estimation can be implemented in SEM. Two empirical examples on latent growth model and meta-analysis are used to illustrate the procedures implemented in OpenMx. Issues related to implementing REML in SEM are discussed.     

Modeling Nonlinear Structural Equation Models: A Comparison of the Two-Stage Generalized Additive Models and the Finite Mixture Structural Equation Model

Researchers have devoted some time and effort to developing methods for fitting nonlinear relationships among latent variables. In particular, most of these have focused on correctly modeling interactions between 2 exogenous latent variables, and quadratic relationships between exogenous and endogenous variables. All of these approaches require prespecification of the nonlinearity by the researcher, and are limited to fairly simple nonlinear relationships. Other work has been done using mixture structural equation models (SEMM) in an attempt to fit more complex nonlinear relationships. This study expands on this earlier work by introducing the 2-stage generalized additive model (2SGAM) approach for fitting regression splines in the context of structural equation models. The model is first described and then investigated through the use of simulated data, in which it was compared with the SEMM approach. Results demonstrate that the 2SGAM is an effective tool for fitting a variety of nonlinear relationships between latent variables, and can be easily and accurately extended to models including multiple latent variables. Implications of these results are discussed.     

Randomized Response Analysis in Mplus

This article describes a technique to analyze randomized response data using available structural equation modeling (SEM) software. The randomized response technique was developed to obtain estimates that are more valid when studying sensitive topics. The basic feature of all randomized response methods is that the data are deliberately contaminated with error. This makes it difficult to relate randomized responses to explanatory variables. In this tutorial, we present an approach to this problem, in which the analysis of randomized response data is viewed as a latent class problem, with different latent classes for the random and the truthful responses. To illustrate this technique, an example is presented using the program Mplus.     

Analyzing Different Types of Moderated Method Effects in Confirmatory Factor Models for Structurally Different Methods

An extension of two confirmatory factor models for multitrait-multimethod measurement designs with structurally different methods to the analysis of latent interaction effects is presented: the nonlinear latent difference (NL-LD) model and the nonlinear correlated trait–correlated method-minus-one (NL-CTC[M – 1]) model. Both models are compared with regard to (a) the psychometric definition of the latent variables, (b) the capabilities of explaining latent method effects, and (c) the analysis of latent interaction effects. Using the latent moderated structural equation approach, we show how moderated method effects can be examined in the NL-CTC(M – 1) model. This fine-grained analysis of method effects is not feasible using the classical NL-LD model. We propose an extended version of the NL-LD model, which recovers the results of the NL-CTC(M – 1) model. The different versions of the nonlinear multimethod models are illustrated using real data from a multirater study. Finally, the advantages and challenges of incorporating latent interaction effects in complex CFA–MTMM models are discussed.     

A Comparison of Latent Growth Models for Constructs Measured by Multiple Items

A multiple testing approach is outlined that can be used to examine the assumption of underlying normal variables in latent variable models with categorical indicators. The method is based on an application of the increasingly popular Benjamini–Hochberg multiple testing procedure, and is readily applicable with widely circulated software. The discussed method is especially useful for ascertaining this assumption that is very often made in research based on structural equation modeling using models containing discrete outcomes. The described approach is illustrated with numerical data.     

Sind Modelle der Item-Response-Theorie (IRT) das „Mittel der Wahl“ für die Modellierung von Kompetenzen?

Item response theory (IRT) models can be subsumed under the larger class of statistical models with latent variables. IRT models are increasingly used for the scaling of the responses derived from standardized assessments of competencies. The paper summarizes the strengths of IRT in contrast to more traditional techniques as well as in contrast to alternative models with latent variables (e. g. structural equation modeling). Subsequently, specific limitations of IRT and cases where other methods might be preferable are lined out.