Nonlinear Structural Equation Models Research Articles

On Bayesian estimation and model comparison of an integrated structural equation model

In this paper, we introduce a Bayesian approach to the estimation and model comparison of an integrated two-level nonlinear structural equation model with mixed continuous, dichotomous, and ordered categorical data that may be missing at random. This general model can accommodate nonlinearities of latent variables and the effects of fixed covariates on measurement and structural equations in within-groups and between-groups models. A sampling-based algorithm that combines the Gibbs sampler and the Metropolis–Hastings algorithm is proposed for posterior simulation. A procedure that utilizes path sampling is implemented to compute the Bayes factor for model comparison under the framework of the proposed integrated model. Empirical performances of Bayesian methodologies are illustrated via analysis of a real example.

A Robust Bayesian Approach for Structural Equation Models with Missing Data

In this paper, normal/independent distributions, including but not limited to the multivariate t distribution, the multivariate contaminated distribution, and the multivariate slash distribution, are used to develop a robust Bayesian approach for analyzing structural equation models with complete or missing data. In the context of a nonlinear structural equation model with fixed covariates, robust Bayesian methods are developed for estimation and model comparison. Results from simulation studies are reported to reveal the characteristics of estimation. The methods are illustrated by using a real data set obtained from diabetes patients.

Bayesian analysis of nonlinear structural equation models with mixed continuous, ordered and unordered categorical, and nonignorable missing data

Bayesian analysis of nonlinear structural equation models with mixed continuous, ordered and unordered categorical, and nonignorable missing data

Quasi-Maximum Likelihood Estimation of Structural Equation Models With Multiple Interaction and Quadratic Effects

In this article, a nonlinear structural equation model is introduced and a quasi-maximum likelihood method for simultaneous estimation and testing of multiple nonlinear effects is developed. The focus of the new methodology lies on efficiency, robustness, and computational practicability. Monte-Carlo studies indicate that the method is highly efficient and that the likelihood ratio test of nonlinear effects is robust and outperforms alternative testing procedures. The new method is applied to empirical data of middle-aged men, where a latent interaction between physical fitness and flexibility in goal adjustment on complaint level is hypothesized. A model with 5 simultaneous nonlinear effects is analyzed, and the hypothesized interaction is quantified and tested positively against an additive model with quadratic and linear effects.

Structural equation modeling: a Bayesian approach

About the Author. Preface. Chapter 1. Introduction. Chapter 2. Some Basic Structural Equation Models. Chapter 3. Covariance Structure Analysis. Chapter 4. Bayesian Estimation of Structural Equation Models. Chapter 5. Model Comparison and Model Checking. Chapter 6. Structural Equation Models with Continuous and Ordered Categorical Variables. Chapter 7. Structural Equation Models with Dichotomous Variables. Chapter 8. Nonlinear Structural Equation Models. Chapter 9. Two-level Nonlinear Structural Equation Models. Chapter 10. Multisample Analysis of Structural Equation Models. Chapter 11. Finite Mixtures in Structural Equation Models. Chapter 12. Structural Equation Models with Missing Data. Chapter 13. Structural Equation Models with Exponential Family of Distributions. Chapter 14. Conclusion. Index.

Exploring Tourist Satisfaction with Nonlinear Structural Equation Modeling and Inferred Causation Analysis

The aim of this paper is to provide two examples of advanced analysis which include (i) a new nonlinear latent variable approach to measuring hygiene and delight factors in tourist satisfaction monitoring, and (ii) an application of Inferred Causation Theory to exploring antecedents of satisfaction. Both demonstration examples are considered exploratory as it is preferred to pursue a rigorous concept of inferential analysis where a conclusive model test always would have to imply new data sets.

Does species diversity limit productivity in natural grassland communities?

Theoretical analyses and experimental studies of synthesized assemblages indicate that under particular circumstances species diversity can enhance community productivity through niche complementarity. It remains unclear whether this process has important effects in mature natural ecosystems where competitive feedbacks and complex environmental influences affect diversity-productivity relationships. In this study, we evaluated diversity-productivity relationships while statistically controlling for environmental influences in 12 natural grassland ecosystems. Because diversity-productivity relationships are conspicuously nonlinear, we developed a nonlinear structural equation modeling (SEM) methodology to separate the effects of diversity on productivity from the effects of productivity on diversity. Meta-analysis was used to summarize the SEM findings across studies. While competitive effects were readily detected, enhancement of production by diversity was not. These results suggest that the influence of small-scale diversity on productivity in mature natural systems is a weak force, both in absolute terms and relative to the effects of other controls on productivity.

A Unified Maximum Likelihood Approach for Analyzing Structural Equation Models With Missing Nonstandard Data

In this article, the authors present a unified approach for maximum likelihood analysis of structural equation models that involve subtle model formulations and nonstandard data structures. Based on the idea of data augmentation, they describe a generic Monte Carlo expectation-maximization algorithm for estimation. They propose path sampling for computing the observed data likelihood functions that usually involve complicated integrals and show how to apply this method for computing the Bayesian information criterion for model comparison. An application of the proposed unified approach to a two-level nonlinear structural equation model with missing continuous and ordered categorical data is presented. An illustrative example with a real data set is given.

Challenges in Nonlinear Structural Equation Modeling

Abstract. Challenges in evaluating nonlinear effects in multiple regression analyses include reliability, validity, multicollinearity, and dichotomization of continuous variables. While reliability and validity issues are solved by employing nonlinear structural equation modeling, multicollinearity remains a problem which may even be aggravated when using latent variable approaches. Further challenges of nonlinear latent analyses comprise the distribution of latent product terms, a problem especially relevant for approaches using maximum likelihood estimation methods based on multivariate normally distributed variables, and unbiased estimates of nonlinear effects under multicollinearity. The only methods that explicitly take the nonnormality of nonlinear latent models into account are latent moderated structural equations (LMS) and quasi-maximum likelihood (QML). In a small simulation study both methods yielded unbiased parameter estimates and correct estimates of standard errors for inferential statistics. The advantages and limitations of nonlinear structural equation modeling are discussed.

Bayesian analysis of structural equation models with multinomial variables and an application to type 2 diabetic nephropathy

There is now increasing evidence proving that many complex diseases can be significantly influenced by correlated phenotype and genotype variables, as well as their interactions. Effective and rigorous assessment of such influence is difficult, because the number of phenotype and genotype variables of interest may not be small, and a genotype variable is an unordered categorical variable that follows a multinomial distribution. To address the problem, we establish a novel nonlinear structural equation model for analysing mixed continuous and multinomial data that can be missing at random. A confirmatory factor analysis model with Kronecker product is proposed for grouping the manifest continuous and multinomial variables into latent variables according to their functions; and a nonlinear structural equation is formulated to assess the linear and interaction effects of the independent latent variables to the dependent latent variables. Bayesian methods for estimation and model comparison are developed through Markov chain Monte Carlo techniques and path sampling. The newly developed methodologies are applied to a case-control cohort of type 2 diabetic patients with nephropathy.

Maximum Likelihood Methods in Treating Outliers and Symmetrically Heavy-Tailed Distributions for Nonlinear Structural Equation Models with Missing Data

By means of more than a dozen user friendly packages, structural equation models (SEMs) are widely used in behavioral, education, social, and psychological research. As the underlying theory and methods in these packages are vulnerable to outliers and distributions with longer-than-normal tails, a fundamental problem in the field is the development of robust methods to reduce the influence of outliers and the distributional deviation in the analysis. In this paper we develop a maximum likelihood (ML) approach that is robust to outliers and symmetrically heavy-tailed distributions for analyzing nonlinear SEMs with ignorable missing data. The analytic strategy is to incorporate a general class of distributions into the latent variables and the error measurements in the measurement and structural equations. A Monte Carlo EM (MCEM) algorithm is constructed to obtain the ML estimates, and a path sampling procedure is implemented to compute the observed-data log-likelihood and then the Bayesian information criterion for model comparison. The proposed methodologies are illustrated with simulation studies and an example.

Bayesian Analysis of Structural Equation Models With Nonlinear Covariates and Latent Variables

In this article, we formulate a nonlinear structural equation model (SEM) that can accommodate covariates in the measurement equation and nonlinear terms of covariates and exogenous latent variables in the structural equation. The covariates can come from continuous or discrete distributions. A Bayesian approach is developed to analyze the proposed model. Markov chain Monte Carlo methods for obtaining Bayesian estimates and their standard error estimates, highest posterior density intervals, and a PP p value are developed. Results obtained from two simulation studies are reported to respectively reveal the empirical performance of the proposed Bayesian estimation in analyzing complex nonlinear SEMs, and in analyzing nonlinearSEMs with the normal assumption of the exogenous latent variables violated. Theproposed methodology is further illustrated by a real example. Detailed interpretation about the interaction terms is presented.

Bayesian Analysis of Nonlinear Structural Equation Models with Nonignorable Missing Data

A Bayesian approach is developed for analyzing nonlinear structural equation models with nonignorable missing data. The nonignorable missingness mechanism is specified by a logistic regression model. A hybrid algorithm that combines the Gibbs sampler and the Metropolis–Hastings algorithm is used to produce the joint Bayesian estimates of structural parameters, latent variables, parameters in the nonignorable missing model, as well as their standard errors estimates. A goodness-of-fit statistic for assessing the plausibility of the posited nonlinear structural equation model is introduced, and a procedure for computing the Bayes factor for model comparison is developed via path sampling. Results obtained with respect to different missing data models, and different prior inputs are compared via simulation studies. In particular, it is shown that in the presence of nonignorable missing data, results obtained by the proposed method with a nonignorable missing data model are significantly better than those that are obtained under the missing at random assumption. A real example is presented to illustrate the newly developed Bayesian methodologies.

A Maximum Likelihood Approach for Multisample Nonlinear Structural Equation Models With Missing Continuous and Dichotomous Data

Structural equation models are widely appreciated in social-psychological research and other behavioral research to model relations between latent constructs and manifest variables and to control for measurement error. Most applications of SEMs are based on fully observed continuous normal data and models with a linear structural equation. However, discrete nonnormal data and missing data are rather common, and sometimes it is necessary to incorporate nonlinear structural equations for assessing the impact of nonlinear terms of the exogenous latent variables to the endogenous latent variables. Moreover, to study the behaviors of different populations, it is necessary to extend from a single sample model to a multisample model. In this article, a maximum likelihood (ML) approach is developed for analyzing a multisample nonlinear structural equation model, in the context of mixed continuous and dichotomous data that involve data that are missing at random. The article demonstrates the newly developed methods for estimation and model comparison by a simulation study, a synthetic data application, and a real application to study latent psychological constructs of job attitude, emotion, and benefit attitude that are related to organizational and management research.

Maximum Likelihood Analysis of a Two-Level Nonlinear Structural Equation Model With Fixed Covariates

In this article, a maximum likelihood (ML) approach for analyzing a rather general two-level structural equation model is developed for hierarchically structured data that are very common in educational and/or behavioral research. The proposed two-level model can accommodate nonlinear causal relations among latent variables as well as effects of fixed covariate in its various components. Methods for computing the ML estimates, and the Bayesian information criterion (BIC) for model comparison are established on the basis of powerful tools in statistical computing such as the Monte Carlo EM algorithm, Gibbs sampler, Metropolis–Hastings algorithm, conditional maximization, bridge sampling, and path sampling. The newly developed procedures are illustrated by results obtained from a simulation study and analysis of a real data set in education.

Assessing local influence for nonlinear structural equation models with ignorable missing data

A method is proposed in this paper to assess the local influence of minor perturbations for a nonlinear structural equation model with missing data that are missing at random. The main idea is to apply Zhu and Lee's (J. Roy. Statist. Soc. Ser. B 63 (2001) 111) approach to the conditional expectation of the complete-data log-likelihood function in the corresponding EM algorithm for deriving the conformal normal curvature. Building blocks for achieving the diagnostic measures are computed via latent variables that are generated by the Gibbs sampler and Metropolis–Hastings algorithm. It is shown that the proposed methodology is feasible for a wide variety of perturbation schemes. To illustrate the methodology, results that are obtained from analyses of some artificial examples, a simulation study, and a real example are presented.

Local influence analysis of nonlinear structural equation models

By regarding the latent random vectors as hypothetical missing data and based on the conditional expectation of the complete-data log-likelihood function in the EM algorithm, we investigate assessment of local influence of various perturbation schemes in a nonlinear structural equation model. The basic building blocks of local influence analysis are computed via observations of the latent variables generated by the Metropolis-Hastings algorithm, while the diagnostic measures are obtained via the conformal normal curvature. Seven perturbation schemes, including some perturbation schemes on latent vectors, are investigated. The proposed procedure is illustrated by a simulation study and a real example.

Bayesian model comparison of nonlinear structural equation models with missing continuous and ordinal categorical data

Missing data are very common in behavioural and psychological research. In this paper, we develop a Bayesian approach in the context of a general nonlinear structural equation model with missing continuous and ordinal categorical data. In the development, the missing data are treated as latent quantities, and provision for the incompleteness of the data is made by a hybrid algorithm that combines the Gibbs sampler and the Metropolis-Hastings algorithm. We show by means of a simulation study that the Bayesian estimates are accurate. A Bayesian model comparison procedure based on the Bayes factor and path sampling is proposed. The required observations from the posterior distribution for computing the Bayes factor are simulated by the hybrid algorithm in Bayesian estimation. Our simulation results indicate that the correct model is selected more frequently when the incomplete records are used in the analysis than when they are ignored. The methodology is further illustrated with a real data set from a study concerned with an AIDS preventative intervention for Filipina sex workers.

Bayesian analysis of two‐level nonlinear structural equation models with continuous and polytomous data

Two-level structural equation models with mixed continuous and polytomous data and nonlinear structural equations at both the between-groups and within-groups levels are important but difficult to deal with. A Bayesian approach is developed for analysing this kind of model. A Markov chain Monte Carlo procedure based on the Gibbs sampler and the Metropolis-Hasting algorithm is proposed for producing joint Bayesian estimates of the thresholds, structural parameters and latent variables at both levels. Standard errors and highest posterior density intervals are also computed. A procedure for computing Bayes factor, based on the key idea of path sampling, is established for model comparison.

An integrated model of attitude and affect: Theoretical foundation and an empirical investigation

Abstract The affect (feelings and emotions) and attitude (evaluative judgment based on brand beliefs) streams of research are combined to propose an integrated model of attitude and choice. The essence of the proposed model is based on the interaction effect between affect and cognition. The predictive validity of the proposed model is tested and compared to several other nested models using a regression and logit framework. Results indicate that the proposed model is significantly better than the traditional multiattribute model both in terms of percentage correctly classified and predictive validity. The proposed interaction model is also tested using structural equations modeling with gratifying results. Building on the Kenny and Judd [Psychol. Bull. 96 (1984) 201] approach, the interaction term is estimated using both the Ping [J. Acad. Mark. Sci. 22 (4) (1994) 364; J. Mark. Res. 32 (3) (1995) 336] method and Joreskog and Yang [Nonlinear structural equation models: the Kenny–Judd model with interaction effects. In: Marcoulides GA, Schumacker RE, editors. Advanced Structural Equation Modeling: Issues and Techniques, Mahwah, NJ: Erlbaum, 1996] method. Important findings and implications are discussed together with directions for future research.