Model comparison of nonlinear structural equation models with fixed covariates

Recently, it has been recognized that the commonly used linear structural equation model is inadequate to deal with some complicated substantive theory. A new nonlinear structural equation model with fixed covariates is proposed in this article. A procedure, which utilizes the powerful path sampling for computing the Bayes factor, is developed for model comparison. In the implementation, the required random observations are simulated via a hybrid algorithm that combines the Gibbs sampler and the Metropolis-Hastings algorithm. It is shown that the proposed procedure is efficient and flexible; and it produces Bayesian estimates of the parameters, latent variables, and their highest posterior density intervals as by-products. Empirical performances of the proposed procedure such as sensitivity to prior inputs are illustrated by a simulation study and a real example.This research is fully supported by a grant from the Research Grant Council of the Hong Kong Special Administrative Region, China (Project No. CUHK 4346/01H). The authors are thankful to the Editor, the Associate Editor, and anonymous reviewers for valuable comments which improve the paper significantly, and grateful to ICPSR and the relevant funding agency for allowing use of the data in the example. The assistance of Michael K.H. Leung and Esther L.S. Tam is gratefully acknowledged.     

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. The research described herein was fully supported by a grant (CUHK 4243/03H) from the Rearch Grants Council of the Hong Kong Special Administration Region. The authors are thankful to the Editor, the Associate Editor, and anonymous reviewers for valuable comments which improve the paper significantly, and are grateful to ICPSR and the relevant funding agency for allowing the use of their data. Requests for reprints should be sent to S. Y. Lee, Department of Statistics, The Chinese University of Hong Kong, Shatin, N. T., Hong Kong.     

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. Acknowledgment: This research is fully supported by a grant (CUHK 4243/02H) from the Research Grant Council of the Hong Kong Special Administration Region. The authors are indebted to ICPSR and the relevant funding agency for allowing use of their data, and to the Editor and reviewers for their valuable comments for improving the paper.     

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 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 nonlinear SEMs with the normal assumption of the exogenous latent variables violated. The proposed methodology is further illustrated by a real example. Detailed interpretation about the interaction terms is presented.     

Analysis of structural equation model with ignorable missing continuous and polytomous data

The main purpose of this article is to develop a Bayesian approach for structural equation models with ignorable missing continuous and polytomous data. Joint Bayesian estimates of thresholds, structural parameters and latent factor scores are obtained simultaneously. The idea of data augmentation is used to solve the computational difficulties involved. In the posterior analysis, in addition to the real missing data, latent variables and latent continuous measurements underlying the polytomous data are treated as hypothetical missing data. An algorithm that embeds the Metropolis-Hastings algorithm within the Gibbs sampler is implemented to produce the Bayesian estimates. A goodness-of-fit statistic for testing the posited model is presented. It is shown that the proposed approach is not sensitive to prior distributions and can handle situations with a large number of missing patterns whose underlying sample sizes may be small. Computational efficiency of the proposed procedure is illustrated by simulation studies and a real example.The work described in this paper was fully supported by a grant from the Research Grants Council of the HKSAR (Project No. CUHK 4088/99H). The authors are greatly indebted to the Editor and anonymous reviewers for valuable comments in improving the paper; and also to D. E. Morisky and J.A. Stein for the use of their AIDS data set.     

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.     

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.     

A bayesian approach to confirmatory factor analysis

Confirmatory factor analysis is considered from a Bayesian viewpoint, in which prior information on parameter is incorporated in the analysis. An iterative algorithm is developed to obtain the Bayes estimates. A numerical example based on longitudinal data is presented. A simulation study is designed to compare the Bayesian approach with the maximum likelihood method.Computer facilities were provided by the Computer Services Center, The Chinese University of Hong Kong.     

Maximum Likelihood Analysis of Nonlinear Structural Equation Models With Dichotomous Variables

In this article, a maximum likelihood approach is developed to analyze structural equation models with dichotomous variables that are common in behavioral, psychological and social research. To assess nonlinear causal effects among the latent variables, the structural equation in the model is defined by a nonlinear function. The basic idea of the development is to augment the observed dichotomous data with the hypothetical missing data that involve the latent underlying continuous measurements and the latent variables in the model. An EM algorithm is implemented. The conditional expectation in the E-step is approximated via observations simulated from the appropriate conditional distributions by a Metropolis-Hastings algorithm within the Gibbs sampler, whilst the M-step is completed by conditional maximization. Convergence is monitored by bridge sampling. Standard errors are also obtained. Results from a simulation study and a real example are presented to illustrate the methodology.     

A bayesian estimation of factor score in confirmatory factor model with polytomous, censored or truncated data

The posterior analysis in estimating factor score in a confirmatory factor analysis model with polytomous, censored or truncated data is investigated in this paper. For the above three types of data, posterior distributions of the factor score are studied, and the estimators of the factor score are obtained to be the location parameters of the posterior distributions. The accuracy of Bayesian estimates is studied via simulation studies.This research was supported by a Hong Kong UGC grant.     

Bayesian Inference for Growth Mixture Models with Latent Class Dependent Missing Data

Growth mixture models (GMMs) with nonignorable missing data have drawn increasing attention in research communities but have not been fully studied. The goal of this article is to propose and to evaluate a Bayesian method to estimate the GMMs with latent class dependent missing data. An extended GMM is first presented in which class probabilities depend on some observed explanatory variables and data missingness depends on both the explanatory variables and a latent class variable. A full Bayesian method is then proposed to estimate the model. Through the data augmentation method, conditional posterior distributions for all model parameters and missing data are obtained. A Gibbs sampling procedure is then used to generate Markov chains of model parameters for statistical inference. The application of the model and the method is first demonstrated through the analysis of mathematical ability growth data from the National Longitudinal Survey of Youth 1997 (Bureau of Labor Statistics, U.S. Department of Labor, 1997). A simulation study considering 3 main factors (the sample size, the class probability, and the missing data mechanism) is then conducted and the results show that the proposed Bayesian estimation approach performs very well under the studied conditions. Finally, some implications of this study, including the misspecified missingness mechanism, the sample size, the sensitivity of the model, the number of latent classes, the model comparison, and the future directions of the approach, are discussed.     

Influence analysis of structural equation models with polytomous variables

The estimation of model parameters in structural equation models with polytomous variables can be handled by several computationally efficient procedures. However, sensitivity or influence analysis of the model is not well studied. We demonstrate that the existing influence analysis methods for contingency tables or for normal theory structural equation models cannot be applied directly to structural equation models with polytomous variables; and we develop appropriate procedures based on the local influence approach of Cook (1986). The proposed procedures are computationally efficient, the necessary bits of the proposed diagnostic measures are readily available following an usual fit of the model. We consider the influence of an individual cell frequency with respect to three cases: when all parameters in an unstructured model are of interest, when the unstructured polychoric correlations are of interest, and when the structural parameters are of interest. We also consider the sensitivity of the parameters estimates. Two examples based on real data are presented for illustration.The work described in this paper was partially supported by a Chinese University of Hong Kong Postdoctoral Fellows Scheme and a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (RGC Ref. No. CUHK4186/98P). We are indebted to P.M. Bentler and M.D. Newcomb for providing the data set, Michael Leung for his assistance, and the Editor and the referees for some very valuable comments.     

Bayesian analysis of structural equation models with mixed exponential family and ordered categorical data

Structural equation models are very popular for studying relationships among observed and latent variables. However, the existing theory and computer packages are developed mainly under the assumption of normality, and hence cannot be satisfactorily applied to non‐normal and ordered categorical data that are common in behavioural, social and psychological research. In this paper, we develop a Bayesian approach to the analysis of structural equation models in which the manifest variables are ordered categorical and/or from an exponential family. In this framework, models with a mixture of binomial, ordered categorical and normal variables can be analysed. Bayesian estimates of the unknown parameters are obtained by a computational procedure that combines the Gibbs sampler and the Metropolis–Hastings algorithm. Some goodness‐of‐fit statistics are proposed to evaluate the fit of the posited model. The methodology is illustrated by results obtained from a simulation study and analysis of a real data set about non‐adherence of hypertension patients in a medical treatment scheme.     

Bayesian analysis of mixtures in structural equation models with non‐ignorable missing data

Structural equation models (SEMs) have become widely used to determine the interrelationships between latent and observed variables in social, psychological, and behavioural sciences. As heterogeneous data are very common in practical research in these fields, the analysis of mixture models has received a lot of attention in the literature. An important issue in the analysis of mixture SEMs is the presence of missing data, in particular of data missing with a non‐ignorable mechanism. However, only a limited amount of work has been done in analysing mixture SEMs with non‐ignorable missing data. The main objective of this paper is to develop a Bayesian approach for analysing mixture SEMs with an unknown number of components and non‐ignorable missing data. A simulation study shows that Bayesian estimates obtained by the proposed Markov chain Monte Carlo methods are accurate and the Bayes factor computed via a path sampling procedure is useful for identifying the correct number of components, selecting an appropriate missingness mechanism, and investigating various effects of latent variables in the mixture SEMs. A real data set on a study of job satisfaction is used to demonstrate the methodology.     

Bayesian model selection for mixtures of structural equation models with an unknown number of components

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 efficiently 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 by‐products. The properties and merits of the proposed procedure are discussed and illustrated by means of a simulation study and a real example.     

Estimation for structural equation models with missing data

A direct method in handling incomplete data in general covariance structural models is investigated. Asymptotic statistical properties of the generalized least squares method are developed. It is shown that this approach has very close relationships with the maximum likelihood approach. Iterative procedures for obtaining the generalized least squares estimates, the maximum likelihood estimates, as well as their standard error estimates are derived. Computer programs for the confirmatory factor analysis model are implemented. A longitudinal type data set is used as an example to illustrate the results.This research was supported in part by Research Grant DAD1070 from the U.S. Public Health Service. The author is indebted to anonymous reviewers for some very valuable suggestions. Computer funding is provided by the Computer Services Centre, The Chinese University of Hong Kong.     

Covariance structure analysis in several populations

This paper is concerned with the study of covariance structural models in several populations. Estimation theory of the parameters that are subject to general functional restraints is developed based on the generalized least squares approach. Asymptotic properties of the constrained estimator are studied; and asymptotic chi-square tests are presented to evaluate appropriate model comparisons. The method of multipliers and the standard reparametrization technique are discussed in obtaining the estimates. The methodology is demonstrated by a set of real data.Computer facilities were provided by the Computer Services Center, The Chinese University of Hong Kong. The authors are indebted to several anonymous reviewers for suggestions for improvement of this paper.     

Generalized graded unfolding model with structural equation for subject parameters

The generalized graded unfolding model (GGUM) is capable of analyzing polytomous scored, unfolding data such as agree‐disagree responses to attitude statements. In the present study, we proposed a GGUM with structural equation for subject parameters, which enabled us to evaluate the relation between subject parameters and covariates and/or latent variables simultaneously, in order to avoid the influence of attenuation. Additionally, an algorithm for parameter estimation is newly implemented via the Markov Chain Monte Carlo (MCMC) method, based on Bayesian statistics. In the simulation, we compared the accuracy of estimates of regression coefficients between the proposed model and a conventional method using a GGUM (where regression coefficients are estimated using estimates of θ). As a result, the proposed model performed much better than the conventional method in terms of bias and root mean squared errors of estimates of regression coefficients. The study concluded by verifying the efficacy of the proposed model, using an actual data example of attitude measurement.