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.     

Bayesian estimation and testing of structural equation models

The Gibbs sampler can be used to obtain samples of arbitrary size from the posterior distribution over the parameters of a structural equation model (SEM) given covariance data and a prior distribution over the parameters. Point estimates, standard deviations and interval estimates for the parameters can be computed from these samples. If the prior distribution over the parameters is uninformative, the posterior is proportional to the likelihood, and asymptotically the inferences based on the Gibbs sample are the same as those based on the maximum likelihood solution, for example, output from LISREL or EQS. In small samples, however, the likelihood surface is not Gaussian and in some cases contains local maxima. Nevertheless, the Gibbs sample comes from the correct posterior distribution over the parameters regardless of the sample size and the shape of the likelihood surface. With an informative prior distribution over the parameters, the posterior can be used to make inferences about the parameters underidentified models, as we illustrate on a simple errors-in-variables model.We thank David Spiegelhalter for suggesting applying the Gibbs sampler to structural equation models to the first author at a 1994 workshop in Wiesbaden. We thank Ulf Böckenholt, Chris Meek, Marijtje van Duijn, Clark Glymour, Ivo Molenaar, Steve Klepper, Thomas Richardson, Teddy Seidenfeld, and Tom Snijders for helpful discussions, mathematical advice, and critiques of earlier drafts of this paper.     

A three-stage estimation procedure for structural equation models with polytomous variables

This paper is concerned with the analysis of structural equation models with polytomous variables. A computationally efficient three-stage estimator of the thresholds and the covariance structure parameters, based on partition maximum likelihood and generalized least squares estimation, is proposed. An example is presented to illustrate the method.This research was supported in part by a research grant DA01070 from the U.S. Public Health Service. The production assistance of Julie Speckart is gratefully acknowledged.     

On structural models for analyzing change

The recommendation to base the analysis of multi-wave data upon explicit models for change is advocated. Several univariate and multivariate models are described, which emerge from an interaction between the classical test theory and the structural equation modeling approach. The resulting structural models for analyzing change reflect in some of their parameters substantively interesting aspects of intra- and interindividual change in follow-up studies. The models are viewed as an alternative to an ANOVA-based analysis of longitudinal data, and are illustrated on data from a cognitive intervention study of old adults (Bakes et al , 1986). The approach presents a useful means of analyzing change over time, and is applicable for purposes of (latent) growth curve analysis when analysis of variance assumptions are violated (e.g., Schaie & Hertzog, 1982; Morrison, 1976).     

Non‐linear structural equation models with correlated continuous and discrete data

Structural equation models (SEMs) have been widely applied to examine interrelationships among latent and observed variables in social and psychological research. Motivated by the fact that correlated discrete variables are frequently encountered in practical applications, a non‐linear SEM that accommodates covariates, and mixed continuous, ordered, and unordered categorical variables is proposed. Maximum likelihood methods for estimation and model comparison are discussed. One real‐life data set about cardiovascular disease is used to illustrate the methodologies.     

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.     

Testing mediational models with longitudinal data: questions and tips in the use of structural equation modeling

R. M. Baron and D. A. Kenny (1986; see record 1987-13085-001) provided clarion conceptual and methodological guidelines for testing mediational models with cross-sectional data. Graduating from cross-sectional to longitudinal designs enables researchers to make more rigorous inferences about the causal relations implied by such models. In this transition, misconceptions and erroneous assumptions are the norm. First, we describe some of the questions that arise (and misconceptions that sometimes emerge) in longitudinal tests of mediational models. We also provide a collection of tips for structural equation modeling (SEM) of mediational processes. Finally, we suggest a series of 5 steps when using SEM to test mediational processes in longitudinal designs: testing the measurement model, testing for added components, testing for omitted paths, testing the stationarity assumption, and estimating the mediational effects.     

Assessment of generalised Bayesian structural equation models for continuous and binary data

The paper proposes a novel model assessment paradigm aiming to address shortcoming of posterior predictive $p$-values, which provide the default metric of fit for Bayesian structural equation modelling (BSEM). The model framework presented in the paper focuses on the approximate zero approach (Psychological Methods, 17 , 2012, 313), which involves formulating certain parameters (such as factor loadings) to be approximately zero through the use of informative priors, instead of explicitly setting them to zero. The introduced model assessment procedure monitors the out-of-sample predictive performance of the fitted model, and together with a list of guidelines we provide, one can investigate whether the hypothesised model is supported by the data. We incorporate scoring rules and cross-validation to supplement existing model assessment metrics for BSEM. The proposed tools can be applied to models for both continuous and binary data. The modelling of categorical and non-normally distributed continuous data is facilitated with the introduction of an item-individual random effect. We study the performance of the proposed methodology via simulation experiments as well as real data on the ‘Big-5’ personality scale and the Fagerstrom test for nicotine dependence.     

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 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.     

Sensitivity analysis of structural equation models

The main purpose of this paper is to investigate the sensitivity analysis of structural equation model when minor perturbation is introduced. Some influence measure that based on the general case weight perturbation is derived for the generalized least squares estimation. An influence measure that related to the Cook's distance is also developed for the special case deletion perturbation scheme. Using the proposed methodology, the influential observation in a data set can be detected. Moreover, the general theory can be applied to detect the influential parameters in a model. Finally, some illustrative artificial and real examples are presented. The research of the first author was supported by a Hong Kong UPGC grant. The authors are greatly indebted to two reviewers for some very valuable comments for improvement of the paper.     

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.     

缺失数据的结构方程建模:全息极大似然估计时辅助变量的作用

本研究通过蒙特卡洛模拟考查了采用全息极大似然估计进行缺失数据建模时辅助变量的作用。具体考查了辅助变量与研究变量的共缺机制、共缺率、相关程度、辅助变量数目与样本量等因素对参数估计结果精确性的影响。结果表明,当辅助与研究变量共缺时:(1)对于完全随机缺失的辅助变量,结果更容易出现偏差;(2)对于MAR-MAR组合机制,纳入单个辅助变量是有益的;对于MAR-MCAR或MAR-MNAR组合机制,纳入多于一个辅助变量的效果更好;(3)纳入与研究变量低相关的辅助变量对结果也是有益的。     

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.     

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.     

Constrained versus unconstrained estimation in structural equation modeling

Recently, R. D. Stoel, F. G. Garre, C. Dolan, and G. van den Wittenboer (2006) reviewed approaches for obtaining reference mixture distributions for difference tests when a parameter is on the boundary. The authors of the present study argue that this methodology is incomplete without a discussion of when the mixtures are needed and show that they only become relevant when constrained difference tests are conducted. Because constrained difference tests can hide important model misspecification, a reliable way to assess global model fit under constrained estimation would be needed. Examination of the options for assessing model fit under constrained estimation reveals that no perfect solutions exist, although the conditional approach of releasing a degree of freedom for each active constraint appears to be the most methodologically sound one. The authors discuss pros and cons of constrained and unconstrained estimation and their implementation in 5 popular structural equation modeling packages and argue that unconstrained estimation is a simpler method that is also more informative about sources of misfit. In practice, researchers will have trouble conducting constrained difference tests appropriately, as this requires a commitment to ignore Heywood cases. Consequently, mixture distributions for difference tests are rarely appropriate.     

On the heritability of inspection time and its covariance with IQ: a twin study

Using the classical twin design, this study investigates the influence of genetic factors on the large phenotypic variance in inspection time (IT), and whether the well established IT–IQ association can be explained by a common genetic factor. Three hundred ninety pairs of twins (184 monozygotic, MZ; 206 dizygotic, DZ) with a mean age of 16 years participated, and 49 pairs returned approximately 3 months later for retesting. As in many IT studies, the pi figure stimulus was used and IT was estimated from the cumulative normal ogive. IT ranged from 39.4 to 774.1 ms (159±110.1 ms) with faster ITs (by an average of 26.9 ms) found in the retest session from which a reliability of .69 was estimated. Full-scale IQ (FIQ) was assessed by the Multidimensional Aptitude Battery (MAB) and ranged from 79 to 145 (111±13). The phenotypic association between IT and FIQ was confirmed (−.35) and bivariate results showed that a common genetic factor accounted for 36% of the variance in IT and 32% of the variance in FIQ. The maximum likelihood estimate of the genetic correlation was −.63. When performance and verbal IQ (PIQ & VIQ) were analysed with IT, a stronger phenotypic and genetic relationship was found between PIQ and IT than with VIQ. A large part of the IT variance (64%) was accounted for by a unique environmental factor. Further genetic factors were needed to explain the remaining variance in IQ with a small component of unique environmental variance present. The separability of a shared genetic factor influencing IT and IQ from the total genetic variance in IQ suggests that IT affects a specific subcomponent of intelligence rather than a generalised efficiency.