On the Performance of Likelihood-Based Difference Tests in Nonlinear Structural Equation Models

This article investigates likelihood-based difference statistics for testing nonlinear effects in structural equation modeling using the latent moderated structural equations (LMS) approach. In addition to the standard difference statistic T_D, 2 robust statistics have been developed in the literature to ensure valid results under the conditions of nonnormality or small sample sizes: the robust T_DR and the “strictly positive” T_DRP. These robust statistics have not been examined in combination with LMS yet. In 2 Monte Carlo studies we investigate the performance of these methods for testing quadratic or interaction effects subject to different sources of nonnormality, nonnormality due to the nonlinear terms, and nonnormality due to the distribution of the predictor variables. The results indicate that T_D is preferable to both T_DR and T_DRP. Under the condition of strong nonlinear effects and nonnormal predictors, T_DR often produced negative differences and T_DRP showed no desirable power.     

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.     

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.     

A Note on Structural Equation Modeling Estimates of Reliability

Reliability can be estimated using structural equation modeling (SEM). Two potential problems with this approach are that estimates may be unstable with small sample sizes and biased with misspecified models. A Monte Carlo study was conducted to investigate the quality of SEM estimates of reliability by themselves and relative to coefficient alpha. The SEM approach showed minimal bias when the model was correctly specified if items were relatively well defined by their underlying factor(s). They tended to demonstrate somewhat greater bias when the model was misspecified, particularly underspecified. Overall, SEM estimates were more stable than anticipated. Researchers are more likely to obtain accurate estimates of reliability using SEM by conducting large-sample studies with well-constructed scales and critically assessing model fit.     

Instrumental Variables Two-Stage Least Squares (2SLS) vs. Maximum Likelihood Structural Equation Modeling of Causal Effects in Linear Regression Models

In the presence of omitted variables or similar validity threats, regression estimates are biased. Unbiased estimates (the causal effects) can be obtained in large samples by fitting instead the Instrumental Variables Regression (IVR) model. The IVR model can be estimated using structural equation modeling (SEM) software or using Econometric estimators such as two-stage least squares (2SLS). We describe 2SLS using SEM terminology, and report a simulation study in which we generated data according to a regression model in the presence of omitted variables and fitted (a) a regression model using ordinary least squares, (b) an IVR model using maximum likelihood (ML) as implemented in SEM software, and (c) an IVR model using 2SLS. Coverage rates of the causal effect using regression methods are always unacceptably low (often 0). When using the IVR model, accurate coverage is obtained across all conditions when N = 500. Even when the IVR model is misspecified, better coverage than regression is generally obtained. Differences between 2SLS and ML are small and favor 2SLS in small samples (N ≤ 100).     

The Role of Centering for Interaction of Level 1 Variables in Multilevel Structural Equation Models

This article examined the role of centering in estimating interaction effects in multilevel structural equation models. Interactions are typically represented by product term of 2 variables that are hypothesized to interact. In multilevel structural equation modeling (MSEM), the product term involving Level 1 variables is decomposed into within-cluster and between-cluster random components. The choice of centering affects the decomposition of the product term, and therefore affects the sample variance and covariance associated with the product term used in the maximum likelihood fitting function. The simulation study showed that for an interaction between a Level 1 variable and a Level 2 variable, the product term of uncentered variables or the product term of grand mean centered variables produced unbiased estimates in both Level 1 and Level 2 models. The product term of cluster mean centered variables produced biased estimates in the Level 1 model. For an interaction between 2 Level 1 variables, the product term of cluster mean centered variables produced unbiased estimates in the Level 1 model, whereas the product term of grand mean centered variables produced unbiased estimates for the Level 1 model. Recommendations for researchers who wish to estimate interactions in MSEM are provided.     

A Primer on Maximum Likelihood Algorithms Available for Use With Missing Data

Maximum likelihood algorithms for use with missing data are becoming commonplace in microcomputer packages. Specifically, 3 maximum likelihood algorithms are currently available in existing software packages: the multiple-group approach, full information maximum likelihood estimation, and the EM algorithm. Although they belong to the same family of estimator, confusion appears to exist over the differences among the 3 algorithms. This article provides a comprehensive, nontechnical overview of the 3 maximum likelihood algorithms. Multiple imputation, which is frequently used in conjunction with the EM algorithm, is also discussed.     

A Note on the Use of Mixture Models for Individual Prediction

Mixture models capture heterogeneity in data by decomposing the population into latent subgroups, each of which is governed by its own subgroup-specific set of parameters. Despite the flexibility and widespread use of these models, most applications have focused solely on making inferences for whole or subpopulations, rather than individual cases. This article presents a general framework for computing marginal and conditional predicted values for individuals using mixture model results. These predicted values can be used to characterize covariate effects, examine the fit of the model for specific individuals, or forecast future observations from previous ones. Two empirical examples are provided to demonstrate the usefulness of individual predicted values in applications of mixture models. The first example examines the relative timing of initiation of substance use using a multiple event process survival mixture model, whereas the second example evaluates changes in depressive symptoms over adolescence using a growth mixture model.     

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.     

New Ways to Evaluate Goodness of Fit: A Note on Using Equivalence Testing to Assess Structural Equation Models

Structural equation models are typically evaluated on the basis of goodness-of-fit indexes. Despite their popularity, agreeing what value these indexes should attain to confidently decide between the acceptance and rejection of a model has been greatly debated. A recently proposed approach by means of equivalence testing has been recommended as a superior way to evaluate the goodness of fit of models. The approach has also been proposed as providing a necessary vehicle that can be used to advance the inferential nature of structural equation modeling as a confirmatory tool. The purpose of this article is to introduce readers to key ideas in equivalence testing and illustrate its use for conducting model–data fit assessments. Two confirmatory factor analysis models in which a priori specified latent variable models with known structure and tested against data are used as examples. It is advocated that whenever the goodness of fit of a model is to be assessed researchers should always examine the resulting values obtained via the equivalence testing approach.     

关于代数方程根的研究的注记

利用函数的导数对方程根的存在性、惟一性及个数进行研究和归纳。     

Maximum Likelihood Estimation of Structural Equation Models for Continuous Data: Standard Errors and Goodness of Fit

Classical accounts of maximum likelihood (ML) estimation of structural equation models for continuous outcomes involve normality assumptions: standard errors (SEs) are obtained using the expected information matrix and the goodness of fit of the model is tested using the likelihood ratio (LR) statistic. Satorra and Bentler (1994) introduced SEs and mean adjustments or mean and variance adjustments to the LR statistic (involving also the expected information matrix) that are robust to nonnormality. However, in recent years, SEs obtained using the observed information matrix and alternative test statistics have become available. We investigate what choice of SE and test statistic yields better results using an extensive simulation study. We found that robust SEs computed using the expected information matrix coupled with a mean- and variance-adjusted LR test statistic (i.e., MLMV) is the optimal choice, even with normally distributed data, as it yielded the best combination of accurate SEs and Type I errors.     

缺失数据下线性模型响应变量均值估计的渐近正态性

考虑响应变量随机缺失下线性模型响应变量均值的估计问题,分别获得了基于完全观测样本数据、线性回归插补后的"完全样本"和逆概率加权插补后的"完全样本"得到的响应变量均值估计,并证明了其渐近正态性.     

A Graphical Method for Assessing the Identification of Linear Structural Equation Models

A graphical method is presented for assessing the state of identifiability of the parameters in a linear structural equation model based on the associated directed graph. We do not restrict attention to recursive models. In the recent literature, methods based on graphical models have been presented as a useful tool for assessing the state of identifiability of the parameters of the model. This article proposes the graphical counterpart of the rank condition of the matrix of structural coefficients, which allows for checking of the identifiability through a simple graphical rule. This approach can be used to develop algorithms.     

弱极值原理的一个注记

把文[1]中定理4.2(弱极值原理)关于算子L的系数的条件(2.3)加强为其中p>n,得到定理4.2的结论中常数C只依赖于n,p,Λ／λ,Ω。     

A Simulation Study of Polychoric Instrumental Variable Estimation in Structural Equation Models

Data collected from questionnaires are often in ordinal scale. Unweighted least squares (ULS), diagonally weighted least squares (DWLS) and normal-theory maximum likelihood (ML) are commonly used methods to fit structural equation models. Consistency of these estimators demands no structural misspecification. In this article, we conduct a simulation study to compare the equation-by-equation polychoric instrumental variable (PIV) estimation with ULS, DWLS, and ML. Accuracy of PIV for the correctly specified model and robustness of PIV for misspecified models are investigated through a confirmatory factor analysis (CFA) model and a structural equation model with ordinal indicators. The effects of sample size and nonnormality of the underlying continuous variables are also examined. The simulation results show that PIV produces robust factor loading estimates in the CFA model and in structural equation models. PIV also produces robust path coefficient estimates in the model where valid instruments are used. However, robustness highly depends on the validity of instruments.     

多元函数可微性的一个注记

给出了Henle定理的简单证明，并指出该定理，n≥3时不真，进而又给出了一个当，n≥3时，函数z=f(χ1，χ2，…，χn)在点M0可微的定理及其证明。     

Multilevel Mediation With Small Samples: A Cautionary Note on the Multilevel Structural Equation Modeling Framework

Multilevel structural equation modeling (ML-SEM) for multilevel mediation is noted for its flexibility over a system of multilevel models (MLMs). Sample size requirements are an overlooked limitation of ML-SEM (100 clusters is recommended). We find that 89% of ML-SEM studies have fewer than 100 clusters and the median number is 44. Furthermore, 75% of ML-SEM studies implement 2–1–1 or 1–1–1 models, which can be equivalently fit with MLMs. MLMs theoretically have lower sample size requirements, although studies have yet to assess small sample performance for multilevel mediation. We conduct a simulation to address this pervasive problem. We find that MLMs have more desirable small sample performance and can be trustworthy with 10 clusters. Importantly, many studies lack the sample size and model complexity to necessitate ML-SEM. Although ML-SEM is undeniably more flexible and uniquely positioned for difficult problems, small samples often can be more effectively and simply addressed with MLMs.     

关于回归方程显著性检验的一个注记

针对多元线性回归方程的显著性检验问题,在随机误差服从任意分布的假设下,运用线性模型和随机变量二次型的相关理论,推导出总离差平方和分解式中各分量的数学期望,进而从定量的角度说明回归方程显著性检验中所取拒绝域的合理性。