Constructing Approximate Confidence Intervals for Parameters With Structural Equation Models

Confidence intervals (CIs) for parameters are usually constructed based on the estimated standard errors. These are known as Wald CIs. This article argues that likelihood-based CIs (CIs based on likelihood ratio statistics) are often preferred to Wald CIs. It shows how the likelihood-based CIs and the Wald CIs for many statistics and psychometric indexes can be constructed with the use of phantom variables (Rindskopf, 1984 Rindskopf, D. 1984. Using phantom and imaginary latent variables to parameterize constraints in linear structural models. Psychometrika, 49: 37–47. [Crossref], [Web of Science ®] , [Google Scholar]) in some of the current structural equation modeling (SEM) packages. The procedures to form CIs for the differences in correlation coefficients, squared multiple correlations, indirect effects, coefficient alphas, and reliability estimates are illustrated. A simulation study on the Pearson correlation is used to demonstrate the advantages of the likelihood-based CI over the Wald CI. Issues arising from this SEM approach and extensions of this approach are discussed.     

Approximations to the Distributions of Fit Indexes for Misspecified Structural Equation Models

Approximations to the distributions of goodness-of-fit indexes in structural equation modeling are derived with the assumption of multivariate normality and slight misspecification of models. The fit indexes considered in this article are Joreskog and Sorbom's goodness-of-fit index (GFI) and the adjusted GFI, McDonald's absolute GFI, Steiger and Lind's root mean squared error of approximation, Steiger's Γ₁ and Γ₂, Bentler and Bonett's normed fit index, Bollen's incremental fit index and ρ₁, Tucker and Lewis's index ρ₂, and Bentler's fit index (McDonald and Marsh's relative noncentrality index). An approximation to the asymptotic covariance matrix for the fit indexes is derived by using the delta method. Furthermore, approximations to the densities of the fit indexes are obtained from the transformations of the asymptotically noncentral chi-square distributed variable. A simulation is carried out to confirm the accuracy of the approximations.     

Bayesian Data-Model Fit Assessment for Structural Equation Modeling

Bayesian approaches to modeling are receiving an increasing amount of attention in the areas of model construction and estimation in factor analysis, structural equation modeling (SEM), and related latent variable models. However, model diagnostics and model criticism remain relatively understudied aspects of Bayesian SEM. This article describes and illustrates key features of Bayesian approaches to model diagnostics and assessing data–model fit of structural equation models, discussing their merits relative to traditional procedures.     

Level-Specific Evaluation of Model Fit in Multilevel Structural Equation Modeling

This article examines the structure of several HIV risk behaviors in an ethnically and geographically diverse sample of 8,251 clients from 10 innovative demonstration projects intended for adolescents living with, or at risk for, HIV. Exploratory and confirmatory factor analyses identified 2 risk factors for men (sexual intercourse with men and a general risk factor) and 3 factors for women (sexual intercourse with men, substance abuse, and a high risky sex behavior factor). All factors except women engaging in risky sex with men strongly predicted known HIV status of clients for men and women. The findings from this investigation highlight the use of structural equation modeling for applied problems involving overlapping and complex sets of risk behaviors in youth who present at community health programs.     

Two Rules of Identification for Structural Equation Models

Identification of structural equation models remains a challenge to many researchers. Although empirical tests of identification are readily available in structural equation modeling software, these examine local identification and rely on sample estimates of parameters. Rules of identification are available, but do not include all models encountered in practice. In this article we provide 2 rules of identification: the 2+ emitted paths rule and the exogenous X rule. The former is a necessary condition of identification and the latter is a sufficient condition. We explain and prove each of these rules and provide illustrations of their application. These rules extend the coverage of structural equation models that we can check for identification. We also explain how they can be part of a piecewise identification strategy that extends their use even further.     

An Investigation of the Effect of Sample Size and Specification Error on the Fit of Structural Equation Models

Abstract

Covariance structure analysis provides a useful methodology to test hypotheses about competing structural models. The chi-square goodness of fit test is basically an appropriate test for model evaluation. However, methodologists are particularly concerned about the validity of the test to detect misspecified models in small samples. At the same time, there is the concern of rejecting models with reasonably good fit in large samples. The present Monte Carlo study examined the validity of the chi-square test in different instances of misspecification and sample size. The usefulness of the chi-square difference statistic to compare competing structures and improvement in fit is also addressed.     

Finite Feedback Cycling in Structural Equation Models

In models containing reciprocal effects, or longer causal loops, the usual effect estimates assume that any effect touching a loop initiates an infinite cycling of effects around that loop. The real world, in contrast, might permit only finite feedback cycles. I use a simple hypothetical model to demonstrate that if the world permits only a few effect cycles, many coefficient estimates are substantially biased. If the world permits additional partial-cycle use in addition to full cyclings around the causal loop, some of the effect estimates are proper, and a full set of proper effect estimates can be recovered by hand calculations involving the model total effects. If the world permits no additional partial-cycle use, it might not be possible to recover proper estimates from the usual output.

It is not the equations representing the causal model, but rather the calculations of the covariance implications of the model, that change with limited cycling possibilities. Unfortunately, the features required to permit direct estimation of limited-cycle effects are not under user control in common structural equation programs, so estimation and detailed investigation of models with finite cycling of effects around feedback loops awaits new programming. To obtain unbiased estimates with limited causal cyclings, the researcher must continue to strive to specify the proper effect locations but must also attend to the number of full and partial causal cyclings permitted by the world. Determining the appropriate number of cycles is not a matter to be delegated to a statistician; it is something the researcher must attend to as a matter of substantive theory, methodology, and model interpretation.     

Multiple Imputation Strategies for Multiple Group Structural Equation Models

Although structural equation modeling software packages use maximum likelihood estimation by default, there are situations where one might prefer to use multiple imputation to handle missing data rather than maximum likelihood estimation (e.g., when incorporating auxiliary variables). The selection of variables is one of the nuances associated with implementing multiple imputation, because the imputer must take special care to preserve any associations or special features of the data that will be modeled in the subsequent analysis. For example, this article deals with multiple group models that are commonly used to examine moderation effects in psychology and the behavioral sciences. Special care must be exercised when using multiple imputation with multiple group models, as failing to preserve the interactive effects during the imputation phase can produce biased parameter estimates in the subsequent analysis phase, even when the data are missing completely at random or missing at random. This study investigates two imputation strategies that have been proposed in the literature, product term imputation and separate group imputation. A series of simulation studies shows that separate group imputation adequately preserves the multiple group data structure and produces accurate parameter estimates.     

文化层次结构模型比较研究

文化结构决定了文化功能实现以及文化的自适应发展模式。同心圆模型把文化划分为由内及外的多个层次,由内到外的层次之间是渐次主导的关系,而由外到内是对这种关系的反应。睡莲模型把同心圆模型中的观念层作了严格区分,以强调“所倡导的”与“所共享的”之间的不一致。目前的文化结构模型都有“把文化分层”的思想,从同心圆模型到睡莲模型,对层次性的强调依次减弱,而对文化系统内部的功能实现机制强调依次上升。不使用“固定层”的概念,而使用“动态块”的概念或许能弥补当前文化层次结构模型的不足。     

Analyzing Mixed-Dyadic Data Using Structural Equation Models

Mixed-dyadic data, collected from distinguishable (nonexchangeable) or indistinguishable (exchangeable) dyads, require statistical analysis techniques that model the variation within dyads and between dyads appropriately. The purpose of this article is to provide a tutorial for performing structural equation modeling analyses of cross-sectional and longitudinal models for mixed independent variable dyadic data, and to clarify questions regarding various dyadic data analysis specifications that have not been addressed elsewhere. Artificially generated data similar to the Newlywed Project and the Swedish Adoption Twin Study on Aging were used to illustrate analysis models for distinguishable and indistinguishable dyads, respectively. Due to their widespread use among applied researchers, the AMOS and Mplus statistical analysis software packages were used to analyze the dyadic data structural equation models illustrated here. These analysis models are presented in sufficient detail to allow researchers to perform these analyses using their preferred statistical analysis software package.     

Testing Complex Correlational Hypotheses With Structural Equation Models

It is often of interest to estimate partial or semipartial correlation coefficients as indexes of the linear association between 2 variables after partialing one or both for the influence of covariates. Squaring these coefficients expresses the proportion of variance in 1 variable explained by the other variable after controlling for covariates. Methods exist for testing hypotheses about the equality of these coefficients across 2 or more groups, but they are difficult to conduct by hand, prone to error, and limited to simple cases. A unified framework is provided for estimating bivariate, partial, and semipartial correlation coefficients using structural equation modeling (SEM). Within the SEM framework, it is straightforward to test hypotheses of the equality of various correlation coefficients with any number of covariates across multiple groups. LISREL syntax is provided, along with 4 examples.     

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.     

Equivalent Structural Equation Models: A Challenge and Responsibility

Relevant aspects of the example provided by Raykov and Marcoulides (2001) Raykov, T. and Marcoulides, G. A. 2001. Can there be infinitely many models equivalent to a given covariance structure model?. Structural Equation Modeling, 8: 142–149. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar] are emphasized, specifically the distinctiveness of infinitely many members of its sequence of equivalent structural equation models. This emphasis appears to be needed in light of recent statements by Markus (2002) Markus, K. A. 2002. Statistical equivalence, semantic equivalence, eliminative induction and the Raykov–Marcoulides proof of infinite equivalence. Structural Equation Modeling, 9: 503–522. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar], whose intended counterexamples do not present a disconfirmation of any of the developments of Raykov and Marcoulides (2001) Raykov, T. and Marcoulides, G. A. 2001. Can there be infinitely many models equivalent to a given covariance structure model?. Structural Equation Modeling, 8: 142–149. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]. Issues pertaining to differentiation between equivalent models are also discussed.     

An Examination of Statistical Power in Multigroup Dynamic Structural Equation Models

This study used statistical simulation to calculate differential statistical power in dynamic structural equation models with groups (as in McArdle & Prindle, 2008 McArdle, J. J. and Prindle, J. J. 2008. A latent change score analysis of a randomized clinical trial in reasoning training. Psychology and Aging, 23: 702–719. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). Patterns of between-group differences were simulated to provide insight into how model parameters influence power approximations. Chi-square and root mean square error of approximation (RMSEA) power approximation procedures were used to compare the effects of parameter manipulations and how researchers should interpret findings. The chi-square power of perfect fit calls for at least 270 individuals to detect moderate differences, whereas the RMSEA procedure of close fit seems to require as many as 1,450 participants. It is shown that parameters that provide input into the change score that the transfer leads to affect power versus indirect pathways. A discussion of differences in approximation values and future research directions follows.     

The Incorporation of Sample Weights Into Multilevel Structural Equation Models

Over the past decade and a half, methodologists working with structural equation modeling (SEM) have developed approaches for accommodating multilevel data. These approaches are particularly helpful when modeling data that come from complex sampling designs. However, most data sets that are associated with complex sampling designs also include observation weights, and methods to incorporate these sampling weights into multilevel SEM analyses have not been addressed. This article investigates the use of different weighting techniques and finds, through a simulation study, that the use of an effective sample size weight provides unbiased estimates of key parameters and their sampling variances. Also, a popular normalization technique of scaling weights to reflect the actual sample size is shown to produce negatively biased sampling variance estimates, as well as negatively biased within-group variance parameter estimates in the small group size case.     

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.     

基于正态分布的分组数据置信区间问题

主要研究了分组数据的区间估计问题.一个密度函数的支撑集(不依赖于参数)被分成一些互不相交区间,则分组数据是样本落入每一区间的观测值的个数.本文主要讨论了正态分布下分组数据均值的区间估计问题.     

Using Mx to Analyze Cross-Level Effects in Two-Level Structural Equation Models

Two-level data sets are frequently encountered in social and behavioral science research. They arise when observations are drawn from a known hierarchical structure, such as when individuals are randomly drawn from groups that are randomly drawn from a target population. Although 2-level data analysis in the context of structural equation modeling can be conducted by easily accessible software such as LISREL, the group- and individual-level effects are usually treated as though they are uncorrelated. When extra group variables are measured and their relationships with individual-level variables are studied, the analysis of cross-level covariance structures is of interest. In this article, we propose a model setup framework in Mx that allows the analysis of cross-level covariance structures. An illustrative example is given and a small-scale simulation study is conducted to examine the performance of the proposed procedure. The results show that the proposed method can produce reliable parameter and standard error estimates, and the goodness-of-fit statistics also follow the chi-square distribution in large samples.     

Transformation Structural Equation Models With Highly Nonnormal and Incomplete Data

As useful multivariate techniques, structural equation models have attracted significant attention from various fields. Most existing statistical methods and software for analyzing structural equation models have been developed based on the assumption that the response variables are normally distributed. Several recently developed methods can partially address violations of this assumption, but still encounter difficulties in analyzing highly nonnormal data. Moreover, the presence of missing data is a practical issue in substantive research. Simply ignoring missing data or improperly treating nonignorable missingness as ignorable could seriously distort statistical influence results. The main objective of this article is to develop a Bayesian approach for analyzing transformation structural equation models with highly nonnormal and missing data. Different types of missingness are discussed and selected via the deviance information criterion. The empirical performance of our method is examined via simulation studies. Application to a study concerning people’s job satisfaction, home life, and work attitude is presented.     

Rosenau-Burgers方程的线性化差分方法

本文对Rosenau-Burgers方程的初边值问题进行了数值研究,提出了一个线性化的差分格式,证明了差分解的存在唯一性,并利用离散泛函分析方法分析了该格式的二阶收敛性与稳定性,数值试验验证了方法的有效性。