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.     

Assessing Structural Equation Models by Equivalence Testing With Adjusted Fit Indexes

Conventional null hypothesis testing (NHT) is a very important tool if the ultimate goal is to find a difference or to reject a model. However, the purpose of structural equation modeling (SEM) is to identify a model and use it to account for the relationship among substantive variables. With the setup of NHT, a nonsignificant test statistic does not necessarily imply that the model is correctly specified or the size of misspecification is properly controlled. To overcome this problem, this article proposes to replace NHT by equivalence testing, the goal of which is to endorse a model under a null hypothesis rather than to reject it. Differences and similarities between equivalence testing and NHT are discussed, and new “T-size” terminology is introduced to convey the goodness of the current model under equivalence testing. Adjusted cutoff values of root mean square error of approximation (RMSEA) and comparative fit index (CFI) corresponding to those conventionally used in the literature are obtained to facilitate the understanding of T-size RMSEA and CFI. The single most notable property of equivalence testing is that it allows a researcher to confidently claim that the size of misspecification in the current model is below the T-size RMSEA or CFI, which gives SEM a desirable property to be a scientific methodology. R code for conducting equivalence testing is provided in an appendix.     

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.     

Bayesian Quantile Structural Equation Models

Structural equation modeling is a common multivariate technique for the assessment of the interrelationships among latent variables. Structural equation models have been extensively applied to behavioral, medical, and social sciences. Basic structural equation models consist of a measurement equation for characterizing latent variables through multiple observed variables and a mean regression-type structural equation for investigating how explanatory latent variables influence outcomes of interest. However, the conventional structural equation does not provide a comprehensive analysis of the relationship between latent variables. In this article, we introduce the quantile regression method into structural equation models to assess the conditional quantile of the outcome latent variable given the explanatory latent variables and covariates. The estimation is conducted in a Bayesian framework with Markov Chain Monte Carlo algorithm. The posterior inference is performed with the help of asymmetric Laplace distribution. A simulation shows that the proposed method performs satisfactorily. An application to a study of chronic kidney disease is presented.     

正定线性方程组的一种迭代解法与分解

运用行列式、分块矩阵运算、正定矩阵的性质与Sherman-Morrison公式证明了正定矩阵的相关结论,结合正定矩阵性质得到了正定线性方程组的一种新的迭代解法和分解,相关的数值实验表明其算法计算量小,至多步比最速下降法快,比共轭梯度法效率高.     

高阶线性微分方程特解的一种代数解法

提出了高阶常系数线性微分方程Pn(D)x=Pm(t)e^λs特解的一种代数解法。     

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.     

A Structural Equation Model at the Individual and Group Level for Assessing Faking-Related Change

This article proposes a comprehensive approach based on structural equation modeling for assessing the amount of trait-level change derived from faking-motivating situations. The model is intended for a mixed 2-wave 2-group design, and assesses change at both the group and the individual level. Theoretically the model adopts an integrative approach that relates the 2 main current conceptualizations of faking, and models the amount of trait change as an individual-differences variable. The model and procedures are used in an empirical study based on 512 participants. Some of the results are interesting and warrant further research. Overall, the methodology that is proposed provides new resources for the theoretical and applied assessment of faking. In particular, it provides the practitioner with new tools for clearly assessing faking at the individual level.     

Multivariate Meta-Analysis as Structural Equation Models

Multivariate meta-analysis has become increasingly popular in the educational, social, and medical sciences. It is because the outcome measures in a meta-analysis can involve more than one effect size. This article proposes 2 mathematically equivalent models to implement multivariate meta-analysis in structural equation modeling (SEM). Specifically, this article shows how multivariate fixed-, random- and mixed-effects meta-analyses can be formulated as structural equation models. metaSEM (a free R package based on OpenMx) and Mplus are used to implement the proposed procedures. A real data set is used to illustrate the procedures. Formulating multivariate meta-analysis as structural equation models provides many new research opportunities for methodological development in both meta-analysis and SEM. Issues related to and extensions on the SEM-based meta-analysis are discussed.     

一元线性结构关系EV模型的假设检验

给出了一元线性结构关系EV模型y=a+bx,Y=y+ε,X=x+u中关于线性关系y=a+bx的假设检验法则.     

Assessing Model Similarity in Structural Equation Modeling

Two models can be nonequivalent, but fit very similarly across a wide range of data sets. These near-equivalent models, like equivalent models, should be considered rival explanations for results of a study if they represent plausible explanations for the phenomenon of interest. Prior to conducting a study, researchers should evaluate plausible models that are alternatives to those hypothesized to evaluate whether they are near-equivalent or equivalent and, in so doing, address the adequacy of the study’s methodology. To assess the extent to which alternative models for a study are empirically distinguishable, we propose 5 indexes that quantify the degree of similarity in fit between 2 models across a specified universe of data sets. These indexes compare either the maximum likelihood fit function values or the residual covariance matrices of models. Illustrations are provided to support interpretations of these similarity indexes.     

用残差法拟合线性方程

给出一种利用残差二次作图来修正作图法拟合线性方程的方法,利用该方法,可以使作图法拟合的线性方程更接近于用最小二乘法拟合的方程,从而减小作图法求直线截距和斜率的不确定度.     

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.     

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.     

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.     

Fixed-Effects Meta-Analyses as Multiple-Group Structural Equation Models

Meta-analysis is the statistical analysis of a collection of analysis results from individual studies, conducted for the purpose of integrating the findings. Structural equation modeling (SEM), on the other hand, is a multivariate technique for testing hypothetical models with latent and observed variables. This article shows that fixed-effects meta-analyses with the following characteristics can be modeled in the SEM framework: (a) using any type of effect size; (b) including categorical and continuous moderators; and (c) including multivariate effect sizes. Empirical examples in LISREL syntax are used to demonstrate the equivalence between the meta-analytic and SEM approaches. Future directions for and extensions to 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.     

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.