Modeling Dynamic Functional Neuroimaging Data Using Structural Equation Modeling

The aims of this study were to present a method for developing a path analytic network model using data acquired from positron emission tomography. Regions of interest within the human brain were identified through quantitative activation likelihood estimation meta-analysis. Using this information, a “true” or population path model was then developed using Bayesian structural equation modeling. To evaluate the impact of sample size on parameter estimation bias, proportion of parameter replication coverage, and statistical power, a 2 group (clinical/control) × 6 (sample size: N = 10, N = 15, N = 20, N = 25, N = 50, N = 100) Markov chain Monte Carlo study was conducted. Results indicate that using a sample size of less than N = 15 per group will produce parameter estimates exhibiting bias greater than 5% and statistical power below .80.     

Using Structural Equation Modeling to Examine Group Differences in Assessment Booklet Designs with Sparse Data

The current research demonstrates the effectiveness of using structural equation modeling (SEM) for the investigation of subgroup differences with sparse data designs where not every student takes every item. Simulations were conducted that reflected missing data structures like those encountered in large survey assessment programs (e.g., National Assessment of Educational Progress). A maximum likelihood method of estimation was implemented that allowed all data to be used without performing any imputation. A multiple indicators multiple causes (MIMIC) model was used to examine group differences. There was no detriment to the estimation of the MIMIC model parameters under sparse data design conditions when compared to the design without missing data. The overall size of samples had more influence on the variability of estimates than did the data design.     

Variance Estimation Using Replication Methods in Structural Equation Modeling With Complex Sample Data

This article discusses replication sampling variance estimation techniques that are often applied in analyses using data from complex sampling designs: jackknife repeated replication, balanced repeated replication, and bootstrapping. These techniques are used with traditional analyses such as regression, but are currently not used with structural equation modeling (SEM) analyses. This article provides an extension of these methods to SEM analyses, including a proposed adjustment to the likelihood ratio test, and presents the results from a simulation study suggesting replication estimates are robust. Finally, a demonstration of the application of these methods using data from the Early Childhood Longitudinal Study is included. Secondary analysts can undertake these more robust methods of sampling variance estimation if they have access to certain SEM software packages and data management packages such as SAS, as shown in the article.     

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.     

Effect Analysis Using Nonlinear Structural Equation Mixture Modeling

In this article, we present an approach for comprehensive analysis of the effectiveness of interventions based on nonlinear structural equation mixture models (NSEMM). We provide definitions of average and conditional effects and show how they can be computed. We extend the traditional moderated regression approach to include latent continous and discrete (mixture) variables as well as their higher order interactions, quadratic or more general nonlinear relationships. This new approach can be considered a combination of the recently proposed EffectLiteR approach and the NSEMM approach. A key advantage of this synthesis is that it gives applied researchers the opportunity to gain greater insight into the effectiveness of the intervention. For example, it makes it possible to consider structural equation models for situations where the treatment is noneffective for extreme values of a latent covariate but is effective for medium values, as we illustrate using an example from the educational sciences.     

Evaluating Interventions with Multimethod Data: A Structural Equation Modeling Approach

In many intervention and evaluation studies, outcome variables are assessed using a multimethod approach comparing multiple groups over time. In this article, we show how evaluation data obtained from a complex multitrait–multimethod–multioccasion–multigroup design can be analyzed with structural equation models. In particular, we show how the structural equation modeling approach can be used to (a) handle ordinal items as indicators, (b) test measurement invariance, and (c) test the means of the latent variables to examine treatment effects. We present an application to data from an evaluation study of an early childhood prevention program. A total of 659 children in intervention and control groups were rated by their parents and teachers on prosocial behavior and relational aggression before and after the program implementation. No mean change in relational aggression was found in either group, whereas an increase in prosocial behavior was found in both groups. Advantages and limitations of the proposed approach are highlighted.     

Modeling Latent Growth Curves With Incomplete Data Using Different Types of Structural Equation Modeling and Multilevel Software

This article offers different examples of how to fit latent growth curve (LGC) models to longitudinal data using a variety of different software programs (i.e., LISREL, Mx, Mplus, AMOS, SAS). The article shows how the same model can be fitted using both structural equation modeling and multilevel software, with nearly identical results, even in the case of models of latent growth fitted to incomplete data. The general purpose of this article is to provide a demonstration that integrates programming features from different software. The most immediate goal is to help researchers implement these LGC models as a useful way to test hypotheses of growth.     

Comparing Squared Multiple Correlation Coefficients Using Structural Equation Modeling

In social science research, a common topic in multiple regression analysis is to compare the squared multiple correlation coefficients in different populations. Existing methods based on asymptotic theories (Olkin & Finn, 1995) and bootstrapping (Chan, 2009) are available but these can only handle a 2-group comparison. Another method based on structural equation modeling (SEM) has been proposed recently. However, this method has three disadvantages. First, it requires the user to explicitly specify the sample R² as a function in terms of the basic SEM model parameters, which is sometimes troublesome and error prone. Second, it requires the specification of nonlinear constraints, which is not available in some popular SEM software programs. Third, it is for a 2-group comparison primarily. In this article, a 2-stage SEM method is proposed as an alternative. Unlike all other existing methods, the proposed method is simple to use, and it does not require any specific programming features such as the specification of nonlinear constraints. More important, the method allows a simultaneous comparison of 3 or more groups. A real example is given to illustrate the proposed method using EQS, a popular SEM software program.     

Effects of Missing Data Methods in Structural Equation Modeling With Nonnormal Longitudinal Data

The purpose of this study is to investigate the effects of missing data techniques in longitudinal studies under diverse conditions. A Monte Carlo simulation examined the performance of 3 missing data methods in latent growth modeling: listwise deletion (LD), maximum likelihood estimation using the expectation and maximization algorithm with a nonnormality correction (robust ML), and the pairwise asymptotically distribution-free method (pairwise ADF). The effects of 3 independent variables (sample size, missing data mechanism, and distribution shape) were investigated on convergence rate, parameter and standard error estimation, and model fit. The results favored robust ML over LD and pairwise ADF in almost all respects. The exceptions included convergence rates under the most severe nonnormality in the missing not at random (MNAR) condition and recovery of standard error estimates across sample sizes. The results also indicate that nonnormality, small sample size, MNAR, and multicollinearity might adversely affect convergence rate and the validity of statistical inferences concerning parameter estimates and model fit statistics.     

Regularized Structural Equation Modeling

A new method is proposed that extends the use of regularization in both lasso and ridge regression to structural equation models. The method is termed regularized structural equation modeling (RegSEM). RegSEM penalizes specific parameters in structural equation models, with the goal of creating easier to understand and simpler models. Although regularization has gained wide adoption in regression, very little has transferred to models with latent variables. By adding penalties to specific parameters in a structural equation model, researchers have a high level of flexibility in reducing model complexity, overcoming poor fitting models, and the creation of models that are more likely to generalize to new samples. The proposed method was evaluated through a simulation study, two illustrative examples involving a measurement model, and one empirical example involving the structural part of the model to demonstrate RegSEM’s utility.     

Exploratory Structural Equation Modeling

This article examines the problem of specification error in 2 models for categorical latent variables; the latent class model and the latent Markov model. Specification error in the latent class model focuses on the impact of incorrectly specifying the number of latent classes of the categorical latent variable on measures of model adequacy as well as sample reallocation to latent classes. The results show that the clarity of remaining latent classes, as measured by the entropy statistic depends on the number of observations in the omitted latent class—but this statistic is not reliable. Specification error in the latent Markov model focuses on the transition probabilities when a longitudinal Guttman process is incorrectly specified. The findings show that specifying a longitudinal Guttman process that is not true in the population impacts other transition probabilities through the covariance matrix of the logit parameters used to calculate those probabilities.     

Comparisons of Structural Equation Modeling and Hierarchical Linear Modeling Approaches to Couples' Data

This article compares 2 statistical approaches for the analysis of data obtained from married couples. The article summarizes a current multilevel (or hierarchical) model that has demonstrated considerable utility in marital research; it also extends this formulation in several respects. This model is then respecified into a more familiar structural equation modeling (SEM) formulation, highlighting the similarities and the differences in the 2 approaches. Cross-sectional data on 348 American married couples is used to examine the influence of age, duration of marriage, and number of children on marital satisfaction. Results of the 2 sets of analyses yielded nearly identical findings. The strengths and possible extensions of the SEM approach are discussed.     

Structural Equation Modeling Diagnostics Using R Package semdiag and EQS

Yuan and Hayashi (2010) Yuan, K.-H. and Hayashi, K. 2010. Fitting data to model: Structural equation modeling diagnosis using two scatter plots. Psychological Methods, 15: 335–351. [Crossref], [Web of Science ®] , [Google Scholar] introduced 2 scatter plots for model and data diagnostics in structural equation modeling (SEM). However, the generation of the plots requires in-depth understanding of their underlying technical details. This article develops and introduces an R package semdiag for easily drawing the 2 plots. With a model specified in EQS syntax, one only needs to supply as few as 2 parameters to generate the 2 plots using the semdiag package. Two examples are provided to illustrate the use of the package. Multiple figures are used to explain the elements of data and model diagnostics. Advice on selecting proper estimation methods following the diagnostics is also given.     

Investigating Supervisory Relationships and Therapeutic Alliances Using Structural Equation Modeling

The authors used structural equation modeling to examine the contribution of supervisees' supervisory relationship levels to therapeutic alliance (TA) scores with their clients in practicum. Results showed that supervisory relationship scores positively contributed to the TA. Client and counselor ratings of the TA also differed.     

Multiplicity Control in Structural Equation Modeling

Abstract

Researchers conducting structural equation modeling analyses rarely, if ever, control for the inflated probability of Type I errors when evaluating the statistical significance of multiple parameters in a model. In this study, the Type I error control, power and true model rates of famsilywise and false discovery rate controlling procedures were compared with rates when no multiplicity control was imposed. The results indicate that Type I error rates become severely inflated with no multiplicity control, but also that familywise error controlling procedures were extremely conservative and had very little power for detecting true relations. False discovery rate controlling procedures provided a compromise between no multiplicity control and strict familywise error control and with large sample sizes provided a high probability of making correct inferences regarding all the parameters in the model.     

Model Specification Searches in Structural Equation Modeling with R

This article introduces and demonstrates the application of an R statistical programming environment code for conducting structural equation modeling (SEM) specification searches. The implementation and flexibility of the provided code is demonstrated using the Tabu search procedure, although the underlying code can also be directly modified to implement other search procedures like Ant Colony Optimization, Genetic Algorithms, Ruin-and-Recreate, or Simulated Annealing. The application is illustrated using data with a known common factor structure. The results demonstrate the capabilities of the program for conducting specification searches in SEM. The programming codes are provided as open-source R functions.     

Propensity Score Analysis of Complex Survey Data with Structural Equation Modeling: A Tutorial with Mplus

Propensity score (PS) analysis aims to reduce bias in treatment effect estimates obtained from observational studies, which may occur due to non-random differences between treated and untreated groups with respect to covariates related to the outcome. We demonstrate how to use structural equation modeling (SEM) for PS analysis to remove selection bias due to latent covariates and estimate treatment effects on latent outcomes. Following the discussion of the design and analysis stages of PS analysis with SEM, an example is presented which uses the Mplus software to analyze data from the 1999 School and Staffing Survey (SASS) and 2000 Teacher Follow-up Survey (TFS) to estimate the effects teacher’s participation in a network of teachers on the teacher’s perception of workload manageability.     

A Systematic Evaluation and Comparison Between Exploratory Structural Equation Modeling and Bayesian Structural Equation Modeling

In this study, we contrast two competing approaches, not previously compared, that balance the rigor of CFA/SEM with the flexibility to fit realistically complex data. Exploratory SEM (ESEM) is claimed to provide an optimal compromise between EFA and CFA/SEM. Alternatively, a family of three Bayesian SEMs (BSEMs) replace fixed-zero estimates with informative, small-variance priors for different subsets of parameters: cross-loadings (CL), residual covariances (RC), or CLs and RCs (CLRC). In Study 1, using three simulation studies, results showed that (1) BSEM-CL performed more closely to ESEM; (2) BSEM-CLRC did not provide more accurate model estimation compared with BSEM-CL; (3) BSEM-RC provided unstable estimation; and (4) different specifications of targeted values in ESEM and informative priors in BSEM have significant impacts on model estimation. The real data analysis (Study 2) showed that the differences in estimation between different models were largely consistent with those in Study1 but somewhat smaller.     

Testing Time-Invariance of Variable Specificity in Repeated Measure Designs Using Structural Equation Modeling

A structural equation modeling method for examining time-invariance of variable specificity in longitudinal studies with multiple measures is outlined, which is developed within a confirmatory factor-analytic framework. The approach represents a likelihood ratio test for the hypothesis of stability in the specificity part of the residual term associated with repeated administration of each measure. The procedure can be used in the search for parsimonious versions of multiwave multiple-indicator models, to test for variable specificity in them, and to examine assumptions underlying particular parameter estimation procedures in repeated measure designs. The outlined method is illustrated with empirical data.