Cover
Preface
1 Introduction to structural equation modeling
1. 1.1 Introduction
2. 1.2 Model formulation
3. 1.3 Model identification
4. 1.4 Model estimation
5. 1.5 Model fit evaluation
6. 1.6 Model modification
7. 1.7 Computer programs for SEM
8. Expressing variances and covariances among observed variables as functions of model parameters
9. Maximum likelihood function for SEM
2 Confirmatory factor analysis
1. 2.1 Introduction
2. 2.2 Basics of CFA models
3. 2.3 CFA models with continuous indicators
4. 2.4 CFA models with non‐normal and censored continuous indicators
5. 2.5 CFA models with categorical indicators
6. 2.6 The item response theory (IRT) model and the graded response model (GRM)
7. 2.7 Higher‐order CFA models
8. 2.8 Bifactor models
9. 2.9 Bayesian CFA models
10. 2.10 Plausible values of latent variables
11. Appendix BSI‐18 instrument
12. Appendix Item reliability
13. Appendix Cronbach's alpha coefficient
14. Appendix Calculating probabilities using probit regression coefficients
3 Structural equation models
1. 3.1 Introduction
2. 3.2 Multiple indicators, multiple causes (MIMIC) model
3. 3.3 General structural equation models
4. 3.4 Correcting for measurement error in single indicator variables
5. 3.5 Testing interactions involving latent variables
6. 3.6 Moderated mediating effect models
7. 3.7 Using plausible values of latent variables in secondary analysis
8. 3.8 Bayesian structural equation modeling (BSEM)
9. Appendix Influence of measurement errors
10. Appendix Fraction of missing information (FMI)
4 Latent growth modeling (LGM) for longitudinal data analysis
1. 4.1 Introduction
2. 4.2 Linear LGM
3. 4.3 Nonlinear LGM
4. 4.4 Multiprocess LGM
5. 4.5 Two‐part LGM
6. 4.6 LGM with categorical outcomes
7. 4.7 LGM with individually varying times of observation
8. 4.8 Dynamic structural equation modeling (DSEM)
5 Multigroup modeling
1. 5.1 Introduction
2. 5.2 Multigroup CFA models
3. 5.3 Multigroup SEM
4. 5.4 Multigroup latent growth modeling (LGM)
6 Mixture modeling
1. 6.1 Introduction
2. 6.2 Latent class analysis (LCA) modeling
3. 6.3 Extending LCA to longitudinal data analysis
4. 6.4 Growth mixture modeling (GMM)
5. 6.5 Factor mixture modeling (FMM)
6. 6.A Appendix Including covariates in LTA model
7. Appendix Manually implementing three‐step mixture modeling
7 Sample size for structural equation modeling
1. 7.1 Introduction
2. 7.2 The rules of thumb for sample size in SEM
3. 7.3 The Satorra‐Saris method for estimating sample size
4. 7.4 Monte Carlo simulation for estimating sample sizes
5. 7.5 Estimate sample size for SEM based on model fit indexes
6. 7.6 Estimate sample sizes for latent class analysis (LCA) model
References
Index
End User License Agreement

List of Tables

Chapter 1
1. Table 1.1 Definitions of the variable matrixes in the three basic equat...
2. Table 1.2 Eight fundamental parameter matrices for a general structural...
Chapter 2
1. Table 2.1 Subscales and corresponding items of BSI‐18.
2. Table 2.2 Selected Mplus output: three‐factor CFA.
3. Table 2.3 Selected Mplus output: CFA with effect coding.
4. Table 2.4 Selected Mplus output: non‐normality test.
5. Table 2.5 Selected Mplus output: CFA using robust estimator MLR.
6. Table 2.6 Selected Mplus output: the restricted model.
7. Table 2.7 Calculating scaled difference in chi‐squares test for nested ...
8. Table 2.8 Selected Mplus output: CFA with censored continuous indicator...
9. Table 2.9 Selected Mplus output: CFA with binary indicators.
10. Table 2.10 Selected Mplus output: model test using the DIFFTEST option....
11. Table 2.11 Selected Mplus output: 2PL IRT model.
12. Table 2.12 Selected Mplus output: 2PL GRM model.
13. Table 2.13 Selected Mplus output: second‐order CFA with continuous indi...
14. Table 2.14 Schmid‐Leiman transformation of the second‐order CFA model e...
15. Table 2.15 Selected Mplus output: bifactor model.
16. Table 2.16 Selected Mplus output: BCFA model.
Chapter 3
1. Table 3.1 Selected Mplus output: MIMIC model.
2. Table 3.2 Selected Mplus output: MIMIC model with covariate interaction...
3. Table 3.3 Selected Mplus output: extended MIMIC model – 1.
4. Table 3.4 Selected Mplus output: extended MIMIC model – 2.
5. Table 3.5 Selected Mplus output: structural equation model.
6. Table 3.6 Selected Mplus output: testing indirecteffects.
7. Table 3.7 Selected Mplus output: testing direct, specific indirect, tot...
8. Table 3.8 Comparisons of model results with and without correcting for ...
9. Table 3.9 Selected Mplus output: testing interactions between observed ...
10. Table 3.10 Selected Mplus output: testing the significance of direct an...
11. Table 3.11 Selected Mplus output: bootstrap estimates of the direct and...
12. Table 3.12 Selected Mplus output: SEM using plausible values.
13. Table 3.13 Selected Mplus output: Bayesian path analysis.
Chapter 4
1. Table 4.1 Selected Mplus output: unconditional linear LGM.
2. Table 4.2 Selected Mplus output: linear LGM with time‐invariant covaria...
3. Table 4.3 Selected Mplus output for linear LGM with time‐invariant and ...
4. Table 4.4 Selected Mplus output: quadratic linear LGM.
5. Table 4.5 Selected Mplus output: LGM with time scores centered at the m...
6. Table 4.6 Integer orthogonal second order polynomials.
7. Table 4.7 Selected Mplus output: quadratic linear LGM with integer orth...
8. Table 4.8 SAS output for general orthogonal second‐order polynomials.
9. Table 4.9 Selected Mplus output: quadratic linear LGM with general orth...
10. Table 4.10 Selected Mplus output: piecewise LGM.
11. Table 4.11 Selected Mplus output: LGM with free time scores.
12. Table 4.12 Selected Mplus output: LGM with time‐invariant covariates an...
13. Table 4.13 Selected Mplus output: parallel‐processes LGM.
14. Table 4.14 Selected Mplus output: two‐part LGM.
15. Table 4.15 Selected Mplus output: LGM with categorical outcomes using d...
16. Table 4.16 Selected Mplus output: LGM with categorical outcome using th...
17. Table 4.17 Selected Mplus output: LGM with individually varying time sc...
18. Table 4.18 Selected Mplus output: RDSEM using observed centering for co...
19. Table 4.19 Selected Mplus output: RDSEM using latent variable centering...
Chapter 5
1. Table 5.1 Selected Mplus output: baseline models for Ohio.
2. Table 5.2 Selected Mplus output: baseline models for Kentucky.
3. Table 5.3 Selected Mplus output: configural CFA model.
4. Table 5.4 Selected Mplus output: multigroup CFA model with equality res...
5. Table 5.5 Model χ² statistics of multigroup CFA models with and without...
6. Table 5.6 Selected Mplus output: multigroup CFA model with equality res...
7. Table 5.7 Selected Mplus output: multigroup CFA model with equality res...
8. Table 5.8 Selected Mplus output: testing factor variances invariance.
9. Table 5.9 Selected Mplus output: testing factor covariance invariance....
10. Table 5.10 Selected Mplus output: testing factor means invariance.
11. Table 5.11 Selected Mplus output: second‐order configural CFA model.
12. Table 5.12 Selected Mplus output: testing invariance of second‐order fa...
13. Table 5.13 Selected Mplus output: testing factor mean invariance in a s...
14. Table 5.14 Selected Mplus output: testing factor mean invariance in a s...
15. Table 5.15 Selected Mplus output: testing factor mean invariance in sec...
16. Table 5.16 Selected Mplus output: full measurement invariance model with categor...
17. Table 5.17 Selected Mplus output: partial measurement invariance model ...
18. Table 5.18 Selected Mplus output: the baseline structural equation mode...
19. Table 5.19 Selected Mplus output: multigroup configural LGM.
20. Table 5.20 Model estimated outcome level and change over time by sample...
21. Table 5.21 Selected Mplus output: testing invariance of the growth func...
Chapter 6
1. Table 6.1 Frequency of drug use.
2. Table 6.2 Comparisons of different LCA models.
3. Table 6.3 Selected Mplus output: unconditional three‐class LCA model.
4. Table 6.4 Unconditional and conditional probabilities: the three‐class ...
5. Table 6.5 Saving Mplus output information for further analysis.
6. Table 6.6 Selected Mplus output: conditional three‐class LCA model.
7. Table 6.7 Selected Mplus output: auxiliary variables in LCA using the P...
8. Table 6.8 Selected Mplus output: manual implementation of the BCH metho...
9. Table 6.9 Sample statistics of the observed outcome indicators by time ...
10. Table 6.10 Model fit information of different LCA models fort ₁ and t...
11. Table 6.11 Unconditional and conditional probabilities: three‐class LCA...
12. Table 6.12 Unconditional and conditional probabilities: three‐class LCA...
13. Table 6.13 Comparisons of different LTA models (N = 430).
14. Table 6.14 Selected Mplus output: three‐class LTA model with measuremen...
15. Table 6.15 Transition probabilities from latent classc ₁ to latent cla...
16. Table 6.16 Logits of regressingc ₂ on c ₁.
17. Table 6.17 Saving Mplus output information in a text data file for furt...
18. Table 6.18 Comparisons of different growth mixture models (N = 430).
19. Table 6.19 Selected Mplus output: four‐class unconditional growth mixtu...
20. Table 6.20 Selected Mplus output: growth mixture model with covariates ...
21. Table 6.21 Selected Mplus output: two‐factor/two‐class MFA model.
22. Table 6.22 Selected Mplus output: LTA with a covariate but no interacti...
23. Table 6.23 Logits in a three‐class LTA with interaction between dichoto...
24. Table 6.24 Relationships between structural parameters of Parameterizat...
25. Table 6.25 Structural parameter estimates by different parameterization...
26. Table 6.26 Comparison of structural parameter estimates between Paramet...
27. Table 6.27 New parameters for Parameterization 2 using the last column ...
28. Table 6.28 Log odds ratios of transition with the diagonal elements of ...
29. Table 6.29 Selected results of the three‐step model: Step 1.
30. Table 6.30 Selected results of the three‐step model: Step 3.
Chapter 7
1. Table 7.1 Selected Mplus output: Satorra‐Saris method, Step 1.
2. Table 7.2 Selected Mplus output: Satorra‐Saris method, Step 2.
3. Table 7.3 Estimated statistical power by sample size using Satorra‐Sari...
4. Table 7.4 Estimated statistical power by sample size using the Satorra‐...
5. Table 7.5 Selected Mplus output: Monte Carlo simulation for a CFA model...
6. Table 7.6 Estimated statistical power by sample size using a Monte Carl...
7. Table 7.7 Selected Mplus output for Monte Carlo latent growth model sim...
8. Table 7.8 Estimated statistical power by sample size using a Monte Carl...
9. Table 7.9 Selected Mplus output: Monte Carlo simulation for a latent gr...
10. Table 7.10 Logit by attrition rate.
11. Table 7.11 Critical noncentrality parameters by degrees of freedom and ...
12. Table 7.12 Estimated constantsm based on bootstrap simulation experimen...

List of Illustrations

Chapter 1
1. Figure 1.1 A hypothesized general structural equation model.
2. Figure 1.2 (a) Measurement model 1; (b) measurement model 2; (c) measureme...
3. Figure 1.3 Structural model.
Chapter 2
1. Figure 2.1 (a) CFA of SOM. (b) CFA of DEP. (c) CFA of ANX.
2. Figure 2.2 CFA of BSI‐18.
3. Figure 2.3 Frequencies of selected indicators (x ₁ and x ₂).
4. Figure 2.4 Example of item characteristic curves.
5. Figure 2.5 Anxiety item ICCs.
6. Figure 2.6 Anxiety item information function (IIF) curve.
7. Figure 2.7 Anxiety test information (TIF) curve.
8. Figure 2.8 Item characteristic curves (ICCs) of 2PL GRM: BSI‐18 anxiety it...
9. Figure 2.9 Boundary response function (BRF) of 2PL GRM: anxiety item u ₃....
10. Figure 2.10 Second‐order CFA of BSI‐18.
11. Figure 2.11 Bifactor CFA of BSI‐18.
Chapter 3
1. Figure 3.1 MIMIC model.
2. Figure 3.2 Differential item functioning (DIF) in a MIMIC model.
3. Figure 3.3 Structural equation model.
4. Figure 3.4 Testing indirect and total effects in SEM.
5. Figure 3.5 Correcting for measurement error in a single indicator.
6. Figure 3.6 Testing interactions involving latent variable.
7. Figure 3.7 Moderated mediation analysis.
8. Figure 3.8 MLR symmetric CI of the direct effect of X on ANX.
9. Figure 3.9 Non‐symmetric bootstrap CI of the direct effect of X on ANX.
10. Figure 3.10 (a) Bayesian trace plot of the indirect effect of Gender on DE...
Chapter 4
1. Figure 4.1 Linear unconditional latent growth model.
2. Figure 4.2 Growth curves of a linear latent growth model.
3. Figure 4.3 Latent growth model with time‐invariant covariates.
4. Figure 4.4 Latent growth model with time‐varying and time‐invariant covari...
5. Figure 4.5 Unconditional quadratic latent growth model.
6. Figure 4.6 Observed growth trajectory of BDI scores over time.
7. Figure 4.7 Piecewise latent growth model.
8. Figure 4.8 Unconditional latent growth model with free time scores.
9. Figure 4.9 Latent growth model with time‐invariant covariates and a distal...
10. Figure 4.10 Unconditional parallel latent growth model.
11. Figure 4.11 Unconditional two‐part latent growth model.
12. Figure 4.12 Frequency of crack cocaine using days in the past 30 days by t...
13. Figure 4.13 Dynamic structural equation model using observed centering for...
14. Figure 4.14 Means of outcome measure y over subjects by time.
15. Figure 4.15 Residual dynamic structural equation model using observed cent...
16. Figure 4.16 Residual dynamic structural equation model using latent variab...
17. Figure 4.17 Residual dynamic structural equation model with distal outcome...
Chapter 5
1. Figure 5.1 Baseline model: Ohio.
2. Figure 5.2 Baseline model: Kentucky.
3. Figure 5.3 Second‐order CFA model: Ohio.
4. Figure 5.4 Second‐order CFA model: Kentucky.
5. Figure 5.5 Growth trajectories of crack cocaine use by group.
Chapter 6
1. Figure 6.1 Unconditional LCA model.
2. Figure 6.2 Profiles for three‐class LCA models of drug use.
3. Figure 6.3 Profiles for a three‐class LLCA model of BDI‐defined depression...
4. Figure 6.4 Patterns of drug‐use practices estimated from the three‐class L...
5. Figure 6.5 Outcome growth trajectories in a latent growth model and a grow...
6. Figure 6.6 Unconditional growth mixture model.
7. Figure 6.7 Latent growth trajectories by class: four‐class unconditional g...
8. Figure 6.8 Growth mixture model with covariates and a distal outcome.
9. Figure 6.9 Latent growth trajectories by class: four‐class growth mixture ...
10. Figure 6.10 (a) MFA model; (b) nonparametric factor analysis model – LCFA....
11. Figure 6.11 Auxiliary Model with Covariates and Distal Outcome
Chapter 7
1. Figure 7.1 Hypothesized CFA model for estimating statistical power and sam...
2. Figure 7.2 Hypothesized unconditional latent growth model for estimating s...
3. Figure 7.3 Estimated statistical power for the CFA model by sample size: M...
4. Figure 7.4 Estimated statistical power for the latent growth model by samp...
5. Figure 7.5 Estimating sample size for RMSEA: generate the R code, and subm...
6. Figure 7.6 Creating a plot for power and sample size for RMSEA: generate t...
7. Figure 7.7 Plot of power and sample size for RMSEA.

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This edition first published 2020

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Preface

The first edition of this book was one of a few books that provide detailed instruction for how to commonly fit structural equation models using Mplus – a popular software program for latent variable models. The intent of the book is to provide a resource for learning how to practically conduct structural equation modeling (SEM) using Mplus in real research and a reference guide for structural equation models.

Since the first publication of the book in 2012, Mplus has undergone several major version updates (from Edition 6.12 to Edition 8.2). The updates have added many new features, including some of the latest development in SEM. In the current edition of the book, we expand the first edition to cover more structural equation models, including some that are newly developed. All of the example Mplus programs have also been updated using Mplus 8.2. The following are the updates by chapter.

Chapter 1 . Descriptions of the Bayes estimator and corresponding model fit evaluation are included.

Chapter 2. New topics/models are added, including: effect coding for factor scale, two‐parameter logistic (2PL) item response theory (IRT) models, two‐parameter normal ogive (2PNO) IRT models, two‐parameter logistic form of graded response models (2PL GRM), two‐parameter normal ogive form of graded response models (2PNO GRM), bifactor confirmatory factor analysis (CFA) models, and Bayesian CFA models, as well as approach for estimating plausible values of latent variables.

Chapter 3. Added topics include the moderate mediating effect model, bootstrapp approach for SEM, using plausible values of latent variables in secondary analysis, and Bayesian SEM.

Chapter 4. The new topics/models added in this chapter include: latent growth modeling (LGM) with individually varying times of observations, and dynamic structural equation modeling (DSEM). The former addresses the challenge in longitudinal data in which assessment times and specific time intervals between measurement occasions vary by individuals. The latter is a recently developed model for time series analysis in a SEM framework on intensive longitudinal data (ILD). Both DSEM and its variant – residual DSEM (RDSEM) – with and without a distal outcome are demonstrated using simulated data.

Chapter 5. Added topics include multigroup CFA with categorical indicators to evaluate measurement invariance of scales with categorical indicator variables.

Chapter 6. Several topics/models are added in this chapter, including: various auxiliary variable approaches, such as the pseudo‐class (PC) method, the three‐step method (automatic implementation and manual implementation), Lanza's method, and the BCH method (automatic implementation and manual implementation); latent class analysis (LCA) with residual covariances; and the longitudinal latent class analysis (LLCA) model, which extends LCA to longitudinal data analysis.

Chapter 7. Power analysis and sample size estimation for LCA with dichotomous items are added.

The book covers the basic concepts, methods, and applications of SEM, including some recently developed advanced structural equation models. Written in non‐mathematical terms, a variety of structural equation models for studying both cross‐sectional and longitudinal data are discussed. The book provides step‐by‐step instructions for model specification, estimation, evaluation, and modification for SEM practice, and thus it is very practical. Examples of various structural equation models are demonstrated using real‐world research data. The internationally well‐known computer program Mplus 8.2 is used for model demonstrations, and Mplus program syntax is provided for each example model. While the data sets used for the example models in the book are drawn from public health studies, the methods and analytical methods are applicable to all fields of quantitative social studies. Data sets are available from the first author of the book upon request.

The target readership of the book is teachers, graduate students, and researchers who are interested in understanding the basic ideas, theoretical frameworks, and methods of SEM, as well as how to implement various structural equation models using Mplus. The book can be used as a resource for learning SEM and a reference guide for conducting SEM using Mplus. The original version of the book has also been used by professors as a textbook at universities both within and outside the United States. Many researchers and graduate students have found the book helpful for learning and practicing SEM. We believe the second edition of the book will better serve as a useful reference for SEM with Mplus. Readers are encouraged to contact the first author at jiwang@gwu.edu or jiwang@cnmc.org with regard to feedback, suggestions, and questions.

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Structural Equation Modeling

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Preface