2N Confirmatory Factor Analysis and Structural Equation Modelling

Course content

The course shows how theoretical assumptions concerning measurement models and substantive models can be translated into a structural equation model, and how the model can be estimated and tested with the Mplus 8 computer program. In addition, we provide syntax for all examples in R (lavaan). We will show how to use R procedures that produce lavaan code from Mplus input (mplus2lavaan) and Mplus input, Mplus compatible data and output via R (MplusAutomation). In the first part, we deal with confirmatory factor analysis (CFA), which relates multiple indicators to one (CFA) or several latent variables (Simultaneous Confirmatory Factor Analysis, Bifactor Models and second order confirmatory factor analysis) and multiple-group confirmatory factor analysis. Different specifications of measurement models are tested via confirmatory factor analysis (CFA) as a special case of a structural equation model (SEM) and we will discuss scale building procedures, measurement invariance testing and adequate reliability and validity estimates.

Special emphasis is given to the (cross cultural) analysis of multiple groups (MGCFA) for comparisons between groups and over time points (repeated cross-sections), including the assessment of measurement invariance (i.e., comparability of intercepts of observed variables and factor loadings) to compare regression coefficients and latent means. We also deal with the new procedures of measurement invariance testing (e.g., alignment optimization) to check approximate measurement invariance.

The second part comprises both the structural model and the measurement model. Topics include recursive vs. non-recursive models for the structural part of the model, moderation (interaction effects) and mediation analyses, treatment missing and categorical data and the use of formative vs. reflective indicators in MIMIC Models. Again, special emphasis will be directed toward the use of the multiple-group option for cross-group comparisons of both the measurement and the structural model. A major focus will be the process of model modification and alternative model testing using adequate fit measures and how to report CFA and SEM results.

We will draw all examples are from data of the European Social Survey (ESS) and use the concept of human values as operationalized by the Portrait Value Questionnaire developed by Schwartz for single country or cross-cultural analyses. Nonetheless, participants from all fields are welcome to participate.

Course Objectives

The objectives of this course are

1.to enable participants to use (multiple group) confirmatory factor analysis and (multiple group) structural equation modelling to develop and/or test both measurement models and scales and, furthermore, causal theories with latent variables.
2. to familiarize participants with the Mplus 8 program to handle the most important standard models. Nonetheless, to facilitate the transfer of the course content to other programs, all example syntaxes will also be supplied for R lavaan..
3. to increase participants’ competencies in the applications of the techniques and models with their own data, participants are highly encouraged to bring their own data as cleaned raw data files (see for input specifications of the data, e.g., Byrne 2012) and to use them for developing their models and projects for a dissertation or publications during the course. We will offer consultation hours from the beginning both by the instructor and the teaching assistant to answer individual questions and specially to support participants to perform analyses with their own data. Moreover, we will offer an open colloquium format to present and critically discuss the own research findings.

Course Prerequisites

Participants are expected to have extensive familiarity with Windows applications, good knowledge of exploratory factor analysis and regression analysis is required. The course is most optimal for those who want to apply CFA and SEM within the next twelve months.

Remedial Reading

T. Brown (2015) Confirmatory Factor Analysis for Applied Research, Paperback, Second edition, Guilford Press – this text will be provided by ESS

R.B. Kline (2016) Principles and Practice of Structural Equation Modeling, Paperback, Fourth edition, Guilford Press – this text will be provided by ESS

B. Byrne (2012) Structural Equation Modeling with MPLUS, Routledge

J. Wang and X. Wang (2019) Structural Equation Modeling: Applications Using Mplus. J.Wiley.

K. Gama and G. Broc (2019) Structural Equation Modelling with lavaan. New York, J. Wiley

Representative Background Reading

E.Davidov/P.Schmidt/S.Schwartz(2008): Bringing values back in. The adequacy of the European Social Survey to measure values in 20 countries. Public Opinion Quarterly 72,3, 420-445.
E.Davidov/B.Meuleman/J.Billiet/P.Schmidt(2008) Values and the Support for immigration: A Cross Country Comparison, European Sociological Review,24,5,583-599.
E.Davidov/P.Schmidt/J.Billiet and B.Meuleman(April 2018) Cross-Cultural Analysis,: Methods and Applications.
Paperback.Routledge
E.Davidov/B.Meulemann/J.Ciechuch/P.Schmidt/J.Biliet(2014) Measurement Invariance in cross-national research, Annual Review of Sociology, 40, 55-75.

Background knowledge required

Statistics
OLS = moderate

Maximum Likelihood = elementary

Computer Background

R = elementary

MPlus = elementary

Maths

Calculus = elementary

Linear Regression = moderate

Week 1: CONFIRMATORY FACTOR ANALYSIS for COMPARATIVE RESEARCH

Generally
In the practical exercises, we will focus exclusively on the software Mplus Version 8.7. In addition, we provide syntax for all examples in R (lavaan). We will show how to use R procedures that produce lavaan code from Mplus input (mplus2lavaan) and Mplus input, Mplus compatible data and output via R (MplusAutomation). In case you have your own data available, we strongly encourage you to use them and work with them over the course of the summer school. We offer time for individual consultations concerning your data and projects and we incorporate presentations of your own models and findings at the end of the course.

DAY 1

Theory and illustration: Types of models, causality and data-dets

Overview of the whole course. Types of models, comparative data-set (European Social Survey) and the concept of values in the ESS. Causality and empirical research, notation, the generalized latent variable model and different types of models, theory testing, use of the Mplus manual, Mplus 8 additions, Transformation of Mplus model syntax into R-Lavaan syntax via Mplus Automation and manually. Discussion of the course material.

Practical exercise: CFA with one construct and four indicators

Mplus and the logic of its use, preparing and reading-in data into Mplus. Selected items and constructs: values and attitudes in the three Benelux countries. Confirmatory factor analysis (CFA) with one theoretical construct (factor): tradition and conformity in the Netherlands with four indicators.

Essential Reading: Brown (2015), chapter 3.

Additional reading: Byrne (2012), chapters 1 and 2; Davidov & Schmidt (2007); Davidov, Schmidt, & Schwartz (2008); Marsh et al. (2009); Muthén & Muthén (2012), chapter 1 and 2; Schwartz (2007, 2012).

DAY 2

Theory and illustration: CFA model specification, estimation and model fit

Foundation of CFA: linear causal modelling, raw data as input, assumptions, types of constraints including formalization, equality constraints. Maximum likelihood estimation (ML) and robust maximum likelihood estimation (MLR). Global and detailed model fit. Ordinal indicators and weighted least squares (WLSMV) estimation. List of presentations of individual projects on Day 10.

Practical exercise: Simultaneous CFA with two constructs and model fit

Modelling the values tradition and conformity in the Netherlands with four indicators, estimation and output interpretation. Estimation of 2-factor models. Comparison of factor loadings, measurement errors. Using global and detailed model fit and model modification statistics with ML and MLR.

Essential Reading: Brown (2015), chapters 4 and 7, 207-241.

Additional Reading: Byrne (2012), chapters 3, 4 and 5; Muthén & Muthén (2012), chapters 4 and 5; Davidov, Datler, Schmidt, & Schwartz (2011); Schmidt & Hermann (2011).

DAY 3

Theory and illustration: Simultaneous confirmatory factor analysis, restrictions and identification, comparison of model fit.

Equality constraints. Restrictions, identification, simultaneous confirmatory factor analysis (SCFA) vs. separate confirmatory factor analysis, cross-loadings and measurement error correlations. Comparison of model-fit measures and cut-off values. 

Practical exercise: Test theory models and simultaneous confirmatory factor analysis with three constructs.

CFA with tau-equivalent, parallel and strictly parallel constraints. Output interpretation and comparison of model fit, coefficients and explained variance. SCFA and its modification: tradition/conformity, universalism and attitude toward immigration. Examination of detailed and global model fit.

Essential Reading: Brown (2015), chapters 3, 4 and 5.

Additional reading: Byrne (2012), chapters 3 and 4; Davidov & Schmidt (2007); Davidov, Schmidt, & Schwartz (2008); Marsh, Hau, & Wen (2004); Muthén & Muthén (2012), chapters 4 and 5; Saris et al. (2009); Saris & Knoppen (2009).

DAY 4

 Theory and illustration: Multi-group confirmatory factor analysis (MGCFA) and measurement invariance (MI)

Multiple group confirmatory factor analysis (MGCFA). Configural, metric and scalar invariance in cross cultural research. Classical and new fit measures for testing measurement invariance. The concept of partial invariance.

 Practical exercise: MGCFA and MI in the Benelux countries

MGCFA by hand. Multiple group comparisons with the bottom-up approach (MGCFA) across Benelux countries. Using the Mplus convenience feature for a simultaneous test of measurement invariance. Introduction of partial measurement invariance.

Essential Reading: Brown (2015), chapter 7, 241-284.

Additional Reading: Asparouhov & Muthen (2014); Byrne (2012), chapter 7; Chen (2007); Ciechuch et al. (2016, 2017); Davidov (2008); Davidov et al. (2008); Davidov & De Beuckelaer (2010); Davidov et al. (2014); Davidov et al. (2015); Kotzur et al., (2019); Marsh et al. (2016); Munck et al. (2016); Muthén & Muthén (2012), chapter 5; Steinmetz et al. (2009)

DAY 5

 Theory and illustration: Scalar invariance and latent means. Higher order MTMM and bifactor models.

Scalar invariance in cross cultural research as a prerequisite of comparing observed and latent means. Mean comparisons of items, indices and latent variables. Drawbacks of the t-test. Using the new alignment procedure in Mplus as an alternative invariance testing procedure for comparisons of many groups (fixed and random). Higher order CFA and bifactor Models. General strategy for testing measurement models in comparative research. How to report CFA results in comparative research. Using the new alignment procedure in Mplus as an alternative invariance testing procedure for comparisons of many groups (fixed and random)

Practical exercise: CFA with latent means.

CFA with latent means, output interpretation, consolidation of the contents of the first week. Optional: Alignment optimization.

Essential Reading: Brown (2015), chapter 8.

Additional reading: Byrne (2012), chapters 5, 8 and 10; Byrne & Stewart (2006); Ciecuch et al. (2014); Davidov et al. (2011); Davidov et al. (2014, 2015); Lomazzi & Seddig (2020); McDonald et al. (2002); Muthén & Muthén (2012), chapter 5; Reise et al. 2017; Seddig et al. (2020); Steinmetz et al. (2009); Van der Schoot et al. (2013); Zercher et al. (2015); Zick et al. (2008). 

Week 2: STRUCTURAL EQUATION MODELS

DAY 6

Theory and illustration: full SEM and multiple indicator multiple causes (MIMIC) models

Path modeling foundations and structural equation models (SEM) with latent variables and multiple indicators: specification, identification and estimation. MIMIC models, causality and equivalent models and the “two step strategy” as a typology of model testing. Fit measures and model modification revisited.

Practical exercise: Values and attitudes toward immigration

Preparation of a full SEM: demographic (control) variables, values and attitudes toward immigration. Comparison of the results for the three countries for the separate analyses with (un)constrained measurement models.

Essential Reading: Kline (2016), chapters 6, 7 and 8 (DAY 6 & 7).

Additional Reading: Anderson & Gerbing (1988); Byrne (2012), chapter 6; Davidov et al. (2008b); Muthén & Muthén (2012), chapter 5; Schmidt & Hermann (2011).

DAY 7

 Theory and illustration: model modification, interpretation of parameters, mediation

Model testing and model modification, detailed and global fit measures, interpretation of parameters, feedback models, decomposition of effects, bootstrapping for testing indirect and total causal effects, full and partial mediation.

Practical exercise: values and attitudes toward immigration

Full SEM with decomposition of effects into indirect and total effects with the Sobel test and bootstrapping, full versus partial mediation, output interpretation.

Essential Reading: Kline (2016), chapters 6, 7 and 8 (DAY 6 & 7).

Additional Reading: Bou & Satorra (2010); Marsh et al. (2004); Muthen (2012a); Muthén & Muthén (2012), chapter 5; Muthén, Muthén, & Aspourov (2016); Paxton et al (2011); Saris et al. (2009).

DAY 8 

Theory and illustration: Moderation in SEM

Multiple group structural equation modeling (MGSEM) as screening procedure. Direct test of interaction effects, interaction effects/moderation on the observed, observed/latent, and latent level. Double mean centering product indicator approach vs. direct estimation of interaction effects with QML in Mplus. 

Practical exercise: age as moderator of the effect of values on attitudes toward migration

Double mean centering product indicator approach and direct estimation of interaction (moderation) effects in Mplus, unstandardized and standardized effects, plotting effects.

Essential Reading: Kline (2016), chapters 6 (133-135) and 17.

Additional Reading: Byrne (2012), chapter 9; Davidov et al. (2008b); Davidov et al. (2011); Davidov et al. (2014); Muthén & Muthén (2010), chapter 5; Ganzeboom (2009); Lin et a. 2010; Yang-Wallentin et al. (2006).

DAY 9

Theory and illustration: categorical data analysis and the use of R (lavaan)

Handling of missing values (pairwise/listwise deletion and full information maximum likelihood estimation); analyzing categorical data with SEM.

Practical exercise:

Preparation for individual presentations on Friday.

Essential Reading: Kline (2016), 83-87; 237-238; 458-460; Edwards et al. (2012).

Additional Reading: Byrne (2012), chapter 12; Kuiper et al. (2013); Muthén & Muthén (2010), chapter 6 and 11; Schaefer & Graham (2002).

DAY 10 Presentation of results of participants

Presentation of the SEM models of the participants using datasets from their projects. 15 minutes presentation, 5-10 minutes discussion for every presentation.

References

1. Methodological References

• Anderson, J. C. & Gerbing, D. W. (1988) Structural equation modeling in practice: A review and recommended two-step approach. Psychological Bulletin, 103(3), 411-423.
• Beierlein, C. & Davidov, E. & Schwartz, S. H. & Schmidt, P. & Rammstedt, B. (2012 – erscheint): Testing the discriminant validity of Schwartz‘ Portrait Value Questionnaire items: a replication and extension of Knoppen and Saris (2009). In Survey Research Methods 6(1), 25-36.
• Bollen, K. A. (2002). Latent Variables in Psychology and the Social Sciences. Annual Review of Psychology, 53, 605-634.
• Boomsma, A. (2000). Reporting analyses of covariance structures. Structural Equation Modeling, 7(3), 461-483.
• Brown, T.A. (2015) Confirmatory factor Analysis for Applied Research. Paperback Second Edition, Guilford.
• Byrne, B. M. & Stewart, S. M. (2006) The MACS Approach to Testing for Multigroup Invariance of Second-Order Structure: A Walk Through the Process, Structural Equation Modeling 13(2), 287-321.
• Byrne, B.M. (2012). Structural Equation Modeling with Mplus. Basic Concepts, Applications and Programming.
• Chen, F. F. (2007) Sensitivity of Goodness of Fit Indexes to Lack of Measurement Invariance, Structural Equation Modeling 14(3), 464-504.
• Ciecuch, J.& Davidov, E.& Schmidt, P.& Algesheimer,& S. Schwartz, S.(2014) Comparing results of an exact vs. an approximate (Bayesian) measurement invariance test: a cross country illustration with a scale to measure 19 values, Frontiers in Psychology,5, 59-68.
• Davidov, E. (2011) Nationalism and Constructive Patriotism: A Longitudinal Test of Comparability in 22 Countries with the ISSP. International Journal of Public Opinions. 23(1):, 88-103.
• Davidov, E. & De Beuckelaer, A. (2010) How Harmful are Survey Translations? – A Test with Schwartz´s Human Values Instrument, International Journal of Public Opinions Research, in Press
• Davidov, E. & Thörner, S. & Schmidt, P. & Gosen, S. & Wolf, C. (2011) Level and change of group-focused enmity in Germany: unconditional and conditional latent growth curve models with four panel waves. AStA Advances in Statistical Analysis. 95(4), 481-500.
• Davidov, E. & Datler, G. & Schmidt, P & Schwartz S. H. (2010) Testing the invariance of values in the Benelux countries with the European Social Survey: Accounting for ordinality. In: E. Davidov&P. Schmidt&J. Billiet (Eds.): Cross-Cultural Analysis: Methods and Applications (European Association of Methodology). Taylor and Francis, 2010
• Davidov,E. & Meuleman B.& Ciecuch J. & Schmidt P. & Billiet J. Equivalence in cross-national research, Annual Review of Sociology 2014.
• Davidov, E. &Ciecuch J.&Meulemann B.&Schmidt,P.& Algesheimer R.&Hausherr M.(2015) The comparability of Measurements of Attitudes toward Immigration in the European Social Survey, Public Opinion Quarterly, 79, 244-266.
• Edwards, M. C., Wirth, R. J., Houts, C. R., & Xi, N. (2012). Categorical data in the structural equation modeling framework. In R. H. Hoyle (Ed.), Handbook of structural equation modeling (pp. 195–208). The Guilford Press.
• Hoogland, J. J. & Boomsma, A. (1998). Robustness studies in covariance structure modeling. An overview and a metanalysis. Sociological Methods & Research, 26(3), 329-367.
• Ganzeboom H. B. G. (2009) Multiple Indicators Models for social Background. Paper presented at European Survey Research Association, Warsaw, July 2009
• Kline,R. Structural Equation Modeling(2011), Third edition, Guilford.
• Lomazzi, V., & Seddig, D. (2020). Gender role attitudes in the International Social Survey Programme: Cross-national comparability and relationships to cultural values. Cross-Cultural Research, 54(4), 398–431.
• Marsh, H. W. & Muthén, B. & Asparouhov, T. & Lüdtke, O. & Robitzsch, A. & Morin, J. S. & Trautwein, U. (2009) Exploratory Structural Equation Modeling, Integrating CFA and EFA: Application to Students´ Evaluations of University Teaching. Structural Equation Modeling 16(3) 439 – 476.
• Marsh, H. W. & Hau, K. T. & Wen Z. (2004) In Search of Golden Rules: Comment on Hypothesis-testing Approaches to Setting Cutoff Values for Fit Indexes and Dangers in Overgeneralizing HU and Bentler´s (1999) Finings. Structural Equation Modeling 11(3) 320- 341.
• Muller, D. & Judd, C. M. & Yzerbyt, V. Y. (2005) When Moderation Is Mediated and Mediation Is Moderated. Journal of Personality and Social Psychology. 89 (6), 852–863.
• Muthén, L. K. & Muthén, B. O. (2012) Mplus – Statistical Analysis With Latent Variables User’s Guide. Muthén & Muthén.
• Muthén, B. O. (2011) Applications of Causally Defined Direct and Indirect Effects in Mediation Analysis using SEM in Mplus. Muthén & Muthén.
• Preacher, K. J. & Kelley, K. (2011) Effect size measures for mediation models: Quantitative strategies for communicating indirect effects. Psychological Methods. 16 (2), 93-115.
• Saris, W. E. (2001). Measurement models in sociology and political science. In Structural Equation Modeling: present and future. Robert Cudeck, Stephen Du Toit, Dag Soerbom, editors.
• Saris, W. E. & Satorra, A. & van der Veld, W. M. (2009) Testing Structural Equation Models or detection of Misspecifications? Structural Equation Modeling 16(4) 561- 582.
• Schafer, J. L. & Graham, J. W. (2002): Missing Data: Our View of the State of the Art. Psychological Methods. (7(2)) 147–177.
• Scherpenzeel, A.C. & Saris, W. E. (1997). The validity and reliability of survey questions. In Sociological Methods and Research, 25, 347-383.
• Schmidt, P. & Herrmann, J. (2011). Factor Analysis, in International Encyclopedia of Political science Methodology, Sage.
• Schmidt, P. & Herrmann, J. (2011). Structural Equation Models in : International Encyclopedia of Political science Methodology, Sage.
• Seddig, D., Maskileyson, D., & Davidov, E. (2020). The comparability of measures in the ageism module of the fourth round of the European Social Survey, 2008-2009. Survey Research Methods, 14(4), 351-364.
• Sijtsma K. (2009) On the Use, the Misuse, and the very limited Usefulness of Cronbach´s Alpha. Psychometrika 74(1) 107- 120.
• Steinmetz, H. & Schmidt P. & Tina-Booh A. & Wieczorek S. & Schwartz S. H.(2009). Testing invariance using multigroup CFA: differences between educational groups in human values measurement, Quality and Quantity, 599-616.
• Steinmetz, H. & Davidov, E. & Schmidt, P. (2011). Three Approaches to Estimate Latent Interaction Effects: Intention and Perceived Behavioral Control in the Theory of Planned Behavior. Methodological Innovations Online. 6(1), 95-110.
• Van de Schoot, R.& Kluytmans A.&,Tummers,L.& Lugtig,P.& Hox, J.&Muthen,B. Facing off Scylla and Charybdis: a comparison of scalar, partial and the novel possibility of approximate invariance, Frontiers in Psychology ,5, 173 – 187.
• Van der Schoot, R.&Schmidt,P.& Beuckelaer, A.(2015) Measurement Invariance, Special Issue in Frontiers in Psychology, 5.
• Yang-Wallentin, F., Davidov, E., Schmidt, P. &/ Bamberg, S.: Is there any interaction effect between intention and perceived behavioral control?, Methods of Psychological Research Online 2004, 8(2), 127-157.
• Zercher,F.&Schmidt,P.&Ciecuch, J.& Davidov E.(2015) The comparability of the universalism value over time and countries in the European Social Survey: exact vs. approximate invariance, Frontiers in Psychology , 6.733 doi:10.3389/fpsyg.2015.00733

2. Substantive References

• Davidov, E. & Schmidt, P. (2007). Working Paper: Are values in the Benelux countries comparable? Testing for equivalence with the European Social Survey 2004-5.
• Davidov, E. & Schmidt, P. & Schwartz, S. H. (2008). Bringing values back in: The adequacy of the European Social Survey to measure values in 20 countries. Public Opinion Quarterly. 72(3). 420-445.
• Davidov, E. & Meuleman, B. & Billiet, J. & Schmidt, P. (2008b). Values and Support for Immigration: A Cross-Country Comparison. European Sociological Review – Oxford Univ Press. 24(5). 583-599.
• D. & Saris,W. (2009) Do we have to combine values in the Schwartz ` Human Values Scale? A Comment on the Davidov Studies. Survey Research Methods, 3, 91-103.
• Seddig, D., Maskileyson, D., Davidov, D., Ajzen, I., & Schmidt. (2022). Correlates of COVID-19 vaccination intentions: Attitudes, institutional trust, fear, conspiracy beliefs, and vaccine skepticism. Social Science and Medicine, 302, 114981.
• Wagner, U. & Becker, J. C. & Christ, O. & Pettigrew T. F. & Schmidt, P. (2010) A Longitudinal Test of the Relation between German Nationalism, Patriotism, and Outgroup Derogation. European Sociological Review, 1-14.
• Zick, A. & Wolf, C. & Küpper, B. & Davidov, E. & Schmidt, P. & Heitmeyer W. (2008). The Syndrome of Group-Focused Enmity: The Interrelation of Prejudices Tested with Multiple Cross-Sectional and Panel Data. Journal of Social Issues, 64 (2), 363-383.

3. Relevant internet homepages:

• Mplus software: http://www.statmodel.com/
• European Social Survey data: https://www.europeansocialsurvey.org
• SEMNET discussion group: https://listserv.ua.edu/cgi-bin/wa?INDEX&X=&Y (registration needed)
• R lavaan: https://lavaan.ugent.be