Introduction to Structural Equation Modeling (SEM)

Lecturers: Eldad Davidov & Peter Schmidt 

Modality: Online

Week 2: 19-23 August 2024


Workshop Contents and Objectives

The objective of this course is to show how structural equation modeling can be used to develop and/or test both measurement models (scales) and causal theories between latent variables with survey data. In the first part we deal with building scales by employing confirmatory factor analysis, simultaneous factor analysis, bifactor models, multiple group factor analysis and higher order factor analysis as well as measurement invariance over groups and countries. When discussing full structural equation modeling, we will treat formative and reflective indicators, mediation, indirect effects, moderation and multiple group structural equation modeling. A further important aim is to familiarize participants with the AMOS program. The program will be run by graphical input via path diagrams (AMOS Graphics). A special focus will be given to the analysis of comparative data across groups. This includes how to test for measurement invariance using Multiple group confirmatory factor analysis.

Participants are strongly encouraged to bring their own data, prepare in advance the raw data or correlations and standard deviations according to the specifications necessary for AMOS or R-lavaan  and apply the new procedures besides the prepared examples of the instructors in the practical sessions. Everyone will be able to receive consultation and have the opportunity to present own first results on the last day to receive feedback and recommendations for further analyses.


Workshop design

A combination of either live lectures or recorded video lectures, theoretical exercises as well as applied exercises with the program Amos, group work where applicable, and individual or group work on own models with consultation and a presentation on the last day of the course.


Deatiled lecture plan (daily schedule)

Day 1.
Basic ideas of measurement models

Day 2.
Confirmatory factor analysis (CFA)

Day 3.
Variants of Confirmatory Factor Models(Bifactor, MTMM, Higher Order), Multiple group confirmatory factor analysis (MGCFA) and introduction to Structural Equation Modeling (SEM)

Day 4.

Day 5.
Presentations of participants and summary, open questions

See course outline below for details


Class materials

All the material will be provided online. The used programs will be Amos and SPSS. However those who have pused R-Lavaan can use it also for their own data and presentation.



Some experience with regression analysis techniques is required. Basic knowledge of factor analysis is recommended. Participants who bring their own data will most profit from the course.


Recommended readings or preliminary material

Basic texts/overview

  • Arbuckle, J.L. (2019): AMOS 26.0 User's Guide. Chicago: SPSS/Erlbaum.
  • Brown, Timothy, A. (2015). Confirmatory Factor Analysis for Applied Research. New York: Guilford
  • Byrne, Barbara M. (2016): Structural equation modeling with AMOS. Basic concepts, application, and programming. 3rd Edition. Ney York: Routledge.
  • Davidov, E., P. Schmidt, J. Billiet and B. Meuleman (eds.) (2018). Cross-cultural analysis: Methods and applications. Second edition. NY: Routledge.
  • Kline, R. B. (2023). Principles and Practice of Structural Equation Modeling. Fifth h edition. Guilford Press: New York.

Additional reading

  • Aleman JA, Schmidt P, Meitinger K and Meuleman B (2022) Editorial: Comparative political science and measurement invariance: Basic issues and current applications. Front. Polit. Sci. 4:1039744. doi: 10.3389/fpos.2022.1039744
  • Cieciuch, J., E. Davidov, P. Schmidt and R. Algesheimer (2016). The assessment of cross-cultural comparability. Pp. 630-648 in: C. Wolf, D. Joye, T. W. Smith & Y.-C. Fu (eds.), The Sage Handbook in Survey Methodology. New York: Sage.
  • Davidov, E., J. Cieciuch, B. Meuleman, P. Schmidt and J. Billiet (2014). Measurement equivalence in cross-national research. Annual Review of Sociology, 40, 55-75
  • Davidov, Schmidt and Schwartz (2008). Bringing values back in: the adequacy of the European social survey to measure values in 20 countries. Public opinion quarterly, 72, 420-445.
  • Davidov, Meuleman, Billiet and Schmidt (2008). Values and support for immigration: a cross-country comparison. European sociological review, 24, 583-599.
  • Davidov, E., D. Seddig, A. Gorodzeisky, R. Raijman, P. Schmidt and M. Semyonov (2020). Direct and indirect predictors of opposition to immigration in Europe: Individual values, cultural values, and symbolic threat.  Journal of Ethnic and Migration Studies, 46(3), 553-573.
  • Leitgoeb, H.Seddig, D.,Aspourov T.,Behr, D., Davidov E., De Roover, K., Jak, S., meitinger KL., Menold, N., Muthen, BN.,Rudnev, M., Schmidt, P., van de Schoot R.(2022), Measurement Invariance in the Social Sciences: Historical development, methodological challenges, state of the art, and future perspectives, Social Science Research, 110, 102805


The course will show how a causal theory can be represented by a path diagram and translated into a confirmatory factor and/or a full structural equation model and how the model can be estimated and tested with the AMOS computer program. In the first part we will deal with measurement models relating single or multiple indicators to latent variables. Furthermore, different specifications of measurement models are tested via confirmatory factor analysis as a special case of a structural equation model. The second part comprises both the structural and the measurement model. Topics include treatment of cross-cultural data with multiple-group modeling, Mimic models, moderation and mediation, and missing values. Special attention is given to the process of model modification. We warmly recommend participants to bring their own data with them. Time will be dedicated for consultation on Thursday, and participants will have the opportunity to present their models on the last day of the course to get feedback for their research.



Day 1.
Overview of the whole course. Causality and empirical research, notation, different types of models, theory testing, use of the AMOS manual, SEMNET and course material. Foundation of CFA: Process of linear causal modelling, types of input, assumptions, equality constraints, formalization, formative vs. reflective indicators, typology of models, treatment of missing values (pairwise vs. Full Information Maximum Likelihood - FIML).

Practical session: AMOS and the logic of its use. CFA with one measurement model. Preparation of EXAMPLE 1: (input file: cov_NL2.sav). Conformity/Tradition (COTR) value with four indicators. Computation and interpretation of model 1. Model-Modification. Output interpretation and comparison of models.

Essential Reading:    Arbuckle 2017 Introduction and chapters 22, 26 and 27, Examples 1, 3; Byrne 2010 (ToolBox) and chapter 4; Brown 2015 chapter 3 and 7, 238-265; Davidov & Schmidt 2007, Schafer & Graham 2002, Schmidt & Hermann 2011 (a).

Additional reading:   Heyder & Schmidt 2002 (1-11).


Day 2.
Restrictions, identification, model modifications, global and detailed model fit, Simultaneous Confirmatory Factor Analysis (SCFA).

Practical session: Preparation of EXAMPLES 2: (input File: cov_NL2.sav). SCFA and its modification: Conformity/Tradition and Universalism/Benevolence. Examination of detailed and global model fit. Types of errors, reliability and validity estimates in CFA, variance decomposition, Multiple Group Confirmatory Factor Analysis (MGCFA). Preparation of EXAMPLE 3: (Input Files: cov_NL2.sav, cov_BE2.sav, cov_LU2.sav). Multiple group comparisons across BENELUX countries.    

Essential Reading:    Brown 2015 chapters 3, 4, 5 and 7; Davidov, Meuleman, Billiet & Schmidt 2008;

Additional reading:  Davidov, Schmidt & Schwartz 2008; Davidov & Schmidt 2007; Knoppen & Saris 2009; Heyder & Schmidt 2002 11-13; Arbuckle 2017 Examples 10 and 12.


Day 3.
MGCFA with intercepts and latent means, higher order CFA, MTMM.

Practical session: EXAMPLE 4: (Input Files: cov_NL2.sav, cov_BE2.sav, cov_LU2.sav) MGCFA with means and intercepts: Subgroups Belgium, Netherlands, Luxemburg. Output interpretation.

Essential Reading:    Arbuckle 2017 Examples 15, 24, Appendix E, F and G; Brown 2015 chapters 6, 7 and 8, Podsakoff et al. 2003, Steenkamp and Baumgartner 1998, Thompson & Green 2013.

Additional reading:   Leitgoeb et al. (2023); Meuleman et al.(2022); Steinmetz et al 2009;  Zick et al. 2008.



Day 3.
Structural Equation Models (SEM) with latent variables and multiple indicators: Specification, identification and estimation. Causality and equivalent models. Typology of model testing. “The two step strategy“. Model Modification revisited. Theoretical exercise 6. 

Essential Reading:    Davidov et. al. 2008, Kline (2023, fifth edition, parts II to IV ) Schmidt & Hermann 2011 (b)

Additional Reading: Heyder & Schmidt 2002 (13-16); Arbuckle 2017 chapter 5; Anderson/Gerbing 1988.  


Day 4.
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. Mediation. SEM with multiple groups: Model specification and estimation. MIMIC Models. Moderation/interaction effect (the Little method). MIMIC models with higher order factors, latent means and intercepts.

Practical session: SEM with decomposition of effects and mediation. Preparation of FINAL EXERCISE – READ ONLY (Input File: cov_NL2.sav): Full SEM and a MIMIC model: COTR, UNBE and sociodemographic variables. Using bootstrap to receive standard errors of indirect and total effects. Output interpretation.

Work on own data and consultation.

Essential Reading:    Heyder & Schmidt 2002 (17-23); Arbuckle 2017 Example 7; McKinnon et al. 2007; Davidov et. al. 2008; Kline(2023, parts II - IV); Schmidt & Herrmann 2011, Steinmetz et al. 2011.Additional Reading: Berry 1984; Arbuckle 2009 Example 25; Heyder & Schmidt 2002 (23-24); Yang-Wallentin et. al. 2006.


Day 5.
Participants’ presentations.

Essential Reading:    Arbuckle 2009 Example 17 and 18.



  • Anderson, James C. and David W. Gerbing (1988), "Structural Equation Modeling in Practice: A Review and Recommended Two-Step Approach," Psychological Bulletin, 103, 411-423.
  • Arbuckle, J. L. (2017). AMOS 25.0 User’s Guide. Chicago: SPSS.
  • Berry, W. D. (1984). Nonrecursive causal models. London: Sage.
  • Billiet, J.B. and E. Davidov (2008). Testing the Stability of an Acquiescence Style Factor Behind Two Interrelated Substantive Variables in a Panel Design. Sociological Methods and Research, 36, 542-562.
  • Bollen, K. A. (1989). Structural equation modelling with latent variables. NY: Wiley
  • Bollen, K. A. (2002). Latent Variables in Psychology and the Social Sciences. Annual Review of Psychology, 53, 605-634.
  • Bollen, K.A. (2008). Frequently Asked Questions on Structural Equation Models [FAQs on SEMs]. The Sociological Methodologist, 6-8.
  • Bollen K.A. & P.J. Curran (2006). Latent Curve Models. A Structural Equation Perspective. Hoboken: Wiley.
  • Bollen, K.A. & J. Pearl (2012). Eight myths about causality and structural equation models. In S. Morgan (Ed.), Handbook of Causal Analysis for Social Research. Springer.
  • 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, 2nd edition. New York: The Guilford Press. Byrne, M.B. (2004). Testing for Multigroup Invariance Using AMOS Graphics: A Road Less Traveled. Structural Equation Modeling 11(2), 272-300.
  • Byrne, Barbara M. (2016). Structural equation modelling with AMOS. Basic concepts, application, and programming. London: Lawrence Erlbaum Associates.
  • Byrne, M.B. and Stewart, S.M. (2006). The MACS Approach to Testing for Multigroup Invariance of a Second-Order Structure: A Walk Through the Process. Structural Equation Modeling 13(2), 287-321.
  • Cieciuch, Jan, Eldad Davidov, Peter Schmidt, René Algesheimer, and Shalom H. Schwartz. 2014. “Comparing Results of an Exact Versus an Approximate (Bayesian) Measurement Invariance Test: A Cross-Country Illustration with a Scale to Measure 19 Human Values.” Frontiers in Psychology 5:982. doi: 10.3389/fpsyg.2014.00982.
  • Curran, P. & Bollen, K. (2002). The best of both worlds: combining autoregressive and latent curve models. In A. Sayer & L. Collins (Eds.). New methods for the analysis of change.
  • Davidov, E., J. Cieciuch, B. Meuleman, P. Schmidt, R. Algesheimer, and M. Hausherr (2015). The comparability of measurements of attitudes toward immigration in the European Social Survey. Exact versus approximate measurement equivalence. Public Opinion Quarterly, 79, Special Issue, 244–266.
  • Davidov, E. and A. De Beuckelaer (2010). Testing the equivalence of an instrument to assess Schwartz’ human values: How harmful are translations? International Journal of Public Opinion Research, 22(4), 485-510.
  • Davidov, Eldad, Hermann Dülmer, Elmar Schlüeter, Peter Schmidt, and Bart Meuleman. (2012). “Using a Multilevel Structural Equation Modeling Approach to Explain Cross-Cultural Measurement Noninvariance.” Journal of Cross-Cultural Psychology 43: 558-75. doi: 10.1177/0022022112438397.
  • Davidov, E., Meuleman, B., Billiet, J., & Schmidt, P. (2008). Values and Support for Immigration: A Cross-Country Comparison. European Sociological Review, 24(5). 583-599.
  • Davidov, Eldad, Bart Meuleman, Jan Cieciuch, Peter Schmidt, and Jaak Billiet. 2014. “Measurement Equivalence in Cross-National Research.” Annual Review of Sociology 40:55-75. doi: 10.1146/annurev-soc-071913-043137.
  • Davidov, E. & Schmidt, P. (2007). Are values in the Benelux countries comparable? Testing for equivalence with the European Social Survey 2004-5. In Loosveldt, G., M. Swyngedouw and B. Cambré (Eds.), Measuring meaningful data in social research (Pp.373-386). Leuven: Acco.
  • Davidov, E., Schmidt, P., & Billiet, J. (2011). Cross-Cultural Analysis: Methods and Applications. New York: Routledge.
  • 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., S. Thörner, S. Gosen, P. Schmidt and C. Wolf (in press). Level and Change of Group-Focused Enmity in Germany: Unconditional and Conditional Latent Growth Curve Models with Four Panel Waves- Advances in Statistical Analysis.
  • Duncan, T. E. and Duncan, S. C. (2009), The ABC's of LGM: An Introductory Guide to Latent Variable Growth Curve Modeling. Social & Personality Psychology Compass, 3(6), 979-991.
  • Duncan T.E., Duncan S.C. and Strycker L.A. (2006). An Introduction to Latent Variable Growth Curve Modeling: Concepts, Issues, and Applications. Mahwah: Lawrence Erlbaum Associates.
  • Finkel, S. (1995). Causal Analysis with Panel Data. London: Sage.
  • Heyder, A. and P. Schmidt (2002). Authoritarianism and Ethnocentrism in East and West Germany - Does the system matter? Pp. 187-210 in: R. Alba, P. Schmidt & M. Wasmer (eds.). New York: Palgrave, St. Martins 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.
  • Hoyle, H. R. (2023). Handbook of Structural Equation Modeling. Second Edition. NY: The Guilford Press.
  • Kline, R. (2023) Principles and Practice of Structural Equation Modeling, 5 th edition. New York: Guilford Press.
  • Knoppen.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.
  • Legge, S., E. Davidov and P. Schmidt (2008). Social Structural Effects on the Level and Development of the Individual Experience of Anomie in the German Population. International Journal of conflict and violence, 2, 248-267.
  • Leitgoeb, H. et al. (2023) Measurement Invariance in the social sciences: Historical Development, methodological challenges, state of the art, and future perspectives. Social Science Research, 110, 102805
  • Little, T.D., Slegers, D.W. and Card, N.A. (2006). A Non-arbitrary Method of Identifying and Scaling Latent Variables in SEM and MACS Models. Structural Equation Modeling, 13, 59-72.
  • MacKinnon, D.P. and Fairchild, A.J. (2009). Current directions in mediation analysis. Current directions in psychological analysis, 18, 16-20.
  • MacKinnon, D. P., Fairchild, A. J., Fritz, M. S. (2007). Mediation analysis. Annual Review of Psychology, 58, 593 -614.
  • Meuleman, B., Zoltak T., Pokropek, Davidov, E., Muthen, B., Oberski, D., Billiet J., Schmidt, P.(2022) Why Measurement Invariance is important in comparative research, A Response to Welzel et al. (2021), Sociological Methods and Research, 52, 3, https//  
  • Podsakoff, P.M., S.B. MacKenzie, J.-Y. Lee, and N. P. Podsakoff (2003). Common method biases in behavioural research: A critical review of the literature and recommended remedies. Journal of applied psychology, 88(5), 879-903.
  • 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.
  • 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. Sociological Methods and Research, 25, 347-383.
  • Schlüter, E., Davidov E.& Schmidt P.(2007). Applying Autoregressive and latent growth Curve Models to a three-wave panel study, in K.van Montfort, J.Oud &.A.Satorra (eds.) Longitudinal Models in the Behavioral and Related Sciences. Erlbaum, pp. 315-336.
  • Schmidt, P. & Herrmann, J. (2011 a). Factor Analysis, in International Encyclopedia of Political science Methodology. London: Sage.
  • Schmidt, P. & Herrmann, J. (2011 b). Structural Equation Models, in International Encyclopedia of Political science Methodology. London: Sage.
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  • Steenkamp, J.-E. B.M. and Baumgartner H. (1998). Assessing measurement invariance in cross-national consumer research. Journal of Consumer Research 25, 78-90.
  • 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, 43, 599-616.
  • Steinmetz, H., Davidov, E. & Schmidt, P. (2011). Constrained or unconstrained? A comparison of different approaches to test interactions with latent variables, Methodological Innovations, 6(1), 95-110.
  • Thompson, Marilyn S. & Samuel B. Green (2013). Evaluating between-group differences in latent variable means. In Hancock, Gregory R. & Ralph O. Mueller (Eds.), Structural Equation Modeling. A Second Course, Pp. 163-218. Charlotte, NC: IAP.
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  • 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.
  • 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.
  • Zercher, Florian, Peter Schmidt, Jan Cieciuch, and Eldad Davidov. 2015. “The Comparability of the Universalism Value over Time and Across Countries in the European Social Survey: Exact Versus Approximate Measurement Equivalence.” Frontiers in Psychology.


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