# Structural equation model

The term structural equation model ( English modeling equation structural , short- SEM ) refers to a statistical model that the treasures and testing correlative allows relationships between dependent and independent variables and the hidden structures in between. It can be checked whether the hypotheses assumed for the model agree with the given variables. It is assigned to the structure- checking multivariate procedure and has a confirmatory character. Approaches to structural equation modeling can fundamentally kovarianzbasierte (eg. Amos and LISREL ) and variance-based (such as the. Partial least squares estimation english least partial squares , in short: PLS ) method are distinguished.

## history

Fundamental considerations go back to Sewall Wright (1921) and (1923), Trygve Haavelmo (1943) and Simon (1977). The approach of the covariance structure analysis is essentially based on Karl G. Jöreskog (1973). The partial least squares approach for estimating so-called causal models was originally developed by Herman OA Wold (1982). The covariance-based structural equation modeling was the established and therefore dominant method for many years. In recent years, the use of variance-based structural equation modeling has become increasingly popular, as shown by numerous studies on the use of the method in various disciplines.

## Model elements

Structural equation model with the latent variable "intelligence"
• Indicator (Item): These are observed variables. For example, indicators for “intelligence” are the “final grade in the Abitur”, the “intelligence quotient” and the “number of languages ​​that a person speaks”. The use of at least four indicators is usually recommended in the model.
• Latent variable (factor): This is the unobserved variable that is only measured by its indicators . In the example, “intelligence” is the latent variable . A distinction is made between independent latent (= exogenous) and dependent latent (= endogenous) variables.
• Measurement model (measurement model): This is the core of the structural equation model. In him a will in the sense of confirmatory factor analysis (confirmatory factor analysis) compounds modeled between the indicators and the latent variables. The covariance plays a decisive role here.
• Structural model (structural model): This is the amount exogenous and endogenous variables and their compounds.

## Model elements and basic procedure

### Model elements

Mulaik and Millsap (2000) proposed four steps for modeling. The first step is to perform a factor analysis to determine the number of latent variables. With a confirmatory factor analysis , the measurement model is confirmed in the second step. In the third step, the structural model is tested. In the fourth step, nested models are tested to identify the most economical.

However, it should be noted that the literature warns against modifying models until they “fit” ( overfitting ). Rather, a new sample must always be collected to check changed or new hypotheses.

### Basic procedure

1. Theoretical foundation and hypothesis formation
2. Choice of method
3. Model formulation
4. Empirical survey
5. Parameter estimation
6. Assessment of the estimation results
7. possibly modification of the model structure

## Applications

Structural equation models play an important role in empirical social research and psychology , among others . A special feature of structural equation models is the checking of latent (not directly observable) variables. Path analysis , factor analysis and regression analysis can be viewed as special cases of structural equation models . A structural equation model in turn represents a special case of a so-called causal model.

## software

Structural equation models are supported by many popular statistical programs. There is also a range of specialized software that can either be run as a standalone program or as a supplement to existing software. Since the different programs often have different capabilities and use different algorithms to perform similar named analyzes, it is good practice to state both the name and the version of the program you are working with.

Mplus commercially Windows, Mac, Linux Standalone statmodel.com
AMOS commercially Windows Standalone ibm.com
Lavaan Open source Windows, Mac, Linux Addition to R lavaan.ugent.be
Lisrel commercially Windows Standalone ssicentral.com
EQS commercially Windows, Mac, Linux Standalone mvsoft.com
Stata commercially Windows, Mac, Linux Standalone stata.com
SAS commercially Windows, Mac, Linux Standalone sas.com
OpenMX Open source Windows, Mac, Linux Addition to R openmx.ssri.psu.edu
Ωnyx Freeware Windows, Mac, Linux Standalone onyx.brandmaier.de
SmartPLS 2 Freeware Windows, Linux Standalone smartpls.com
SmartPLS 3 commercially Windows, Mac Standalone smartpls.com
PLSGraph commercially Windows Standalone plsgraph.com
WarpPLS commercially Windows Standalone warppls.com
ADANCO commercially Windows, Mac Standalone composite-modeling.com
LVPLS Freeware MS Dos Standalone www2.kuas.edu.tw
matrixpls Open source Windows, Mac, Linux Addition to R cran.r-project.org

## literature

• K. Arzheimer: Structural equation models for political scientists. An application-oriented introduction. Springer VS., Wiesbaden 2015, ISBN 978-3-658-09608-3 .
• R. Bagozzi, Y. Yi: Specification, evaluation, and interpretation of structural equation models. In: Journal of the Academy of Marketing Science. Volume 40, No. 1, 2012, pp. 8-34, doi: 10.1007 / s11747-011-0278-x .
• BM Byrne: Structural Equation Modeling with EQS and EQS / Windows. Basic Concepts, Applications, and Programming. Thousand Oaks 1994.
• O Christ; Schlüter, E .: Structural equation models with Mplus. A practical introduction. Oldenbourg Wissenschaftsverlag, Munich 2012, ISBN 978-3-486-59046-3 .
• JF Hair, GTM Hult, CM Ringle, M. Sarstedt: A Primer on Partial Least Squares Structural Equation Modeling (PLS-SEM). 2nd Edition. Sage, Thousand Oaks, CA 2017, ISBN 978-1-4833-7744-5 ( pls-sem.com ).
• JF Hair, GTM Hult, CM Ringle, M. Sarstedt, NF Richter, S. Hauff: Partial Least Squares Structural Equation Modeling (PLS-SEM). An application-oriented introduction. Vahlen, Munich 2017, ISBN 978-3-8006-5360-7 .
• JF Hair, M. Sarstedt, CM Ringle, SP Gudergan: Advanced Issues in Partial Least Squares Structural Equation Modeling. Sage, Thousand Oaks 2018, ISBN 978-1-4833-7739-1
• RH Hoyle (Ed.): Handbook of structural equation modeling. Guilford Press, 2012, ISBN 978-1-4625-1679-7 .
• J.-B. Lohmöller: Latent Variable Path Modeling with Partial Least Squares. Physica, Heidelberg 1989, ISBN 978-3-642-52512-4 .
• Weiber, Mühlhaus: Structural Equation Modeling: An application-oriented introduction to causal analysis with the help of AMOS, SmartPLS and SPSS. Springer, 2009, ISBN 3-642-02876-4 .

## Individual evidence

1. JF Hair, GTM Hult, CM Ringle, M. Sarstedt: A Primer on Partial Least Squares Structural Equation Modeling (PLS-SEM) . Sage, Thousand Oaks, CA 2014, ISBN 978-1-4522-1744-4 ( pls-sem.com ).
2. ^ Wright, Sewall: The theory of path coefficients a reply to Niles's criticism. In: Genetics 8.3, 1923, p. 239.
3. ^ Trygve Haavelmo: The statistical implications of a system of simultaneous equations. In: Econometrica, Journal of the Econometric Society 1943, pp. 1-12.
4. ^ Karl G. Jöreskog: Analysis of covariance structures. In: Multivariate analysis. 3, 1973, pp. 263-285.
5. ^ HOA Wold: Soft Modeling: The Basic Design and Some Extensions. In: KG Jöreskog, HOA Wold (ed.): Systems Under Indirect Observations: Part II. Amsterdam, 1982, pp. 1-54.
6. ^ Nicole Franziska Richter, Rudolf R. Sinkovics, Christian M. Ringle, Christopher Schlägel: A critical look at the use of SEM in international business research . In: International Marketing Review . tape 33 , no. 3 , May 9, 2016, ISSN  0265-1335 , p. 376-404 , doi : 10.1108 / IMR-04-2014-0148 .
7. ^ Joseph F. Hair, Marko Sarstedt, Torsten M. Pieper, Christian M. Ringle: The Use of Partial Least Squares Structural Equation Modeling in Strategic Management Research: A Review of Past Practices and Recommendations for Future Applications . In: Long Range Planning . tape 45 , no. 5-6 , pp. 320-340 , doi : 10.1016 / j.lrp.2012.09.008 .
8. Joe F. Hair, Marko Sarstedt, Christian M. Ringle, Jeannette A. Mena: An assessment of the use of partial least squares structural equation modeling in marketing research . In: Journal of the Academy of Marketing Science . tape 40 , no. 3 , May 1, 2012, ISSN  0092-0703 , p. 414-433 , doi : 10.1007 / s11747-011-0261-6 .
9. Stanley A. Mulaik, Roger E. Millsap: Doing the four-step right . In: Structural Equation Modeling . tape 7 , no. 1 , 2000, pp. 36-73 , doi : 10.1207 / S15328007SEM0701_02 .
10. Klaus Backhaus, Bernd Erichson, Wulff Plinke, Rolf Weiber: Multivariate Analysis Methods - An Application-Oriented Introduction . 12th edition. Springer / Heidelberg, Berlin 2008, ISBN 978-3-540-85044-1 .
11. Jürgen Bortz , René Weber: Statistics for human and social scientists . 6th edition. Springer, Heidelberg 2005, ISBN 3-540-21271-X , doi : 10.1007 / b137571 .
12. Klaus Backhaus, Wulff Plinke, Bernd Erichson, Rolf Weiber: Multivariate Analysis Methods - An Application-Oriented Introduction . 11th edition. Springer, Berlin / Heidelberg 2006, ISBN 3-540-29932-7 .
13. Ronald D. Anderson, Gyula Vastag: Causal modeling alternatives in operations research: Overview and application . In: European Journal of Operational Research . tape 156 , 2004, pp. 92–109 ( home.kelley.iupui.edu [PDF; accessed October 18, 2011]).
14. John Fox: Teacher's Corner: Structural Equation Modeling with-the sem package in R . In: Structural Equation Modeling . tape 13 , no. 3 , 2006, p. 465–486 ( socserv.mcmaster.ca [PDF; accessed October 18, 2011]).
15. ^ Kline: Principles and Practice of Structural Equation Modeling . 2011, p. 79-88 .