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SEM involves a model representing how various aspects of some [[phenomenon]] are thought to [[Causality|causally]] connect to one another. Structural equation models often contain postulated causal connections among some latent variables (variables thought to exist but which can't be directly observed). Additional causal connections link those latent variables to observed variables whose values appear in a data set. The causal connections are represented using [[equation]]s, but the postulated structuring can also be presented using diagrams containing arrows as in Figures 1 and 2. The causal structures imply that specific patterns should appear among the values of the observed variables. This makes it possible to use the connections between the observed variables' values to estimate the magnitudes of the postulated effects, and to test whether or not the observed data are consistent with the requirements of the hypothesized causal structures.<ref name="Pearl09">{{cite journal |doi=10.1017/CBO9780511803161 }}{{pn}}</ref>
The boundary between what is and is not a structural equation model is not always clear, but SE models often contain postulated causal connections among a set of latent variables (variables thought to exist but which can't be directly observed, like an attitude, intelligence, or mental illness) and causal connections linking the postulated latent variables to variables that can be observed and whose values are available in some data set. Variations among the styles of latent causal connections, variations among the observed variables measuring the latent variables, and variations in the statistical estimation strategies result in the SEM toolkit including [[confirmatory factor analysis]] (CFA), [[confirmatory composite analysis]], [[Path analysis (statistics)|path analysis]], multi-group modeling, longitudinal modeling, [[partial least squares path modeling]], [[latent growth modeling]] and hierarchical or multilevel modeling.<ref name="kline_2016">{{Cite book|last=Kline|first=Rex B. |title=Principles and practice of structural equation modeling|date=2016 |isbn=978-1-4625-2334-4|edition=4th |___location=New York|oclc=934184322}}</ref><ref name="Hayduk87">Hayduk, L. (1987) Structural Equation Modeling with LISREL: Essentials and Advances. Baltimore, Johns Hopkins University Press. ISBN 0-8018-3478-3</ref><ref>{{Cite book |last=Bollen |first=Kenneth A. |title=Structural equations with latent variables |date=1989 |publisher=Wiley |isbn=0-471-01171-1 |___location=New York |oclc=18834634}}</ref><ref>{{Cite book |last=Kaplan |first=David |title=Structural equation modeling: foundations and extensions |date=2009 |publisher=SAGE |isbn=978-1-4129-1624-0 |edition=2nd |___location=Los Angeles |oclc=225852466}}</ref><ref>{{
SEM researchers use computer programs to estimate the strength and sign of the coefficients corresponding to the modeled structural connections, for example the numbers connected to the arrows in Figure 1. Because a postulated model such as Figure 1 may not correspond to the worldly forces controlling the observed data measurements, the programs also provide model tests and diagnostic clues suggesting which indicators, or which model components, might introduce inconsistency between the model and observed data. Criticisms of SEM methods include disregard of available model tests, problems in the model's specification, a tendency to accept models without considering external validity, and potential philosophical biases.<ref>{{cite journal |last1=Tarka |first1=Piotr |year=2017 |title=An overview of structural equation modeling: Its beginnings, historical development, usefulness and controversies in the social sciences |journal=Quality & Quantity |volume=52 |issue=1 |pages=313–54 |doi=10.1007/s11135-017-0469-8 |pmc=5794813 |pmid=29416184}}</ref>
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* [[Latent growth modeling]] {{citation needed|date=July 2023}}
* Link functions {{citation needed|date=July 2023}}
* Longitudinal models
* [[Measurement invariance]] models <ref>{{
* [[Mixture model]] {{citation needed|date=July 2023}}
* [[Multilevel models]], hierarchical models (e.g. people nested in groups) <ref>{{Citation |last1=Sadikaj |first1=Gentiana |title=Multilevel structural equation modeling for intensive longitudinal data: A practical guide for personality researchers |date=2021 |url=https://linkinghub.elsevier.com/retrieve/pii/B9780128139950000339 |work=The Handbook of Personality Dynamics and Processes |pages=855–885 |access-date=2023-11-03 |publisher=Elsevier |language=en |doi=10.1016/b978-0-12-813995-0.00033-9 |isbn=978-0-12-813995-0 |last2=Wright |first2=Aidan G.C. |last3=Dunkley |first3=David M. |last4=Zuroff |first4=David C. |last5=Moskowitz |first5=D.S.}}</ref>
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<ref name="MacCallum1996">{{cite journal |doi=10.1037/1082-989X.1.2.130 |title=Power analysis and determination of sample size for covariance structure modeling |journal=Psychological Methods |volume=1 |issue=2 |pages=130–49 |year=1996 |last1=MacCallum |first1=Robert C |last2=Browne |first2=Michael W |last3=Sugawara |first3=Hazuki M }}</ref>
<ref name="Bentler2016">{{cite journal |doi=10.1177/0049124187016001004 |title=Practical Issues in Structural Modeling |journal=Sociological Methods & Research |volume=16 |issue=1 |pages=78–117 |year=2016 |last1=Bentler |first1=P. M |last2=Chou |first2=Chih-Ping
<ref name="Browne1993">{{cite book|last1=Browne|first1=M. W.|last2=Cudeck|first2=R.|editor1-last=Bollen|editor1-first=K. A.|editor2-last=Long|editor2-first=J. S.|title=Testing structural equation models|date=1993|publisher=Sage|___location=Newbury Park, CA|chapter=Alternative ways of assessing model fit}}</ref>
<ref name="Loehlin2004">{{cite journal |doi=10.4324/9781315643199 }}{{pn}}</ref>
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*{{cite book|last1=Kline|first1=Rex|title=Principles and Practice of Structural Equation Modeling| publisher=Guilford| isbn=978-1-60623-876-9|date=2011 |edition=Third}}
* {{cite journal |last1=MacCallum |first1=Robert C. |last2=Austin |first2=James T. |title=Applications of Structural Equation Modeling in Psychological Research |journal=Annual Review of Psychology |date=February 2000 |volume=51 |issue=1 |pages=201–226 |doi=10.1146/annurev.psych.51.1.201 |pmid=10751970 }}
*{{cite journal|last1=Quintana|first1=Stephen M.|last2=Maxwell|first2=Scott E.|date=1999|title=Implications of Recent Developments in Structural Equation Modeling for Counseling Psychology|journal=The Counseling Psychologist|volume=27|issue=4|pages=485–527|doi=10.1177/0011000099274002
== Further reading ==
*{{cite journal |doi=10.1007/s11747-011-0278-x |title=Specification, evaluation, and interpretation of structural equation models |journal=Journal of the Academy of Marketing Science |volume=40 |issue=1 |pages=8–34 |year=2011 |last1=Bagozzi |first1=Richard P |last2=Yi |first2=Youjae
* Bartholomew, D. J., and Knott, M. (1999) ''Latent Variable Models and Factor Analysis'' Kendall's Library of Statistics, vol. 7, [[Edward Arnold (publisher)|Edward Arnold Publishers]], {{ISBN|0-340-69243-X}}
* {{cite journal |last1=Bentler |first1=P. M. |last2=Bonett |first2=Douglas G. |title=Significance tests and goodness of fit in the analysis of covariance structures. |journal=Psychological Bulletin |date=November 1980 |volume=88 |issue=3 |pages=588–606 |id={{ProQuest|614302171}} |doi=10.1037/0033-2909.88.3.588 }}
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