Structural equation modeling: Difference between revisions

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[[File:Example Structural equation model.svg|alt= An example structural equation model|thumb|336x336px|Figure 1. An example structural equation model after estimation. Latent variables are sometimes indicated with ovals while observed variables are shown in rectangles. Residuals and variances are sometimes drawn as double-headed arrows (shown here) or single arrows and a circle (as in Figure 2). The latent IQ variance is fixed at 1 to provide scale to the model. Figure 1 depicts measurement errors influencing each indicator of latent intelligence and each indicator of latent achievement. Neither the indicators nor the measurement errors of the indicators are modeled as influencing the latent variables.<ref name="Salkind2007" />]]

[[File:Example SEM of Human Intelligence.png|alt=An example structural equation model pre-estimation|thumb|336x336px|Figure 2. An example structural equation model before estimation. Similar to Figure 1 but without standardized values and with fewer items. Because intelligence and academic performance are merely imagined or theory-postulated variables, their precise scale values are unknown, though the model specifies that each latent variable's values must fall somewhere along the observable scale possessed by one of the indicators. The 1.0 effect connecting a latent to an indicator specifies that each real unit increase or decrease in the latent variable's value results in a corresponding unit increase or decrease in the indicator's value. It is hoped a good indicator has been chosen for each latent, but the 1.0 values do not signal perfect measurement because this model also postulates that there are other unspecified entities causally impactingaffecting the observed indicator measurements, thereby introducing measurement error. This model postulates that separate measurement errors influence each of the two indicators of latent intelligence, and each indicator of latent achievement. The unlabeled arrow pointing to academic performance acknowledges that things other than intelligence can also influence academic performance.]]

'''Structural equation modeling''' ('''SEM''') is a diverse set of methods used by scientists for both observational and experimental research. SEM is used mostly in the social and behavioral science fields, but it is also used in epidemiology,<ref name="BM08">{{cite book | doi=10.4135/9781412953948.n443 | chapter=Structural Equation Modeling | title=Encyclopedia of Epidemiology | date=2008 | isbn=978-1-4129-2816-8 }}</ref> business,<ref name="Shelley06">{{cite book | doi=10.4135/9781412939584.n544 | chapter=Structural Equation Modeling | title=Encyclopedia of Educational Leadership and Administration | date=2006 | isbn=978-0-7619-3087-7 }}</ref> and other fields. ABy commona standard definition of, SEM is, "...a class of methodologies that seeks to represent hypotheses about the means, variances, and covariances of observed data in terms of a smaller number of 'structural' parameters defined by a hypothesized underlying conceptual or theoretical model,".<ref>{{Cite web |title=Structural Equation Modeling - an overview {{!}} ScienceDirect Topics |url=https://www.sciencedirect.com/topics/neuroscience/structural-equation-modeling#:~:text=Structural%20equation%20modeling%20can%20be,underlying%20conceptual%20or%20theoretical%20model. |access-date=2024-11-15 |website=www.sciencedirect.com}}</ref>

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">Pearl, J. (2009). Causality: Models, Reasoning, and Inference. Second edition. New York: Cambridge University Press.</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>{{Cite journal|last=Curran|first=Patrick J.|date=2003-10-01|title=Have Multilevel Models Been Structural Equation Models All Along?|journal=Multivariate Behavioral Research|volume=38|issue=4|pages=529–569|doi=10.1207/s15327906mbr3804_5|issn=0027-3171|pmid=26777445|s2cid=7384127}}</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 hint at: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>

A great advantage of SEM is that all of these measurements and tests occur simultaneously in one statistical estimation procedure, where all the model coefficients are calculated using all information from the observed variables. This means the estimates are more accurate than if a researcher were to calculate each part of the model separately.{{sfn|MacCallum|Austin|2000|p=209}}

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== History ==

Structural equation modeling (SEM) began differentiating itself from correlation and regression when [[Sewall Wright]] provided explicit causal interpretations for a set of regression-style equations based on a solid understanding of the physical and physiological mechanisms producing direct and indirect effects among his observed variables.<ref name="Wright21">Wright, Sewall. (1921) "Correlation and causation". Journal of Agricultural Research. 20: 557-585.</ref><ref name="Wright34">{{cite journal | doi=10.1214/aoms/1177732676 | title=The Method of Path Coefficients | date=1934 | last1=Wright | first1=Sewall | journal=The Annals of Mathematical Statistics | volume=5 | issue=3 | pages=161–215 }}</ref><ref name="Wolfle99">Wolfle, L.M. (1999) "Sewall Wright on the method of path coefficients: An annotated bibliography" Structural Equation Modeling: 6(3):280-291.</ref> The equations were estimated like ordinary regression equations but the substantive context for the measured variables permitted clear causal, not merely predictive, understandings. O. D. Duncan introduced SEM to the social sciences in his 1975 book,<ref name="Duncan75">Duncan, Otis Dudley. (1975). Introduction to Structural Equation Models. New York: Academic Press. ISBN 0-12-224150-9.</ref> and SEM blossomed in the late 1970's and 1980's when increasing computing power permitted practical model estimation. In 1987 Hayduk<ref name="Hayduk87"/> provided the first book-length introduction to structural equation modeling with latent variables, and this was soon followed by Bollen's popular text (1989).<ref name="Bollen89">Bollen, K. (1989). Structural Equations with Latent Variables. New York, Wiley. ISBN 0-471-01171-1.</ref>

Different yet mathematically related modeling approaches developed in psychology, sociology, and economics. Early [[Cowles Foundation|Cowles Commission]] work on [[Simultaneous equations model|simultaneous equations]] estimation centered on Koopman and Hood's (1953) algorithms from [[transport economics]] and optimal routing, with [[maximum likelihood estimation]], and closed form algebraic calculations, as iterative solution search techniques were limited in the days before computers. The convergence of two of these developmental streams (factor analysis from psychology, and path analysis from sociology via Duncan) produced the current core of SEM. One of several programs Karl Jöreskog developed at Educational Testing Services, LISREL<ref name="JGvT70">Jöreskog, Karl; Gruvaeus, Gunnar T.; van Thillo, Marielle. (1970) ACOVS: A General Computer Program for Analysis of Covariance Structures. Princeton, N.J.; Educational Testing Services.</ref><ref name=":0">{{Cite journal|last1=Jöreskog|first1=Karl Gustav|last2=van Thillo|first2=Mariella|date=1972|title=LISREL: A General Computer Program for Estimating a Linear Structural Equation System Involving Multiple Indicators of Unmeasured Variables|url=https://files.eric.ed.gov/fulltext/ED073122.pdf|journal=Research Bulletin: Office of Education|volume=ETS-RB-72-56|via=US Government}}</ref><ref name="JS76">Jöreskog, Karl; Sorbom, Dag. (1976) LISREL III: Estimation of Linear Structural Equation Systems by Maximum Likelihood Methods. Chicago: National Educational Resources, Inc.</ref> embedded latent variables (which psychologists knew as the latent factors from factor analysis) within path-analysis-style equations (which sociologists inherited from Wright and Duncan). The factor-structured portion of the model incorporated measurement errors which permitted measurement-error-adjustment, though not necessarily error-free estimation, of effects connecting different postulated latent variables.