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[[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 affecting 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. By a standard definition, 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
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
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 |last1=Curran |first1=Patrick J. |title=Have Multilevel Models Been Structural Equation Models All Along? |journal=Multivariate Behavioral Research |date=October 2003 |volume=38 |issue=4 |pages=529–569 |doi=10.1207/s15327906mbr3804_5 |pmid=26777445 }}</ref>
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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">{{cite journal |last1=Wright |first1=Sewall |date=1921 |title=Correlation and causation |journal=Journal of Agricultural Research |volume=20 |pages=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">{{cite journal |last1=Wolfle |first1=Lee M. |title=Sewall wright on the method of path coefficients: An annotated bibliography |journal=Structural Equation Modeling: A Multidisciplinary Journal |date=January 1999 |volume=6 |issue=3 |pages=280–291 |doi=10.1080/10705519909540134 }}</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.{{pn}}</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.{{pn}}</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.{{pn}}</ref><ref name=":0">{{cite journal |last1=Jőreskog |first1=Karl G. |last2=van Thiilo |first2=Marielle |title=
Traces of the historical convergence of the factor analytic and path analytic traditions persist as the distinction between the measurement and structural portions of models; and as continuing disagreements over model testing, and whether measurement should precede or accompany structural estimates.<ref name="HG00a">{{cite journal |last1=Hayduk |first1=Leslie A. |last2=Glaser |first2=Dale N. |title=Jiving the Four-Step, Waltzing Around Factor Analysis, and Other Serious Fun |journal=Structural Equation Modeling: A Multidisciplinary Journal |date=January 2000 |volume=7 |issue=1 |pages=1–35 |doi=10.1207/s15328007sem0701_01 }}</ref><ref name="HG00b">{{cite journal |last1=Hayduk |first1=Leslie A. |last2=Glaser |first2=Dale N. |title=Doing the Four-Step, Right-2-3, Wrong-2-3: A Brief Reply to Mulaik and Millsap; Bollen; Bentler; and Herting and Costner |journal=Structural Equation Modeling: A Multidisciplinary Journal |date=January 2000 |volume=7 |issue=1 |pages=111–123 |doi=10.1207/S15328007SEM0701_06 }}</ref> Viewing factor analysis as a data-reduction technique deemphasizes testing, which contrasts with path analytic appreciation for testing postulated causal connections – where the test result might signal model misspecification. The friction between factor analytic and path analytic traditions continue to surface in the literature.
Wright's path analysis influenced Hermann Wold, Wold's student Karl Jöreskog, and Jöreskog's student Claes Fornell, but SEM never gained a large following among U.S. econometricians, possibly due to fundamental differences in modeling objectives and typical data structures. The prolonged separation of SEM's economic branch led to procedural and terminological differences, though deep mathematical and statistical connections remain.<ref name="Westland15">{{cite
[[Judea Pearl]]<ref name="Pearl09" /> extended SEM from linear to nonparametric models, and proposed causal and counterfactual interpretations of the equations. Nonparametric SEMs permit estimating total, direct and indirect effects without making any commitment to linearity of effects or assumptions about the distributions of the error terms.<ref name="BP13" />
SEM analyses are popular in the social sciences because these analytic techniques help us break down complex concepts and understand causal processes, but the complexity of the models can introduce substantial variability in the results depending on the presence or absence of conventional control variables, the sample size, and the variables of interest.<ref>{{cite
Today, SEM forms part of a basis of [[machine learning]] and (interpretable) [[Neural network (machine learning)|neural networks]]. Exploratory and confirmatory factor analyses in classical statistics mirror unsupervised and supervised machine learning.
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Modelers specify each coefficient in a model as being ''free'' to be estimated, or ''fixed'' at some value. The free coefficients may be postulated effects the researcher wishes to test, background correlations among the exogenous variables, or the variances of the residual or error variables providing additional variations in the endogenous latent variables. The fixed coefficients may be values like the 1.0 values in Figure 2 that provide a scales for the latent variables, or values of 0.0 which assert causal disconnections such as the assertion of no-direct-effects (no arrows) pointing from Academic Achievement to any of the four scales in Figure 1. SEM programs provide estimates and tests of the free coefficients, while the fixed coefficients contribute importantly to testing the overall model structure. Various kinds of constraints between coefficients can also be used.<ref name="Kline16"/><ref name="Hayduk87"/><ref name="Bollen89"/> The model specification depends on what is known from the literature, the researcher's experience with the modeled indicator variables, and the features being investigated by using the specific model structure.
There is a limit to how many coefficients can be estimated in a model. If there are fewer data points than the number of estimated coefficients, the resulting model is said to be "unidentified" and no coefficient estimates can be obtained. Reciprocal effect, and other causal loops, may also interfere with estimation.<ref name="Rigdon95">{{cite journal |last1=Rigdon |first1=Edward E. |title=A Necessary and Sufficient Identification Rule for Structural Models Estimated in Practice |journal=Multivariate Behavioral Research |date=July 1995 |volume=30 |issue=3 |pages=359–383 |doi=10.1207/s15327906mbr3003_4 |pmid=26789940 }}</ref><ref name="Hayduk96">{{cite book |last1=Hayduk |first1=Leslie A. |title=LISREL Issues, Debates and Strategies |date=1996 |publisher=JHU Press |isbn=978-0-8018-5336-4 }}{{pn}}</ref><ref name="Kline16"/>
=== Estimation of free model coefficients ===
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A modification index is an estimate of how much a model's fit to the data would "improve" (but not necessarily how much the model's structure would improve) if a specific currently-fixed model coefficient were freed for estimation. Researchers confronting data-inconsistent models can easily free coefficients the modification indices report as likely to produce substantial improvements in fit. This simultaneously introduces a substantial risk of moving from a causally-wrong-and-failing model to a causally-wrong-but-fitting model because improved data-fit does not provide assurance that the freed coefficients are substantively reasonable or world matching. The original model may contain causal misspecifications such as incorrectly directed effects, or incorrect assumptions about unavailable variables, and such problems cannot be corrected by adding coefficients to the current model. Consequently, such models remain misspecified despite the closer fit provided by additional coefficients. Fitting yet worldly-inconsistent models are especially likely to arise if a researcher committed to a particular model (for example a factor model having a desired number of factors) gets an initially-failing model to fit by inserting measurement error covariances "suggested" by modification indices. MacCallum (1986) demonstrated that "even under favorable conditions, models arising from specification serchers must be viewed with caution."<ref name="MacCallum1986" /> Model misspecification may sometimes be corrected by insertion of coefficients suggested by the modification indices, but many more corrective possibilities are raised by employing a few indicators of similar-yet-importantly-different latent variables.<ref name="HL12">{{cite journal | doi=10.1186/1471-2288-12-159 | doi-access=free | title=Should researchers use single indicators, best indicators, or multiple indicators in structural equation models? | date=2012 | last1=Hayduk | first1=Leslie A. | last2=Littvay | first2=Levente | journal=BMC Medical Research Methodology | volume=12 | page=159 | pmid=23088287 | pmc=3506474 }}</ref>
"Accepting" failing models as "close enough" is also not a reasonable alternative. A cautionary instance was provided by Browne, MacCallum, Kim, Anderson, and Glaser who addressed the mathematics behind why the {{math|χ<sup>2</sup>}} test can have (though it does not always have) considerable power to detect model misspecification.<ref name="BMKAG02">{{cite journal |last1=Browne |first1=Michael W. |last2=MacCallum |first2=Robert C. |last3=Kim |first3=Cheong-Tag |last4=Andersen |first4=Barbara L. |last5=Glaser |first5=Ronald |title=When fit indices and residuals are incompatible. |journal=Psychological Methods |date=2002 |volume=7 |issue=4 |pages=403–421 |doi=10.1037
Many researchers tried to justify switching to fit-indices, rather than testing their models, by claiming that {{math|χ<sup>2</sup>}} increases (and hence {{math|χ<sup>2</sup>}} probability decreases) with increasing sample size (N). There are two mistakes in discounting {{math|χ<sup>2</sup>}} on this basis. First, for proper models, {{math|χ<sup>2</sup>}} does not increase with increasing N,<ref name="Hayduk14b"/> so if {{math|χ<sup>2</sup>}} increases with N that itself is a sign that something is detectably problematic. And second, for models that are detectably misspecified, {{math|χ<sup>2</sup>}} increase with N provides the good-news of increasing statistical power to detect model misspecification (namely power to detect Type II error). Some kinds of important misspecifications cannot be detected by {{math|χ<sup>2</sup>}},<ref name="Hayduk14a"/> so any amount of ill fit beyond what might be reasonably produced by random variations warrants report and consideration.<ref name="Barrett07"/><ref name="Hayduk14b"/> The {{math|χ<sup>2</sup>}} model test, possibly adjusted,<ref name="SB94">Satorra, A.; and Bentler, P. M. (1994) “Corrections to test statistics and standard errors in covariance structure analysis”. In A. von Eye and C. C. Clogg (Eds.), Latent variables analysis: Applications for developmental research (pp. 399–419). Thousand Oaks, CA: Sage.</ref> is the strongest available structural equation model test.
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The controversy over model testing declined as clear reporting of significant model-data inconsistency becomes mandatory. Scientists do not get to ignore, or fail to report, evidence just because they do not like what the evidence reports.<ref name="Hayduk14b"/> The requirement of attending to evidence pointing toward model mis-specification underpins more recent concern for addressing “endogeneity” – a style of model mis-specification that interferes with estimation due to lack of independence of error/residual variables. In general, the controversy over the causal nature of structural equation models, including factor-models, has also been declining. Stan Mulaik, a factor-analysis stalwart, has acknowledged the causal basis of factor models.<ref name="Mulaik09">Mulaik, S.A. (2009) Foundations of Factor Analysis (second edition). Chapman and Hall/CRC. Boca Raton, pages 130-131.</ref> The comments by Bollen and Pearl regarding myths about causality in the context of SEM<ref name="BP13" /> reinforced the centrality of causal thinking in the context of SEM.
A briefer controversy focused on competing models. Comparing competing models can be very helpful but there are fundamental issues that cannot be resolved by creating two models and retaining the better fitting model. The statistical sophistication of presentations like Levy and Hancock (2007),<ref name="LH07">{{cite journal |last1=Levy |first1=Roy |last2=Hancock |first2=Gregory R. |title=A Framework of Statistical Tests For Comparing Mean and Covariance Structure Models |journal=Multivariate Behavioral Research |date=29 June 2007 |volume=42 |issue=1 |pages=33–66 |doi=10.1080/00273170701329112 |pmid=26821076 }}</ref> for example, makes it easy to overlook that a researcher might begin with one terrible model and one atrocious model, and end by retaining the structurally terrible model because some index reports it as better fitting than the atrocious model. It is unfortunate that even otherwise strong SEM texts like Kline (2016)<ref name="Kline16"/> remain disturbingly weak in their presentation of model testing.<ref name="Hayduk18">{{cite journal | doi=10.25336/csp29397 | title=Review essay on Rex B. Kline's Principles and Practice of Structural Equation Modeling: Encouraging a fifth edition | date=2018 | last1=Hayduk | first1=Leslie | journal=Canadian Studies in Population | volume=45 | issue=3–4 | page=154 | doi-access=free }}</ref> Overall, the contributions that can be made by structural equation modeling depend on careful and detailed model assessment, even if a failing model happens to be the best available.
An additional controversy that touched the fringes of the previous controversies awaits ignition.{{citation needed|date=March 2024}} Factor models and theory-embedded factor structures having multiple indicators tend to fail, and dropping weak indicators tends to reduce the model-data inconsistency. Reducing the number of indicators leads to concern for, and controversy over, the minimum number of indicators required to support a latent variable in a structural equation model. Researchers tied to factor tradition can be persuaded to reduce the number of indicators to three per latent variable, but three or even two indicators may still be inconsistent with a proposed underlying factor common cause. Hayduk and Littvay (2012)<ref name="HL12"/> discussed how to think about, defend, and adjust for measurement error, when using only a single indicator for each modeled latent variable. Single indicators have been used effectively in SE models for a long time,<ref name="EHR82"/> but controversy remains only as far away as a reviewer who has considered measurement from only the factor analytic perspective.
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<ref name="Barrett07">{{cite journal |last1=Barrett |first1=Paul |title=Structural equation modelling: Adjudging model fit |journal=Personality and Individual Differences |date=May 2007 |volume=42 |issue=5 |pages=815–824 |doi=10.1016/j.paid.2006.09.018 }}</ref>
<ref name="BC92">{{cite journal |last1=Browne |first1=Michael W. |last2=Cudeck |first2=Robert |title=Alternative Ways of Assessing Model Fit |journal=Sociological Methods & Research |date=November 1992 |volume=21 |issue=2 |pages=230–258 |doi=10.1177/0049124192021002005 }}</ref>
<ref name="S90">{{cite journal |last1=Steiger |first1=James H. |title=Structural Model Evaluation and Modification: An Interval Estimation Approach |journal=Multivariate Behavioral Research |date=April 1990 |volume=25 |issue=2 |pages=173–180 |doi=10.1207/s15327906mbr2502_4 |pmid=26794479 }}</ref>
<ref name="SL80">Steiger, J. H.; and Lind, J. (1980) "Statistically Based Tests for the Number of Common Factors." Paper presented at the annual meeting of the Psychometric Society, Iowa City.</ref>
<ref name="MacCallum1986">{{cite journal |doi=10.1037/0033-2909.100.1.107 |title=Specification searches in covariance structure modeling |journal=Psychological Bulletin |volume=100 |pages=107–120 |year=1986 |last1=MacCallum |first1=Robert }}</ref>
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<ref name="Ing2024">{{cite journal |doi=10.1101/2024.06.13.598616 |title=Integrating Multi-Modal Cancer Data Using Deep Latent Variable Path Modelling |date=2024 |last1=Ing |first1=Alex |last2=Andrades |first2=Alvaro |last3=Cosenza |first3=Marco Raffaele |last4=Korbel |first4=Jan O. }}</ref>
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