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'''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 book |doi=10.1016/B0-08-043076-7/00776-2 |quote=Structural equation modeling can be defined as 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. |chapter=Structural Equation Modeling |title=International Encyclopedia of the Social & Behavioral Sciences |date=2001 |last1=Kaplan |first1=D. |pages=15215–15222 |isbn=978-0-08-043076-8 }}</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">{{cite book |doi=10.1017/CBO9780511803161 |title=Causality |date=2009 |last1=Pearl |first1=Judea |isbn=978-0-511-80316-1 }}{{pn|date=June 2025}}</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 |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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== 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">{{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
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|date=June 2025}}</ref><ref name=":0">{{cite journal |last1=Jőreskog |first1=Karl G. |last2=van Thiilo |first2=Marielle |title=Lisrel a General Computer Program for Estimating a Linear Structural Equation System Involving Multiple Indicators of Unmeasured Variables |journal=ETS Research Bulletin Series |date=December 1972 |volume=1972 |issue=2 |id={{ERIC|ED073122}} |doi=10.1002/j.2333-8504.1972.tb00827.x }}</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.{{pn|date=June 2025}}</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.
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
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 book |doi=10.1007/978-3-030-12508-0 |title=Structural Equation Models |series=Studies in Systems, Decision and Control |date=2019 |volume=22 |isbn=978-3-030-12507-3 }}{{pn|date=June 2025}}</ref><ref>{{cite journal |last1=Christ |first1=Carl F. |title=The Cowles Commission's Contributions to Econometrics at Chicago, 1939-1955 |journal=Journal of Economic Literature |date=1994 |volume=32 |issue=1 |pages=30–59 |jstor=2728422 }}</ref> Disciplinary differences in approaches can be seen in SEMNET discussions of endogeneity, and in discussions on causality via directed acyclic graphs (DAGs).<ref name="Pearl09"/> Discussions comparing and contrasting various SEM approaches are available<ref name="Imbens20">{{cite journal |last1=Imbens |first1=Guido W. |title=Potential Outcome and Directed Acyclic Graph Approaches to Causality: Relevance for Empirical Practice in Economics |journal=Journal of Economic Literature |date=December 2020 |volume=58 |issue=4 |pages=1129–1179 |doi=10.1257/jel.20191597 }}</ref><ref name="BP13">{{cite book | doi=10.1007/978-94-007-6094-3_15 | chapter=Eight Myths About Causality and Structural Equation Models | title=Handbook of Causal Analysis for Social Research | series=Handbooks of Sociology and Social Research | date=2013 | last1=Bollen | first1=Kenneth A. | last2=Pearl | first2=Judea | pages=301–328 | isbn=978-94-007-6093-6 }}</ref> highlighting disciplinary differences in data structures and the concerns motivating economic models.
[[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" />
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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|date=June 2025}}</ref><ref name="Kline16"/>
=== Estimation of free model coefficients ===
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a) the coefficients' locations in the model (e.g. which variables are connected/disconnected),
b) the nature of the connections between the variables (covariances or effects; with effects often assumed to be linear),
c) the nature of the error or residual variables (often assumed to be independent of, or causally
and d) the measurement scales appropriate for the variables (interval level measurement is often assumed).
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Replication is unlikely to detect misspecified models which inappropriately-fit the data. If the replicate data is within random variations of the original data, the same incorrect coefficient placements that provided inappropriate-fit to the original data will likely also inappropriately-fit the replicate data. Replication helps detect issues such as data mistakes (made by different research groups), but is especially weak at detecting misspecifications after exploratory model modification – as when confirmatory factor analysis is applied to a random second-half of data following exploratory factor analysis (EFA) of first-half data.
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
"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=10.1037/1082-989x.7.4.403 |pmid=12530701 |pmc=2435310 }}</ref> The probability accompanying a {{math|χ<sup>2</sup>}} test is the probability that the data could arise by random sampling variations if the current model, with its optimal estimates, constituted the real underlying population forces. A small {{math|χ<sup>2</sup>}} probability reports it would be unlikely for the current data to have arisen if the current model structure constituted the real population causal forces – with the remaining differences attributed to random sampling variations. Browne, McCallum, Kim, Andersen, and Glaser presented a factor model they viewed as acceptable despite the model being significantly inconsistent with their data according to {{math|χ<sup>2</sup>}}. The fallaciousness of their claim that close-fit should be treated as good enough was demonstrated by Hayduk, Pazkerka-Robinson, Cummings, Levers and Beres<ref name="HP-RCLB05">{{cite journal |doi=10.1186/1471-2288-5-1|doi-access=free |title=Structural equation model testing and the quality of natural killer cell activity measurements |date=2005 |last1=Hayduk |first1=Leslie A. |last2=Pazderka-Robinson |first2=Hannah |last3=Cummings |first3=Greta G. |last4=Levers |first4=Merry-Jo D. |last5=Beres |first5=Melanie A. |journal=BMC Medical Research Methodology |volume=5 |page=1 |pmid=15636638 |pmc=546216 }} Note the correction of .922 to .992, and the correction of .944 to .994 in the Hayduk, et al. Table 1.</ref> who demonstrated a fitting model for Browne, et al.'s own data by incorporating an experimental feature Browne, et al. overlooked. The fault was not in the math of the indices or in the over-sensitivity of {{math|χ<sup>2</sup>}} testing. The fault was in Browne, MacCallum, and the other authors forgetting, neglecting, or overlooking, that the amount of ill fit cannot be trusted to correspond to the nature, ___location, or seriousness of problems in a model's specification.<ref name="Hayduk14a">{{cite journal | doi=10.1177/0013164414527449 | title=Seeing Perfectly Fitting Factor Models That Are Causally Misspecified | date=2014 | last1=Hayduk | first1=Leslie | journal=Educational and Psychological Measurement | volume=74 | issue=6 | pages=905–926 }}</ref>
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Numerous fit indices quantify how closely a model fits the data but all fit indices suffer from the logical difficulty that the size or amount of ill fit is not trustably coordinated with the severity or nature of the issues producing the data inconsistency.<ref name="Hayduk14a"/> Models with different causal structures which fit the data identically well, have been called equivalent models.<ref name="Kline16"/> Such models are data-fit-equivalent though not causally equivalent, so at least one of the so-called equivalent models must be inconsistent with the world's structure. If there is a perfect 1.0 correlation between X and Y and we model this as X causes Y, there will be perfect fit and zero residual error. But the model may not match the world because Y may actually cause X, or both X and Y may be responding to a common cause Z, or the world may contain a mixture of these effects (e.g. like a common cause plus an effect of Y on X), or other causal structures. The perfect fit does not tell us the model's structure corresponds to the world's structure, and this in turn implies that getting closer to perfect fit does not necessarily correspond to getting closer to the world's structure – maybe it does, maybe it doesn't. This makes it incorrect for a researcher to claim that even perfect model fit implies the model is correctly causally specified. For even moderately complex models, precisely equivalently-fitting models are rare. Models almost-fitting the data, according to any index, unavoidably introduce additional potentially-important yet unknown model misspecifications. These models constitute a greater research impediment.
This logical weakness renders all fit indices "unhelpful" whenever a structural equation model is significantly inconsistent with the data,<ref name="Barrett07"/> but several forces continue to propagate fit-index use. For example, Dag Sorbom reported that when someone asked Karl Joreskog, the developer of the first structural equation modeling program, "Why have you then added GFI?" to your LISREL program, Joreskog replied "Well, users threaten us saying they would stop using LISREL if it always produces such large chi-squares. So we had to invent something to make people happy. GFI serves that purpose."<ref name="Sorbom">{{cite book |editor1-last=Cudeck |editor1-first=Robert |editor2-last=Jöreskog |editor2-first=K. G. |editor3-last=Sörbom |editor3-first=Dag |editor4-last=Toit |editor4-first=S. H. C. Du |title=Structural Equation Modeling: Present and Future : a Festschrift in Honor of Karl Jöreskog |date=2001 |publisher=Scientific Software International |isbn=978-0-89498-049-7 }}{{pn|date=June 2025}}</ref> The {{math|χ<sup>2</sup>}} evidence of model-data inconsistency was too statistically solid to be dislodged or discarded, but people could at least be provided a way to distract from the "disturbing" evidence. Career-profits can still be accrued by developing additional indices, reporting investigations of index behavior, and publishing models intentionally burying evidence of model-data inconsistency under an MDI (a mound of distracting indices). There seems no general justification for why a researcher should "accept" a causally wrong model, rather than attempting to correct detected misspecifications. And some portions of the literature seems not to have noticed that "accepting a model" (on the basis of "satisfying" an index value) suffers from an intensified version of the criticism applied to "acceptance" of a null-hypothesis. Introductory statistics texts usually recommend replacing the term "accept" with "failed to reject the null hypothesis" to acknowledge the possibility of Type II error. A Type III error arises from "accepting" a model hypothesis when the current data are sufficient to reject the model.
Whether or not researchers are committed to seeking the world’s structure is a fundamental concern. Displacing test evidence of model-data inconsistency by hiding it behind index claims of acceptable-fit, introduces the discipline-wide cost of diverting attention away from whatever the discipline might have done to attain a structurally-improved understanding of the discipline’s substance. The discipline ends up paying a real costs for index-based displacement of evidence of model misspecification. The frictions created by disagreements over the necessity of correcting model misspecifications will likely increase with increasing use of non-factor-structured models, and with use of fewer, more-precise, indicators of similar yet importantly-different latent variables.<ref name="HL12"/>
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=== Interpretation ===
Causal interpretations of SE models are the clearest and most understandable but those interpretations will be fallacious/wrong if the model’s structure does not correspond to the world’s causal structure. Consequently, interpretation should address the overall status and structure of the model, not merely the model’s estimated coefficients. Whether a model fits the data, and/or how a model came to fit the data, are paramount for interpretation. Data fit obtained by exploring, or by following successive modification indices, does not guarantee the model is wrong but raises serious doubts because these approaches are prone to incorrectly modeling data features. For example, exploring to see how many factors are required preempts finding the data are not factor structured, especially if the factor model has been “persuaded” to fit via inclusion of measurement error covariances. Data’s ability to speak against a postulated model is progressively eroded with each unwarranted inclusion of a “modification index suggested” effect or error covariance. It becomes exceedingly difficult to recover a proper model if the initial/base model contains several misspecifications.<ref name="HC00">{{cite journal |last1=Herting |first1=Jerald R. |last2=Costner |first2=Herbert L. |title=Another Perspective on 'The Proper Number of Factors' and the Appropriate Number of Steps |journal=Structural Equation Modeling
Direct-effect estimates are interpreted in parallel to the interpretation of coefficients in regression equations but with causal commitment. Each unit increase in a causal variable’s value is viewed as producing a change of the estimated magnitude in the dependent variable’s value given control or adjustment for all the other operative/modeled causal mechanisms. Indirect effects are interpreted similarly, with the magnitude of a specific indirect effect equaling the product of the series of direct effects comprising that indirect effect. The units involved are the real scales of observed variables’ values, and the assigned scale values for latent variables. A specified/fixed 1.0 effect of a latent on a specific indicator coordinates that indicator’s scale with the latent variable’s scale. The presumption that the remainder of the model remains constant or unchanging may require discounting indirect effects that might, in the real world, be simultaneously prompted by a real unit increase. And the unit increase itself might be inconsistent with what is possible in the real world because there may be no known way to change the causal variable’s value. If a model adjusts for measurement errors, the adjustment permits interpreting latent-level effects as referring to variations in true scores.<ref name="BMvH03"/>
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SE model interpretation should connect specific model causal segments to their variance and covariance implications. A single direct effect reports that the variance in the independent variable produces a specific amount of variation in the dependent variable’s values, but the causal details of precisely what makes this happens remains unspecified because a single effect coefficient does not contain sub-components available for integration into a structured story of how that effect arises. A more fine-grained SE model incorporating variables intervening between the cause and effect would be required to provide features constituting a story about how any one effect functions. Until such a model arrives each estimated direct effect retains a tinge of the unknown, thereby invoking the essence of a theory. A parallel essential unknownness would accompany each estimated coefficient in even the more fine-grained model, so the sense of fundamental mystery is never fully eradicated from SE models.
Even if each modeled effect is unknown beyond the identity of the variables involved and the estimated magnitude of the effect, the structures linking multiple modeled effects provide opportunities to express how things function to coordinate the observed variables – thereby providing useful interpretation possibilities. For example, a common cause contributes to the covariance or correlation between two effected variables, because if the value of the cause goes up, the values of both effects should also go up (assuming positive effects) even if we do not know the full story underlying each cause.<ref name="Duncan75"/> (A correlation is the covariance between two variables that have both been standardized to have variance 1.0). Another interpretive contribution might be made by expressing how two causal variables can both explain variance in a dependent variable, as well as how covariance between two such causes can increase or decrease explained variance in the dependent variable. That is, interpretation may involve explaining how a pattern of effects and covariances can contribute to decreasing a dependent variable’s variance.<ref name="Hayduk87p20">Hayduk, L. (1987) Structural Equation Modeling with LISREL: Essentials and Advances, page 20. Baltimore, Johns Hopkins University Press. ISBN 0-8018-3478-3 Page 20</ref> Understanding causal implications implicitly connects to understanding “controlling”, and potentially explaining why some variables, but not others, should be controlled.<ref name="Pearl09"/><ref name="HCSNGDGP-R03">{{cite journal |last1=Hayduk |first1=Leslie |last2=Cummings |first2=Greta |last3=Stratkotter |first3=Rainer |last4=Nimmo |first4=Melanie |last5=Grygoryev |first5=Kostyantyn |last6=Dosman |first6=Donna |last7=Gillespie |first7=Michael |last8=Pazderka-Robinson |first8=Hannah |last9=Boadu |first9=Kwame |title=Pearl's D-Separation: One More Step Into Causal Thinking |journal=Structural Equation Modeling
The statistical insignificance of an effect estimate indicates the estimate could rather easily arise as a random sampling variation around a null/zero effect, so interpreting the estimate as a real effect becomes equivocal. As in regression, the proportion of each dependent variable’s variance explained by variations in the modeled causes are provided by ''R''<sup>2</sup>, though the Blocked-Error ''R''<sup>2</sup> should be used if the dependent variable is involved in reciprocal or looped effects, or if it has an error variable correlated with any predictor’s error variable.<ref name="Hayduk06">{{cite journal |last1=Hayduk |first1=Leslie A. |title=Blocked-Error-R 2: A Conceptually Improved Definition of the Proportion of Explained Variance in Models Containing Loops or Correlated Residuals |journal=Quality & Quantity |date=August 2006 |volume=40 |issue=4 |pages=629–649 |doi=10.1007/s11135-005-1095-4 }}</ref>
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Interpretations become progressively more complex for models containing interactions, nonlinearities, multiple groups, multiple levels, and categorical variables.<ref name="Kline16"/> <!-- For interpretations of coefficients in models containing interactions, see { reference needed }, for multilevel models see { reference needed }, for longitudinal models see, { reference needed }, and for models containing categoric variables see { reference needed }. --> Effects touching causal loops, reciprocal effects, or correlated residuals also require slightly revised interpretations.<ref name="Hayduk87"/><ref name="Hayduk96"/>
Careful interpretation of both failing and fitting models can provide research advancement. To be dependable, the model should investigate academically informative causal structures, fit applicable data with understandable estimates, and not include vacuous coefficients.<ref name="Millsap07">{{cite journal |last1=Millsap |first1=Roger E. |title=Structural equation modeling made difficult |journal=Personality and Individual Differences |date=May 2007 |volume=42 |issue=5 |pages=875–881 |doi=10.1016/j.paid.2006.09.021 }}</ref> Dependable fitting models are rarer than failing models or models inappropriately bludgeoned into fitting, but appropriately-fitting models are possible.<ref name="HP-RCLB05"/><ref name="EHR82">Entwisle, D.R.; Hayduk, L.A.; Reilly, T.W. (1982) Early Schooling: Cognitive and Affective Outcomes. Baltimore: Johns Hopkins University Press.{{pn|date=June 2025}}</ref><ref name="Hayduk94">{{cite journal |last1=Hayduk |first1=Leslie A. |title=Personal space: Understanding the simplex model |journal=Journal of Nonverbal Behavior |date=September 1994 |volume=18 |issue=3 |pages=245–260 |doi=10.1007/BF02170028 }}</ref><ref name="HSR97">{{cite journal |last1=Hayduk |first1=Leslie A. |last2=Stratkotter |first2=Rainer F. |last3=Rovers |first3=Martin W. |title=Sexual Orientation and the Willingness of Catholic Seminary Students to Conform to Church Teachings |journal=Journal for the Scientific Study of Religion |date=1997 |volume=36 |issue=3 |pages=455–467 |doi=10.2307/1387861 |jstor=1387861 }}</ref>
The multiple ways of conceptualizing PLS models<ref name="RSR17">{{cite journal | doi=10.15358/0344-1369-2017-3-4 | title=On Comparing Results from CB-SEM and PLS-SEM: Five Perspectives and Five Recommendations | date=2017 | last1=Rigdon | first1=Edward E. | last2=Sarstedt | first2=Marko | last3=Ringle | first3=Christian M. | journal=Marketing ZFP | volume=39 | issue=3 | pages=4–16 | doi-access=free }}</ref> complicate interpretation of PLS models. Many of the above comments are applicable if a PLS modeler adopts a realist perspective by striving to ensure their modeled indicators combine in a way that matches some existing but unavailable latent variable. Non-causal PLS models, such as those focusing primarily on ''R''<sup>2</sup> or out-of-sample predictive power, change the interpretation criteria by diminishing concern for whether or not the model’s coefficients have worldly counterparts. The fundamental features differentiating the five PLS modeling perspectives discussed by Rigdon, Sarstedt and Ringle<ref name="RSR17"/> point to differences in PLS modelers’ objectives, and corresponding differences in model features warranting interpretation.
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* Deep Path Modelling <ref name="Ing2024"/>
* Exploratory Structural Equation Modeling<ref>{{cite journal |last1=Marsh |first1=Herbert W. |last2=Morin |first2=Alexandre J.S. |last3=Parker |first3=Philip D. |last4=Kaur |first4=Gurvinder |title=Exploratory Structural Equation Modeling: An Integration of the Best Features of Exploratory and Confirmatory Factor Analysis |journal=Annual Review of Clinical Psychology |date=28 March 2014 |volume=10 |issue=1 |pages=85–110 |doi=10.1146/annurev-clinpsy-032813-153700 |pmid=24313568 }}</ref>
* Fusion validity models<ref name="HEH19">{{doi|10.3389/psyg.2019.01139|doi-access=free}}{{dead link|date=June 2025}}</ref>
* [[Item response theory]] models {{citation needed|date=July 2023}}
* [[Latent class models]] {{citation needed|date=July 2023}}
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* [[Measurement invariance]] models <ref>{{cite journal |last1=Leitgöb |first1=Heinz |last2=Seddig |first2=Daniel |last3=Asparouhov |first3=Tihomir |last4=Behr |first4=Dorothée |last5=Davidov |first5=Eldad |last6=De Roover |first6=Kim |last7=Jak |first7=Suzanne |last8=Meitinger |first8=Katharina |last9=Menold |first9=Natalja |last10=Muthén |first10=Bengt |last11=Rudnev |first11=Maksim |last12=Schmidt |first12=Peter |last13=van de Schoot |first13=Rens |title=Measurement invariance in the social sciences: Historical development, methodological challenges, state of the art, and future perspectives |journal=Social Science Research |date=February 2023 |volume=110 |pages=102805 |doi=10.1016/j.ssresearch.2022.102805 |pmid=36796989 |hdl=1874/431763 |hdl-access=free }}</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.|url-access=subscription }}</ref>
* Multiple group modelling with or without constraints between groups (genders, cultures, test forms, languages, etc.) {{citation needed|date=July 2023}}
* Multi-method multi-trait models {{citation needed|date=July 2023}}
* Random intercepts models {{citation needed|date=July 2023}}
* Structural Equation Model Trees {{citation needed|date=July 2023}}
* Structural Equation [[Multidimensional scaling]]<ref>{{cite journal |last1=Vera |first1=José Fernando |last2=Mair |first2=Patrick |title=SEMDS: An R Package for Structural Equation Multidimensional Scaling |journal=Structural Equation Modeling
== Software ==
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<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>
<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
<ref name="Chou1995">{{cite book|last1=Chou|first1=C. P.|last2=Bentler|first2=Peter|editor1-last=Hoyle|editor1-first=Rick|editor1-link=H|title=Structural equation modeling: Concepts, issues, and applications|date=1995|publisher=Sage|___location=Thousand Oaks, CA|pages=37–55|chapter=Estimates and tests in structural equation modeling}}</ref>
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<ref name="Ing2024">{{cite
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