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Structural equation modeling (SEM) is a label for a diverse set of methods used by scientists doing both observational and experimental research. SEM is used mostly in the social and behavioral sciences but it is also used in epidemiology, business, and other fields. A definition of SEM is difficult without reference to technical language, but a good starting place is the name itself.

SEM involves a model representing how various aspects of some phenomenon are thought to 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 equations 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.

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, confirmatory composite analysis, path analysis, multi-group modeling, longitudinal modeling, partial least squares path modeling, latent growth modeling and hierarchical or multilevel modeling.

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: disregard of available model tests, problems in the model’s specification, a tendency to accept models without considering external validity, and potential philosophical biases.

A great advantage of SEM is that all of these measurements and tests occur simultaneously in one statistical estimation procedure, where all the model coefficient 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.

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. 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 and SEM blossomed in the late 1970’s and 1980’s when increasing computing power permitted practical model estimation. In 1987 Hayduk provided the first book-length introduction to structural equation modeling with latent variables, and this was soon followed by Bollen’s popular text (1989).

Different yet mathematically related modeling approaches developed in psychology, sociology, and economics. Early Cowles Commission work on 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 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. 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 and on SEMNET – a free listserve circulating SEM postings to more than 3,000 registrants, supported by a University of Alabama listserv. The SEMNET archive can be searched for user-specified topics, and many of the 2023 updates to this Misplaced Pages page were previously circulated on SEMNET for knowledgeable review and comment.

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. The economic version of SEM can be seen in SEMNET discussions of endogeneity, and in the heat produced as Judea Pearl’s approach to causality via directed acyclic graphs (DAG’s) rubs against economic approaches to modeling. Discussions comparing and contrasting various SEM approaches are available but disciplinary differences in data structures and the concerns motivating economic models make reunion unlikely. Pearl 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.

SEM analyses are popular in the social sciences because computer programs make it possible to estimate complicated causal structures, but the complexity of the models introduces substantial variability in the quality of the results. Some, but not all, results are obtained without the "inconvenience" of understanding experimental design, statistical control, the consequences of sample size, and other features contributing to good research design.

General SEM Steps and Considerations

The following considerations apply to the construction and assessment of many structural equation models.

Model Specification

Building or specifying a model requires attending to:

• the set of variables to be employed,
• what is known about the variables,
• what is presumed or hypothesized about the variables’ causal connections and disconnections,
• what the researcher seeks to learn from the modeling,
• and the cases for which values of the variables will be available (kids? workers? companies? countries? cells? accidents? cults?).

Structural equation models attempt to mirror the worldly forces operative for causally homogeneous cases – namely cases enmeshed in the same worldly causal structures but whose values on the causes differ and who therefore possess different values on the outcome variables. Causal homogeneity can be facilitated by case selection, or by segregating cases in a multi-group model. A model’s specification is not complete until the researcher specifies:

• which effects and/or correlations/covariances are to be included and estimated,
• which effects and other coefficients are forbidden or presumed unnecessary,
• and which coefficients will be given fixed/unchanging values (e.g. to provide measurement scales for latent variables as in Figure 2).

The latent level of a model is composed of endogenous and exogenous variables. The endogenous latent variables are the true-score variables postulated as receiving effects from at least one other modeled variable. Each endogenous variable is modeled as the dependent variable in a regression-style equation. The exogenous latent variables are background variables postulated as causing one or more of the endogenous variables and are modeled like the predictor variables in regression-style equations. Causal connections among the exogenous variables are not explicitly modeled but are usually acknowledged by modeling the exogenous variables as freely correlating with one another. The model may include intervening variables – variables receiving effects from some variables but also sending effects to other variables. As in regression, each endogenous variable is assigned a residual or error variable encapsulating the effects of unavailable and usually unknown causes. Each latent variable, whether exogenous or endogenous, is thought of as containing the cases’ true-scores on that variable, and these true-scores causally contribute valid/genuine variations into one or more of the observed/reported indicator variables.

The LISREL program assigned Greek names to the elements in a set of matrices to keep track of the various model components. These names became relatively standard notation, though the notation has been extended and altered to accommodate a variety of statistical considerations. Texts and programs “simplifying” model specification via diagrams or by using equations permitting user-selected variable names, re-convert the user’s model into some standard matrix-algebra form in the background. The “simplifications” are achieved by implicitly introducing default program “assumptions” about model features with which users supposedly need not concern themselves. Unfortunately, these default assumptions easily obscure model components that leave unrecognized issues lurking within the model’s structure, and underlying matrices.

Two main components of models are distinguished in SEM: the structural model showing potential causal dependencies between endogenous and exogenous latent variables, and the measurement model showing the causal connections between the latent variables and the indicators. Exploratory and confirmatory factor analysis models, for example, focus on the causal measurement connections, while path models more closely correspond to SEMs latent structural connections.

Modelers specify each coefficient in a model as being free to be estimated, or fixed at some value. The free coefficeints 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. 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 speific 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.

Estimation of free parameters

Parameter estimation is done by comparing the actual covariance matrices representing the relationships between variables and the estimated covariance matrices of the best fitting model. This is obtained through numerical maximization via expectation–maximization of a fit criterion as provided by maximum likelihood estimation, quasi-maximum likelihood estimation, weighted least squares or asymptotically distribution-free methods. This is often accomplished by using a specialized SEM analysis program, of which several exist.

Evaluation of models via model fit

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An important task is to examine how well an estimated model “fits” the available data. The output from SEM programs includes a matrix reporting the relationships between the observed variables that would be observed if the estimated effects in the model actually controlled the observed variables’ values. The “fit” of a model reports match or mismatch between the model-implied relationships (often covariances) and the observed relationships among the variables. Large and significant differences between the data and the model’s implications signal problems. The probability accompanying a χ2 (chi-square) test is the probability that the data could arise by random sampling variations if the estimated model constituted the real underlying population forces. A small χ2 probability reports it would be unlikely for the current data to have arisen if the model structure constituted the real population causal forces – with the remaining differences attributed to random sampling variations.

Numerous fit indices quantify how closely a model fits the data but the SEM literature is divided on how to respond to varying amounts of ill fit. One complication is that the size or amount of ill fit is not assuredly coordinated with the severity or nature of the issues producing the data inconsistency. Models with different causal structures which fit the data identically well have been called equivalent models. 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 variables X and Y and the model claims X causes Y, there will be perfect fit between the data and the model’s implication. But that model might not match the world because Y might cause X, or both X and Y might be responding to common-cause Z, or the world might contain a mixture of these effects (such as a common cause plus an effect of Y on X). The perfect fit of the X causes Y model does not guarantee the model’s structure corresponds to the world’s structure – maybe it does, maybe it doesn’t. And getting closer to perfect fit similarly does not guarantee the model is getting closer to matching the world’s structure – maybe it is, maybe it isn’t.

This logical difficulty is especially pronounced whenever a structural equation model is significantly inconsistent with the data, because there is no general justification for why a researcher should “accept” a causally wrong model, rather than correct detected misspecifications. Several forces continue to propagate reporting “close” fit. 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.” The χ2 evidence of model-data inconsistency was too statistically solid to be disregarded, but the GFI and other fit indices could still distract from unwelcome evidence of model-data inconsistency.

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, but is especially weak at detecting misspecifications after exploratory model modification – as when confirmatory factor analysis (CFA) is applied to a random second-half of data following exploratory factor analysis (EFA) of first-half data.

A cautionary instance was provided by Browne, MacCallum, Kim, Anderson, and Glaser (2002) who addressed the mathematics behind why the χ2 test can have (though it does not always have) considerable power to detect model misspecification. They presented a factor model as acceptable despite that model being significantly inconsistent with their data. Incorporating an overlooked experimental feature provided a model fitting the same data and contradicting the original model. The fault was not in the math of the indices or in the over-sensitivity of χ2 testing. The fault was in forgetting, neglecting, or overlooking, that the amount of ill fit cannot be trusted to correspond to the nature or seriousness of problems in a model’s specification. Reporting fit-index values to distract from evidence of model-data inconsistency introduces discipline-wide costs. The discipline pays the opportunity cost of not having pursued a structurally improved understanding of the discipline’s data.

The considerations relevant to assessing fit include checking: 1) whether data concerns have been addressed (To ensure data mistakes are not driving model-data inconsistency.); 2) whether criterion values for the index have been investigated for models structured like the researcher’s model (Index criterion based on factor structured models are only appropriate if the researcher’s model actually is factor structured.); 3) whether the kinds of potential misspecifications in the current model correspond to the kinds of misspecifications on which the index criterion are based (Criteria based on simulation of omitted factor loadings may not be appropriate for misspecifications resulting from failure to include appropriate control variables.); 4) whether the researcher knowingly agrees to disregard evidence pointing to the kinds of misspecifications on which the index criteria were based. (If the index criterion is based on missing a factor loading or two, using that criterion acknowledges the researcher’s willingness to accept a model missing a factor loading or two.); 5) whether the latest, not outdated, index criteria are being used (The criteria for the indices are subject to debate. and have tightened over time.); 6) whether satisfying criterion values on pairs of indices are required (Hu and Bentler report that some common indices function inappropriately unless they are assessed together.); 7) whether a model test is, or is not, available. (A χ2 value, degrees of freedom, and probability will be available for models reporting indices based on χ2.) and 8), whether the researcher has considered both alpha (Type I) and beta (Type II) errors in making their index-based decisions (The “tolerable” amount of data-inconsistency is likely to be lower in medical contexts.).

Some of the more commonly used fit statistics include

• Chi-square
  • A fundamental test of fit used in the calculation of many other fit measures. It is a function of the discrepancy between the observed covariance matrix and the model-implied covariance matrix. Chi-square increases with sample size only if the model is detectably misspecified.
• Akaike information criterion (AIC)
  • A test of relative model fit: The preferred model is the one with the lowest AIC value.
  • ${\displaystyle {\mathit {AIC}}=2k-2\ln(L)\,}$
  • where k is the number of parameters in the statistical model, and L is the maximized value of the likelihood of the model.
• Root Mean Square Error of Approximation (RMSEA)
  • Fit index where a value of zero indicates the best fit. While the guideline for determining a "close fit" using RMSEA is highly contested, most researchers concur that an RMSEA of .1 or more indicates poor fit.
• Standardized Root Mean Squared Residual (SRMR)
  • The SRMR is a popular absolute fit indicator. Hu and Bentler (1999) suggested .08 or smaller as a guideline for good fit. Kline (2011) suggested .1 or smaller as a guideline for good fit.
• Comparative Fit Index (CFI)
  • In examining baseline comparisons, the CFI depends in large part on the average size of the correlations in the data. If the average correlation between variables is not high, then the CFI will not be very high. A CFI value of .95 or higher is desirable.

For each measure of fit, a decision as to what represents a good-enough fit between the model and the data reflects the researcher's modeling objective (perhaps challenging some else's model, or improving measurement); whether or not the model is to be claimed as having been tested; and whether the researcher is comfortable "disregarding" evidence of the index-documented degree of ill fit.

 Approximation

Model modification

The model may need to be modified in order to more closely match the world's structure and thereby improve the fit. Many programs provide modification indices which report the change in χ² that would result from freeing fixed parameters: usually through adding a coeficient to a model which is currently set to zero. Modifications that improve model fit may be flagged as potential changes that can be made to the model. Modifications to a model are changes to the theory claimed to be true. Modifications therefore must make sense in terms of the theory being tested, or be acknowledged as limitations of that theory. Changes to measurement model are effectively claims that the items/data are impure indicators of the latent variables specified by theory.

A modification index is an estimate of how much a model’s chi-square 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 fit improvements. 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, which may not be correctable through addition of coefficients to the current model. Consequently, these models remain misspecified despite their closer fit. 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) converts an initially-failing model into fitting 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.” 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.

Sample size and power

While researchers agree that large sample sizes are required to provide sufficient statistical power and precise estimates using SEM, there is no general consensus on the appropriate method for determining adequate sample size. Considerations for determining sample size include the number of observations per parameter, the number of observations required for fit indexes to perform adequately, and the number of observations per degree of freedom. Researchers have proposed guidelines based on simulation studies, professional experience, and mathematical formulas. Sample size requirements to achieve a particular significance and power in SEM hypothesis testing are similar to requirements for similar sized multiple regressions.

In the past, researchers frequently justified switching to employing fit-indices, rather than testing their models, by claiming χ2 increases (and hence χ2’s probability decreases) with increasing sample size (N). There are two mistakes in discounting χ2 on this basis. First, for proper models, χ2 does not increase with increasing N, so if χ2 increases with N that itself is a sign something is detectably problematic. Second, for models that are detectably misspecified, χ2’s increase with N provides the good statistical news of increasing power to detect model misspecification (namely power to detect Type II error). Some important misspecifications cannot be detected by χ2 irrespective of N, so any amount of ill fit beyond what might be reasonably produced by random variations warrants report and consideration. The χ2 model test, possibly adjusted, is the strongest available structural equation model test.

Coordinating the Multiple Model Assessment Components

Assessing models and their coefficient estimates depends on the data, the theory, the model, and the estimation strategy. Hence model assessments consider:

    whether the data contain reasonable measurements of appropriate variables,
    whether the modeled case are causally homogeneous, It makes no sense to estimate one model if the data cases reflect two or more different causal networks.
    whether the model appropriately represents the theory or features of interest, Models are unpersuasive if they omit features required by a theory, or contain coefficients inconsistent with that theory.
    whether the estimates are statistically justifiable, Substantive assessments may be devastated: by violating assumptions, by using an inappropriate estimator, and/or by encountering non-convergence of iterative estimators.
    the substantive reasonableness of the estimates, Negative variances, and correlations exceeding 1.0 or -1.0, are impossible. Statistically possible estimates that are inconsistent with theory may also challenge theory, and our understanding.

and any remaining inconsistency between the model and data.

The estimation process minimizes the differences between the model and data but important and informative differences may remain. Research claiming to test or “investigate” a theory requires attending to beyond-chance model-data inconsistency. Estimation adjusts the model’s free coefficients to provide the best possible fit to the data. If a model remains inconsistent with the data despite selection of optimal coefficient estimates, an honest research response reports and attends to this evidence. Beyond-chance model-data inconsistency challenges both the coefficient estimates and the model’s capacity for adjudicating the model’s structure, irrespective of whether the inconsistency originates in problematic data, inappropriate statistical estimation, or incorrect model specification.

Coefficient estimates in data-inconsistent (“failing”) models are interpretable, as reports of how the world would appear to someone believing a model that conflicts with the available data. The estimates in data-inconsistent models do not necessarily become “obviously wrong” by becoming statistically strange, or wrongly-signed according to theory. The estimates may even closely match a theory’s requirements, but the remaining data inconsistency renders the match between the estimates and theory unpersuasive. Failing models remain interpretable, but only as interpretations that conflict with available evidence, even if the model is the best of several alternative models.

Caution should be taken when making claims of causality even when experimentation or time-ordered studies have been done. The term causal model must be understood to mean "a model that conveys causal assumptions", not necessarily a model that produces validated causal conclusions. Collecting data at multiple time points and using an experimental or quasi-experimental design can help rule out certain rival hypotheses but even a randomized experiment cannot rule out all such threats to causal inference. No research design, no matter how clever, can fully ascertain causal structures.

Advanced uses

• Measurement invariance
• Multiple group modelling: This is a technique allowing joint estimation of multiple models, each with different sub-groups. Applications include behavior genetics, and analysis of differences between groups (e.g., gender, cultures, test forms written in different languages, etc.).
• Latent growth modeling
• Nonlinear mixed-effects model
• Hierarchical/multilevel models; item response theory models
• Mixture model (latent class) SEM
• Alternative estimation and testing techniques
• Robust inference
• Survey sampling analyses
• Multi-method multi-trait models
• Structural Equation Model Trees

SEM-specific software

Numerous software packages exist for fitting structural equation models. LISREL was the first such software, initially released in the 1970s. Frequently used software implementations among researchers include Mplus, R packages lavaan and sem, LISREL, OpenMx, SPSS AMOS, and Stata. Barbara M. Byrne published multiple instructional books for using a variety of these softwares as part of the Society of Multivariate Experimental Psychology's Multivariate Applications book series.

Scholars consider it good practice to report which software package and version was used for SEM analysis because they have different capabilities and may use slightly different methods to perform similarly named techniques.

See also

• Causal model
• Graphical model
• Multivariate statistics
• Partial least squares path modeling
• Partial least squares regression
• Simultaneous equations model
• Structural Equations with Latent Variables
• Causal map
• Bayesian Network

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Bibliography

• Hu, Li‐tze; Bentler, Peter M (1999). "Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives". Structural Equation Modeling. 6: 1–55. doi:10.1080/10705519909540118. hdl:2027.42/139911.
• Kaplan, D. (2008). Structural Equation Modeling: Foundations and Extensions (2nd ed.). SAGE. ISBN 978-1412916240.
• Kline, Rex (2011). Principles and Practice of Structural Equation Modeling (Third ed.). Guilford. ISBN 978-1-60623-876-9.
• MacCallum, Robert; Austin, James (2000). "Applications of Structural Equation Modeling in Psychological Research" (PDF). Annual Review of Psychology. 51: 201–226. doi:10.1146/annurev.psych.51.1.201. PMID 10751970. Retrieved 25 January 2015.
• Quintana, Stephen M.; Maxwell, Scott E. (1999). "Implications of Recent Developments in Structural Equation Modeling for Counseling Psychology". The Counseling Psychologist. 27 (4): 485–527. doi:10.1177/0011000099274002. S2CID 145586057.

Further reading

• Bagozzi, Richard P; Yi, Youjae (2011). "Specification, evaluation, and interpretation of structural equation models". Journal of the Academy of Marketing Science. 40 (1): 8–34. doi:10.1007/s11747-011-0278-x. S2CID 167896719.
• Bartholomew, D. J., and Knott, M. (1999) Latent Variable Models and Factor Analysis Kendall's Library of Statistics, vol. 7, Edward Arnold Publishers, ISBN 0-340-69243-X
• Bentler, P.M. & Bonett, D.G. (1980), "Significance tests and goodness of fit in the analysis of covariance structures", Psychological Bulletin, 88, 588–606.
• Bollen, K. A. (1989). Structural Equations with Latent Variables. Wiley, ISBN 0-471-01171-1
• Byrne, B. M. (2001) Structural Equation Modeling with AMOS - Basic Concepts, Applications, and Programming.LEA, ISBN 0-8058-4104-0
• Goldberger, A. S. (1972). Structural equation models in the social sciences. Econometrica 40, 979- 1001.
• Haavelmo, Trygve (January 1943). "The Statistical Implications of a System of Simultaneous Equations". Econometrica. 11 (1): 1–12. doi:10.2307/1905714. JSTOR 1905714.
• Hoyle, R H (ed) (1995) Structural Equation Modeling: Concepts, Issues, and Applications. SAGE, ISBN 0-8039-5318-6
• Jöreskog, Karl G.; Yang, Fan (1996). "Non-linear structural equation models: The Kenny-Judd model with interaction effects". In Marcoulides, George A.; Schumacker, Randall E. (eds.). Advanced structural equation modeling: Concepts, issues, and applications. Thousand Oaks, CA: Sage Publications. pp. 57–88. ISBN 978-1-317-84380-1.
• Lewis-Beck, Michael; Bryman, Alan E.; Bryman, Emeritus Professor Alan; Liao, Tim Futing (2004). "Structural Equation Modeling". The SAGE Encyclopedia of Social Science Research Methods. doi:10.4135/9781412950589.n979. hdl:2022/21973. ISBN 978-0-7619-2363-3.
• Schermelleh-Engel, K.; Moosbrugger, H.; Müller, H. (2003), "Evaluating the fit of structural equation models" (PDF), Methods of Psychological Research, 8 (2): 23–74.

External links

• Structural equation modeling page under David Garson's StatNotes, NCSU
• Issues and Opinion on Structural Equation Modeling, SEM in IS Research
• The causal interpretation of structural equations (or SEM survival kit) by Judea Pearl 2000.
• Structural Equation Modeling Reference List by Jason Newsom: journal articles and book chapters on structural equation models
• Handbook of Management Scales, a collection of previously used multi-item scales to measure constructs for SEM

• Graphical models
• Latent variable models
• Regression models
• Structural equation models