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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) 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.

Use of SEM is commonly justified because it helps identify latent variables that are believed to exist, but cannot be directly observed (like an attitude, intelligence or mental illness). Although there are not always clear boundaries of what is and what is not SEM, it generally involves path models (see also path analysis) and measurement models (see also factor analysis) and always employs statistical models and computer programs to investigate the structural connections between latent variables underlying the actual variables taken from observed data. Researchers using SEM employ software programs to estimate the strength and sign of a coefficient for each modeled arrow (the numbers shown in Figure 1 for example), and to provide diagnostic clues suggesting which indicators or model components might produce inconsistency between the model and the data. Criticisms of SEM methods hint at mathematical formulation problems, a tendency to accept models without establishing external validity, and potential philosophical bias.

A SEM suggesting that intelligence (as measured by four questions) can predict academic performance (as measured by SAT, ACT, and high school GPA) is shown in Figure 1. The concept of human intelligence cannot be measured directly in the way that one could measure height or weight. Instead, researchers have a theory and conceptualization of intelligence and then design measurement instruments such as a questionnaire or test that provides them with multiple indicators of intelligence. These indicators are then combined in a model to create a plausible way of measuring intelligence as a latent variable (the circle for intelligence in Figure 1) from the indicators (square boxes with scale 1–4 in Figure 1). Figure 1 is presented as a final model, after running it and obtaining all estimates (the numbers on the arrows). There is no consensus on the best symbolic notation to represent SEMs, for example Figure 2 represents a similar model as Figure 1 without as many arrows and in a format that might occur prior to running the model.

A great advantage of SEM is that all of these measurements and tests occur simultaneously in one statistical estimation procedure, where the errors throughout the model are calculated using all information from the model. This means the errors are more accurate than if a researcher were to calculate each part of the model separately.

History

Structural equation modeling (SEM) has its roots in the work of Sewall Wright who applied explicit causal interpretations to regression equations based on direct and indirect effects of observed variables in population genetics. Lee M. Wolfle compiled an annotated bibliographic history of Sewall Wright's path coefficient method which we know today as path modeling. Wright added two important elements to the standard practice of using regression to predict an outcome. These were (1) to combine information from more than one regression equation using (2) a causal approach to regression modeling rather than merely predictive. Sewall Wright consolidated his method of path analysis in his 1934 article "The Method of Path Coefficients".

Otis Dudley Duncan introduced SEM to the social sciences in 1975 and it flourished throughout the 1970s and 80s. Different yet mathematically related modeling approaches developed in psychology, sociology, and economics. 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 although there is great overlap with econometric practices employing simultaneous equations and exogenous (causal variables).

One of several programs Karl Gustav Jöreskog developed in the early 1970s 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 had inherited from Wright and Duncan). The factor-structured portion of the model incorporated measurement errors and thereby permitted measurement-error-adjusted estimation of effects connecting latent variables.

Loose and confusing terminology has been used to obscure weaknesses in the methods. In particular, PLS-PA (also known as PLS-PM) has been conflated with partial least squares regression PLSR, which is a substitute for ordinary least squares regression and has nothing to do with path analysis. PLS-PA has been erroneously promoted as a method that works with small datasets when other estimation approaches fail; in fact, it has been shown that minimum required sample sizes for this method are consistent with those required in multiple regression.

Both LISREL and PLS-PA were conceived as iterative computer algorithms, with an emphasis from the start on creating an accessible graphical and data entry interface and extension of Wright's (1921) path analysis. 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.

Anderson and Rubin (1949, 1950) developed the limited information maximum likelihood estimator for the parameters of a single structural equation, which indirectly included the two-stage least squares estimator and its asymptotic distribution (Anderson, 2005) (Farebrother, 1999). Two-stage least squares was originally proposed as a method of estimating the parameters of a single structural equation in a system of linear simultaneous equations, being introduced by Theil (1953a, 1953b, 1961) and more or less independently by Basmann (1957) and Sargan (1958). Anderson's limited information maximum likelihood estimation was eventually implemented in a computer search algorithm, where it competed with other iterative SEM algorithms. Of these, two-stage least squares was by far the most widely used method in the 1960s and the early 1970s.

Systems of regression equation approaches were developed at the Cowles Commission from the 1950s on, extending the transportation modeling of Tjalling Koopmans. Sewall Wright and other statisticians attempted to promote path analysis methods at Cowles (then at the University of Chicago). University of Chicago statisticians identified many faults with path analysis applications to the social sciences; faults which did not pose significant problems for identifying gene transmission in Wright's context, but which made path methods such as PLS-PA and LISREL problematic in the social sciences. Freedman (1987) summarized these objections in path analyses: "failure to distinguish among causal assumptions, statistical implications, and policy claims has been one of the main reasons for the suspicion and confusion surrounding quantitative methods in the social sciences" (see also Wold's (1987) response). Wright's path analysis never gained a large following among US econometricians, but was successful in influencing Hermann Wold and his student Karl Jöreskog. Jöreskog's student Claes Fornell promoted LISREL in the US.

Advances in computers made it simple for novices to apply structural equation methods in the computer-intensive analysis of large datasets in complex, unstructured problems. The most popular solution techniques fall into three classes of algorithms: (1) ordinary least squares algorithms applied independently to each path, such as applied in the so-called PLS path analysis packages which estimate with OLS; (2) covariance analysis algorithms evolving from seminal work by Wold and his student Karl Jöreskog implemented in LISREL, AMOS, and EQS; and (3) simultaneous equations regression algorithms developed at the Cowles Commission by Tjalling Koopmans.

Pearl has extended SEM from linear to nonparametric models, and proposed causal and counterfactual interpretations of the equations. For example, excluding a variable Z from the arguments of an equation asserts that the dependent variable is independent of interventions on the excluded variable, once we hold constant the remaining arguments. Nonparametric SEMs permit the estimation of total, direct and indirect effects without making any commitment to the form of the equations or to the distributions of the error terms. This extends mediation analysis to systems involving categorical variables in the presence of nonlinear interactions. Bollen and Pearl survey the history of the causal interpretation of SEM and why it has become a source of confusions and controversies.

SEM path analysis methods are popular in the social sciences because of their accessibility; packaged computer programs allow researchers to obtain results without the inconvenience of understanding experimental design and control, effect and sample sizes, and numerous other factors that are part of good research design. Supporters say that this reflects a holistic, and less blatantly causal, interpretation of many real world phenomena – especially in psychology and social interaction – than may be adopted in the natural sciences; detractors suggest that many flawed conclusions have been drawn because of this lack of experimental control.

Direction in the directed network models of SEM arises from presumed cause-effect assumptions made about reality. Social interactions and artifacts are often epiphenomena – secondary phenomena that are difficult to directly link to causal factors. An example of a physiological epiphenomenon is, for example, time to complete a 100-meter sprint. A person may be able to improve their sprint speed from 12 seconds to 11 seconds, but it will be difficult to attribute that improvement to any direct causal factors, like diet, attitude, weather, etc. The 1 second improvement in sprint time is an epiphenomenon – the holistic product of interaction of many individual factors.

General approach to SEM

Although each technique in the SEM family is different, the following aspects are common to many SEM methods, as it can be summarized as a 4E framework by many SEM scholars like Alex Liu, that is 1) Equation (model or equation specification), 2) Estimation of free parameters, 3) Evaluation of models and model fit, 4) Explanation and communication, as well as execution of results.

Model specification

Two main components of models are distinguished in SEM: the structural model showing potential causal dependencies between endogenous and exogenous variables, and the measurement model showing the relations between latent variables and their indicators. Exploratory and confirmatory factor analysis models, for example, contain only the measurement part, while path diagrams can be viewed as SEMs that contain only the structural part.

In specifying pathways in a model, the modeler can posit two types of relationships: (1) free pathways, in which hypothesized causal (in fact counterfactual) relationships between variables are tested, and therefore are left 'free' to vary, and (2) relationships between variables that already have an estimated relationship, usually based on previous studies, which are 'fixed' in the model.

A modeler will often specify a set of theoretically plausible models in order to assess whether the model proposed is the best of the set of possible models. Not only must the modeler account for the theoretical reasons for building the model as it is, but the modeler must also take into account the number of data points and the number of parameters that the model must estimate to identify the model.

An identified model is a model where a specific parameter value uniquely identifies the model (recursive definition), and no other equivalent formulation can be given by a different parameter value. A data point is a variable with observed scores, like a variable containing the scores on a question or the number of times respondents buy a car. The parameter is the value of interest, which might be a regression coefficient between the exogenous and the endogenous variable or the factor loading (regression coefficient between an indicator and its factor). If there are fewer data points than the number of estimated parameters, the resulting model is "unidentified", since there are too few reference points to account for all the variance in the model. The solution is to constrain one of the paths to zero, which means that it is no longer part of the model.

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 must reflect other contextual factors such as sample size, the ratio of indicators to factors, and the overall complexity of the model. For example, very large samples make the Chi-squared test overly sensitive and more likely to indicate a lack of model-data 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