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Structural equation modeling (SEM) is a statistical technique for testing and estimating causal relationships using a combination of statistical data and qualitative causal assumptions. This view of SEM was articulated by the geneticist Sewal Wright (1921), the economists Trygve Haavelmo (1943) and Herbert Simon (1953), and formally defined by Judea Pearl (2000) using a calculus of counterfactuals.

SEM encourages confirmatory rather than exploratory modeling; thus, it is suited to theory testing rather than theory development. It usually starts with a hypothesis, represents it as a model, operationalises the constructs of interest with a measurement instrument, and tests the model. The causal assumptions embedded in the model often have falsifiable implications which can be tested against the data. With an accepted theory or otherwise confirmed model, SEM can also be used inductively by specifying the model and using data to estimate the values of free parameters. Often the initial hypothesis requires adjustment in light of model evidence, but SEM is rarely used purely for exploration.

Among its strengths is the ability to model constructs as latent variables (variables which are not measured directly, but are estimated in the model from measured variables which are assumed to 'tap into' the latent variables). This allows the modeller to explicitly capture the unreliability of measurement in the model, which in theory allows the structural relations between latent variables to be accurately estimated.

SEM should be distinguished from regression models, which are purely predictive tools, making no empirical claims whatsoever. In SEM, the qualitative causal assumptions are represented by the missing variables in each equation, as well as vanishing covariances among some error terms. These assumptions are testable in experimental studies and must be confirmed judgmentally in observational studies.

An alternative technique for specifying Structural Models using partial least squares has been implemented in software such as LVPLS (Latent Variable Partial Least Square), PLSGraph and SmartPLS (Ringle et al. 2005). Some feel this is better suited to data exploration. More ambitiously, The TETRAD project aims to develop a way to automate the search for possible causal models from data.

Steps in performing SEM analysis

Model specification

Since SEM is a confirmatory technique, the model must be specified correctly based on the type of analysis that the modeller is attempting to confirm. There are usually two main parts to SEM: the structural model showing dependencies between endogenous and exogeneous variables, and the measurement model showing the relations between the latent variables and their indicators. Confirmatory factor analysis models, for example, contain only the measurement part; while linear regression can be viewed as an SEM that only has the structural part. Specifying the model delineates relationships between variables that are thought to be related (and therefore want to be 'free' to vary) and those relationships between variables that already have an estimated relationship, which can be gathered from previous studies (these relationships are 'fixed' in the model).

Estimation of free parameters

Parameter estimation is done 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 of a fit criterion as provided by maximum likelihood, weighted least squares or asymptotically distribution-free methods.

This is best accomplished by using a specialized SEM analysis program, such as SPSS' AMOS, EQS, LISREL, Mplus, Mx, the sem package in R, or SAS PROC CALIS. More information about SAS PROC CALIS:

• at UCLA
• at UCR.

Assessment of fit

Using a SEM analysis program, one can compare the estimated matrices representing the relationships between variables in the model to the actual matrices. Formal statistical tests and fit indices have been developed for this purposes. Individual parameters of the model can also be examined within the estimated model in order to see how well the proposed model fits the driving theory. Most, though not all, estimation methods make such tests of the model possible.

However, the model tests are only correct provided that the model is correct. Although this problem exists in all statistical hypothesis tests, its existence in SEM has led to a large body of discussion among SEM experts, leading to a large variety of different recommendations on the precise application of the various fit indices and hypothesis tests.

Model modification

The model may need to be modified in order to maximize the fit, thereby estimating the most likely relationships between variables.

Interpretation and communication

The model is then interpreted and claims about the constructs are made based on the best fitting model.

Caution should always be taken when making claims of causality even when experimentation or time-ordered studies have been done. The term causal model can be misleading because SEM is most commonly used with data collected at one time point through passive observation. 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. Good fit by a model consistent with one causal hypothesis does not rule out equally good fit by another model consistent with a different causal hypothesis. However careful research design can help distinguish such rival hypotheses.

Replication and revalidation

All model modifications should be replicated and revalidated before interpreting and communicating the results.

Comparison to other methods

In machine learning, SEM may be viewed as a generalization of Linear-Gaussian Bayesian networks which drops the acyclicity constraint, i.e. which allows causal cycles.

Advanced uses

• Invariance
• Multiple group comparison
• Modeling growth
• Relations to other types of advanced models (multilevel models; item response theory models)
• Alternative estimation and testing techniques
• Robust inference
• Interface with survey estimation

See also

• List of publications in statistics
• List of statistical topics
• List of statisticians
• Multivariate statistics
• Misuse of statistics
• Regression analysis

External links

• A special issue of the journal Personality and Individual Differences that focuses specifically on Structural Equation Modeling
• EQS homepage at Multivariate Software
• GNU PSPP - a free software program designed as a replacement for SPSS
• Lisrel Homepage
• MPLUS Homepage
• Mx homepage of the cross-platform Mx software for Structural Equation Modeling.
• sem package for R.
• SEMNET, the main mailing list
• SPSS Inc Homepage
• Structural Equation Modeling: An Multidisciplinary Journal pulished by LEA/Taylor & Francis Group
• Structural equation modeling page under David Garson's StatNotes, NCSU
• student version of AMOS software for Structural Equation Modeling.
• Issues and Opinion on Structural Equation Modeling, SEM in IS Research
• What is Structural Equation Modeling, Ed Rigon, SEM FAQ

References

Books

• Bartholomew, D J, and Knott, M (1999) Latent Variable Models and Factor Analysis Kendall's Library of Statistics, vol. 7. Arnold publishers, ISBN 0-340-69243-X
• Bollen, K A (1989). Structural Equations with Latent Variables. Wiley, ISBN 0-471-01171-1
• Bollen, K A, and Long, S J (1993) Testing Structural Equation Models. SAGE Focus Edition, vol. 154, ISBN 0-8039-4507-8
• Byrne, B. M. (2001) Structural Equation Modeling with AMOS - Basic Concepts, Applications, and Programming.LEA, ISBN 0-8058-4104-0
• Haavelmo, T. (1943) "The statistical implications of a system of simultaneous equations," Econometrica 11:1-2. Reprinted in D.F. Hendry and M.S. Morgan (Eds.), The Foundations of Econometric Analysis, Cambridge University Press, 477--490, 1995.
• Hoyle, R H (ed) (1995) Structural Equation Modeling: Concepts, Issues, and Applications. SAGE, ISBN 0-8039-5318-6
• Kaplan, D (2000) Structural Equation Modeling: Foundations and Extensions. SAGE, Advanced Quantitative Techniques in the Social Sciences series, vol. 10, ISBN 0-7619-1407-2
• Kline, R. B. (2005) Principles and Practice of Structural Equation Modeling. The Guilford Press, ISBN 1-57230-690-4
• Pearl, Judea (1988). Probabilistic Reasoning in Intelligent Systems. Morgan Kaufmann. ISBN 0-934613-73-7.
• Simon, Herbert (1953), "Causal ordering and identifiability", in Hood, W.C.; Koopmans, T.C. (eds.), Studies in Econometric Method, New York: Wiley, pp. 49–74
• Wright, Sewal S. (1921). "Correlation of causation". Journal of Agricultural Research. 20: 557–85.

Software

• Ringle, C. M./Wende, S./Will, A. (2005) SmartPLS 2.0 Beta. Hamburg, http://www.smartpls.de

• Statistics