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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^{[clarification needed]} of SEM was articulated by the geneticist Sewall Wright (1921) ^[1], the economist Trygve Haavelmo (1943) and the cognitive scientist Herbert Simon (1953) ^[2], and formally defined by Judea Pearl (2000) using a calculus of counterfactuals.^[3]

Structural Equation Models (SEM) allow both confirmatory and exploratory modeling, meaning they are suited to both theory testing and theory development. Confirmatory modeling usually starts out with a hypothesis that gets represented in a causal model. The concepts used in the model must then be operationalized to allow testing of the relationships between the concepts in the model. The model is tested against the obtained measurement data to determine how well the model fits the data. The causal assumptions embedded in the model often have falsifiable implications which can be tested against the data. ^[4]

With an initial theory SEM can be used inductively by specifying a corresponding model and using data to estimate the values of free parameters. Often the initial hypothesis requires adjustment in light of model evidence. When SEM is used purely for exploration, this is usually in the context of exploratory factor analysis as in psychometric design.

Among the strengths of SEM is the ability to construct latent variables: variables which are not measured directly, but are estimated in the model from several measured variables each of which is predicted to 'tap into' the latent variables. This allows the modeler to explicitly capture the unreliability of measurement in the model, which in theory allows the structural relations between latent variables to be accurately estimated. Factor analysis, path analysis and regression all represent special cases of SEM.

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 Equation Models is partial least squares path modeling, which some feel is better suited to data exploration ^{[citation needed]}. 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. When building the correct model, the modeller uses two different kinds of variables, namely exogenous and endogenous variables. The distinction between these two types of variables is whether the variable regresses on another variable or not. As in regression the dependent variable (DV) regresses on the independent variable (IV), meaning that the DV is being predicted by the IV. In SEM terminology, other variables regress on exogenous variables. Exogenous variables can be recognized in a graphical version of the model, as the variables sending out arrowheads, denoting which variable it is predicting. A variable that regresses on a variable is always an endogenous variable, even if this same variable is also used as a variable to be regressed on. Endogenous variables are recognized as the receivers of an arrowhead in the model.

It is important to note that SEM is more general than regression. In particular a variable can act as both independent and dependent variable.

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 an SEM that only has the structural part.

In specifying pathways in a model, the modeler can posit two types of relationships: (1) free pathways, in which hypothesised 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 modeller 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 modeller account for the theoretical reasons for building the model as it is, but the modeller 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, 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 a 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 of a fit criterion as provided by 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.

Assessment of fit

Assessment of fit is a basic task in SEM modeling: forming the basis for accepting or rejecting models and, more usually, accepting one competing model over another. The output of SEM programs includes matrices of the estimated relationships between variables in the model. Assessment of fit essentially calculates how similar the predicted data are to matrices containing the relationships in the actual data.

Formal statistical tests and fit indices have been developed for these 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.

Of course as in all statistical hypothesis tests, SEM model tests are based on the assumption that the correct and complete relevant data have been modeled. In the SEM literature, discussion of fit has led to a variety of different recommendations on the precise application of the various fit indices and hypothesis tests.

Measures of fit differ in several ways. Traditional approaches to modeling start from a null hypothesis, rewarding more parsimonious models (i.e. those with fewer free parameters), to others such as AIC that focus on how little the fitted values deviate from a saturated model ^{[citation needed]} (i.e. how well they reproduce the measured values), taking into account the number of free parameters used. Because different measures of fit capture different elements of the fit of the model, it is appropriate to report a selection of different fit measures.

Some of the more commonly used measures of fit include:

Chi-Square A fundamental measure of fit used in the calculation of many other fit measures. Conceptually it is a function of the sample size and the difference between the observed covariance matrix and the model covariance matrix.
Akaike information criterion (AIC)
- A test of relative model fit: The preferred model is the one with the lowest AIC value.
- ${\mathit {AIC}}=2k-2\ln(L)\,$
- 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)^{[clarification needed (short description needed)]}
Standardized Root Mean Residual (SRMR)^{[clarification needed (short description needed)]}
Comparative Fit Index (CFI)^{[clarification needed (short description needed)]}

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 (very large samples make the Chi-square test overly sensitive, for instance), the ratio of indicators to factors, and the overall complexity of the model.

Model modification

The model may need to be modified in order to improve the fit, thereby estimating the most likely relationships between variables. Many programs provide modification indices which report the improvement in fit that results from adding an additional path to the model. Modifications that improve model fit are then flagged as potential changes that can be made to the model. In addition to improvements in model fit, it is important that the modifications also make theoretical sense.

Interpretation and communication

The model is then interpreted so that claims about the constructs can be 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 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. Good fit by a model consistent with one causal hypothesis invariably entails equally good fit by another model consistent with an opposing causal hypothesis. No research design, no matter how clever, can help distinguish such rival hypotheses, save for interventional experiments ^[3].

As in any science, subsequent replication and perhaps modification will proceed from the initial finding.

Comparison to other methods

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

Advanced uses

Invariance
Multiple group comparison: This is a technique for assessing whether certain aspects of a structural equation model or confirmatory factor analysis are the same across groups (e.g., gender, different cultures, test forms written in different languages, etc).
Latent growth modeling
Relations to other types of advanced models (hierarchical/multilevel models; item response theory models)
Mixture model (latent class) SEM
Alternative estimation and testing techniques
Robust inference
Interface with survey estimation
Multi-method multi-trait models

References

^ Wright, Sewall S. (1921). "Correlation and causation". Journal of Agricultural Research. 20: 557–85.
^ 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
^ ^a ^b Pearl, Judea (2000). Causality: Models, Reasoning, and Inference. Cambridge University Press. ISBN 0 521 77362 8.
^ Bollen, K A, and Long, S J (1993) Testing Structural Equation Models. SAGE Focus Edition, vol. 154, ISBN 0-8039-4507-8

External links

OpenMx homepage of the open-source, cross-platform software for Structural Equation Modeling.
NEUSREL homepage of the PLS-type software, that introduces new exploratory features.
sem package for R
SEMNET, the main mailing list
Structural equation modeling page under David Garson's StatNotes, NCSU
Issues and Opinion on Structural Equation Modeling, SEM in IS Research
Academic SEM-Workshops in Europe
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
Ed Rigdon's Structural Equation Modeling Page: people, software and sites
GLLAMM: generalized linear, latent and mixed models, a flexible SEM-type modeling package for Stata
Path Analysis in AFNI: The open source (GPL) AFNI package contains SEM code

[1] Wright, Sewall S. (1921). "Correlation and causation". Journal of Agricultural Research. 20: 557–85.

[2] 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

[Pearl-3] Pearl, Judea (2000). Causality: Models, Reasoning, and Inference. Cambridge University Press. ISBN 0 521 77362 8.

[4] Bollen, K A, and Long, S J (1993) Testing Structural Equation Models. SAGE Focus Edition, vol. 154, ISBN 0-8039-4507-8

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