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Structural equation modeling

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Structural equation modeling (SEM) is a statistical technique for building and testing statistical model, which are often causal models. It is a hybrid technique that encompasses aspects of confirmatory factor analysis, path analysis and regression. All of these can be seen as special cases of SEM.

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

SEM encourages a confirmatory, as opposed to exploratory, approach to modelling. In other words, it is normal to start with a hypothesis, specify a model that reflects it and then begin operationalising the constructs of interest with a measurement instrument and test the model. Often the initial hypothesis requires adjustment in light of model evidence, but it is rare to see SEM used in a purely for exploration.

With an accepted theory or otherwise confirmed model, one can also use SEM inductively by specifying a model and using data to estimate the values of free parameters.

SEM is an extension of the general linear model that simultaneously estimates relationships between multiple independent, dependent and latent variables. SEM is best applied to theory testing, as opposed to the more exploratory areas of theory development.

TETRAD and partial least squares are alternatives to SEM for exploratory modeling.

Basic concepts

Basic steps in performing SEM analysis

  1. Model specification—Since SEM is a confirmatory technique, it is imperative that the model is 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 latent 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).
  2. Estimation of free parameters—parameter estimation is made comparing the actual variance/covariance matrices representing the relationships between variables and the estimated variance/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 AMOS, EQS, LISREL, Mplus, Mx, or SAS PROC CALIS.
  3. Assessment of fit— Using an SEM analysis program, one can compare the estimated matrices representing the relationships between variables in the model to the actual matrices. Individual factors within the model are also examined within the estimated model in order to see how well the proposed model fits the driving theory.
  4. Model modification— The model may need to be modified in order to maximize the fit, thereby estimating the most likely relationships between variables.
  5. Interpretation and communication—The model is then interpreted and claims about the constructs are made based on the best fitting model. Because SEM is limited to correlational data, caution should always be taken when making claims of causality unless further experimentation or time-ordered studies have been done.
  6. Replication and revalidation — All model modifications should be replicated and revalidated before interpreting and communicating the results.

Advanced Uses of SEM

  • 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

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 034069243X
  • Bollen, K A (1989). Structural Equations with Latent Variables. Wiley, ISBN 0471011711
  • Bollen, K A, and Long, S J (1993) Testing Structural Equation Models. SAGE Focus Edition, vol. 154, ISBN 0803945078
  • Byrne, B. M. (2001) Structural Equation Modeling with AMOS - Basic Concepts, Applications, and Programming.LEA, ISBN 0805841040
  • Hoyle, R H (ed) (1995) Structural Equation Modeling: Concepts, Issues, and Applications. SAGE, ISBN 0803953186
  • Kaplan, D (2000) Structural Equation Modeling: Foundations and Extensions. SAGE, Advanced Quantitative Techniques in the Social Sciences series, vol. 10, ISBN 0761914072
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