 Structural equation modeling

Structural equation modeling (SEM) is a statistical technique for testing and estimating causal relations using a combination of statistical data and qualitative causal assumptions. This definition 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.
Contents
Steps in performing SEM analysis
Model specification
When SEM is used as a confirmatory technique, the model must be specified correctly based on the type of analysis that the researcher is attempting to confirm. When building the correct model, the researcher 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 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, 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 of a fit criterion as provided by maximum likelihood estimation, weighted least squares or asymptotically distributionfree 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:
 ChiSquared 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.
 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)
 Another test of model fit, good models are considered to have a RMSEA of .05 or less. Models whose RMSEA is .1 or more have a poor fit.
 Standardized Root Mean Residual (SRMR)
 The SRMR is a popular absolute fit indicator. A good model should have an SRMR smaller than .05.
 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.
For each measure of fit, a decision as to what represents a goodenough fit between the model and the data must reflect other contextual factors such as sample size (very large samples make the Chisquared test overly sensitive, for instance^{[citation needed]}), 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.
Sample size
Where the proposed SEM is the basis for a research hypothesis, ad hoc rules of thumb requiring the choosing of 10 observations per indicator in setting a lower bound for the adequacy of sample sizes have been widely used since their original articulation by Nunnally(1967).^{[5]} Being linear in model constructs, these are easy to compute, but have been found to result in sample sizes that are too small. One study found that sample sizes in a particular stream of SEM literature averaged only 50% of the minimum needed to draw the conclusions the studies claimed. Overall, 80% of the research articles in the study drew conclusions from insufficient samples.^{[6]} Complexities which increase information demands in structural model estimation increase with the number of potential combinations of latent variables; while the information supplied for estimation increases with the number of measured parameters times the number of observations in the sample size – both are nonlinear. Sample size in SEM can be computed through two methods: the first as a function of the ratio of indicator variables to latent variables, and the second as a function of minimum effect, power and significance. Software and methods for computing both have been developed by Westland (2010).^{[6]}
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 timeordered 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 quasiexperimental 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.
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
 Multimethod multitrait models
See also
 List of publications in statistics
 List of statistical topics
 List of statisticians
 Multivariate statistics
 Misuse of statistics
 Partial least squares regression
 Regression analysis
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.. 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 0803945078
 ^ Nunnally, J. C. (1967). "Psychometric Theory". McGrawHill, New York: 355.
 ^ ^{a} ^{b} Westland, J. Christopher (2010). "Lower bounds on sample size in structural equation modeling". Electron. Comm. Res. Appl. 9 (6): 476–487. doi:10.1016/j.elerap.2010.07.003.
Further reading
 Bartholomew, D J, and Knott, M (1999) Latent Variable Models and Factor Analysis Kendall's Library of Statistics, vol. 7. Arnold publishers, ISBN 034069243X
 Bentler, P.M. & Bonett, D.G. (1980). Significance tests and goodness of fit in the analysis of covariance structures. Psychological Bulletin, 88, 588606.
 Bollen, K A (1989). Structural Equations with Latent Variables. Wiley, ISBN 0471011711
 Byrne, B. M. (2001) Structural Equation Modeling with AMOS  Basic Concepts, Applications, and Programming.LEA, ISBN 0805841040
 Goldberger, A. S. (1972). Structural equation models in the social sciences. Econometrica 40, 979 1001.
 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 0803953186
 Kaplan, D (2000) Structural Equation Modeling: Foundations and Extensions. SAGE, Advanced Quantitative Techniques in the Social Sciences series, vol. 10, ISBN 0761914072
 Kline, R. B. (2010) Principles and Practice of Structural Equation Modeling (3rd Edition). The Guilford Press, ISBN 9781606238776
 Jöreskog, K. and F. Yang (1996). Nonlinear structural equation models: The KennyJudd model with interaction effects. In G. Marcoulides and R. Schumacker, (eds.), Advanced structural equation modeling: Concepts, issues, and applications. Thousand Oaks, CA: Sage Publications.
SEMspecific software
 SEPATH in STATISTICA (Electronic Statistics Textbook)
 LISREL
 Mplus
 Packages in R:
 The GLLAMM package for Stata
 EQS
 Procedures in SAS (software)
 CALIS
 TCALIS
 AMOS in SPSS
External links
 NEUSREL homepage of the PLStype software, that introduces new exploratory features.
 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 SEMWorkshops 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
 Path Analysis in AFNI: The open source (GPL) AFNI package contains SEM code
Least squares and regression analysis Computational statistics Least squares · Linear least squares · Nonlinear least squares · Iteratively reweighted least squaresCorrelation and dependence Regression analysis Regression as a
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