= Structural Equation Modeling = '''Structural equation modeling''' ('''SEM''') is a modeling framework that makes use of multiple prediction equations. It is also also known as '''covariance structure analysis''', '''analysis of moment structures''', or '''analysis of linear structural relationships'''. <> ---- == Description == SEM is used for measurement error adjustment. The first component is the '''measurement model''', which is essentially a [[Statistics/FactorAnalysis|CFA]]. Terminology is also common between the two, e.g. '''factors''', '''factor loadings''', '''indicators''', and so on. The most important distinction is that a causal direction is assumed; note the arrows below: {{attachment:path.png}} This is equivalent to a formulation like: * ''X = α,,1,, + β,,1,,x,,1,, + e,,1,,'' * ''X = α,,2,, + β,,2,,x,,2,, + e,,2,,'' * ''X = α,,3,, + β,,3,,x,,3,, + e,,3,,'' The second component is the '''structural model''' which specifies the [[Statistics/MediationAnalysis|mediation]] relationship of the latent factors. If a factor is predicted by other variables in the system, it is '''endogenous'''; otherwise it is '''exogenous'''. The simplest formulation of a structural model might be ''Y = α,,Y,, + β,,Y,,X + e,,Y,,'', but... * there can be multiple predictive factors, e.g. `Y ~ X + Z` * the outcome can itself have a measurement model, e.g. `Y ~= y1 + y2 + y3` Most model estimation strategies require assuming: 1. a uniform variance for the outcome construct's [[Statistics/Residuals|residual]], i.e. ''e,,Y,,''. 2. a uniform variance for the latent constructs, e.g. ''X''. See [[https://www.youtube.com/watch?v=NOWdrfQVWAI&t=2045s]] for an demonstration of why assumptions are required, and how many are needed. All variables are assumed to be [[Analysis/NormalDistribution|jointly normal]]. Failures of this assumption are sometimes addressed through deleting outlier observations or transforming variables. ---- == Reading Notes == * [[AsymptoticallyDistributionFreeMethodsForTheAnalysisOfCovarianceStructures|Asymptotically distribution-free methods for the analysis of covariance structures]], M. W. Browne, 1984 * [[AComparisonOfMethodologiesForTheFactorAnalysisOfNonnormalLikertVariables|A comparison of some methodologies for the factor analysis of non‐normal Likert variables]]; Bengt O. Muthén and David E. Kaplan; 1985 * [[LatentVariableModelingInHeterogeneousPopulations|Latent Variable Modeling in Heterogeneous Populations]], Bengt O. Muthén, 1989 * [[MultilevelCovarianceStructureAnalysis|Multilevel Covariance Structure Analysis]], Bengt O. Muthén, 1994 * [[CorrectionsToTestStatisticsAndStandardErrorsInCovarianceStructureAnalysis|Corrections to Test Statistics and Standard Errors in Covariance Structure Analysis]], Albert Satorra and Peter M. Bentler, 1994 * [[StructuralEquationModelingWithRobustCovariances|Structural equation modeling with robust covariances]], Ke-Hai Yuan and Peter M. Bentler, 1998 * [[AComparativeReviewOfInteractionAndNonlinearModeling|A Comparative Review of Interaction and Nonlinear Modeling]]; Edward E. Rigdon, Randall E. Schumacker, and Werner Wothke; 1998 * [[RobustTransformationWithApplicationsToStructuralEquationModeling|Robust transformation with applications to structural equation modelling]]; Ke-Hai Yuan, Wai Chan, and Peter M. Bentler; 2000 * [[NonnormalityOfDataInStructuralEquationModels|Non-normality of Data in Structural Equation Models]]; Shengyi Gao, Patricia L. Mokhtarian, and Robert A. Johnston; 2008 * [[GeneralRandomEffectLatentVariableModeling|General Random Effect Latent Variable Modeling: Random Subjects, Items, Contexts, and Parameters]], Tihomir Asparouhov and Bengt Muthén, 2014 * [[ACloserLookAtRandomAndFixedEffectsPanelRegressionInStructuralEquationModelingUsingLavaan|A closer look at random and fixed effects panel regression in structural equation modeling using lavaan]], Henrik Kenneth Andersen, 2021 ---- CategoryRicottone