SEM Stas Kolenikov U of Missouri Structural Equation Modeling Introduction Structural Using gllamm , confa and gmm equation models Formulation Path diagrams Identification Estimation Stas Kolenikov Stata tools for SEM Department of Statistics sem gllamm University of Missouri-Columbia confa The World Bank, Washington, DC gmm NHANES Joint work with Kenneth Bollen (UNC) daily functioning Ecology July 1, 2011 example: observed variables References
SEM Goals of the talk Stas Kolenikov U of Missouri 1 Introduce structural equation models Introduction 2 Describe Stata packages to fit them: Structural • confa : a 13mm hex wrench equation models • gllamm : a Swiss-army tomahawk Formulation • gmm : do-it-yourself kit Path diagrams Identification • sem : the promised land? Estimation Stata tools for 3 Example 1: daily functioning in NHANES SEM sem 4 Example 2: experimental ecology data set gllamm confa gmm NHANES daily functioning Ecology example: observed variables References
SEM First, some theory Stas Kolenikov U of Missouri 1 Introduction Introduction Structural equation models 1 Structural equation Formulation models Path diagrams Formulation Path diagrams Identification Identification Estimation Estimation Stata tools for SEM 2 Stata tools for SEM sem gllamm sem confa gmm gllamm NHANES daily confa functioning gmm Ecology example: observed NHANES daily functioning 3 variables References Ecology example: observed variables 4 References 5
SEM Structural equation modeling Stas Kolenikov U of Missouri (SEM) Introduction • Standard multivariate technique in social sciences Structural • Incorporates constructs that cannot be directly equation models observed: Formulation Path diagrams • psychology: level of stress Identification Estimation • sociology: quality of democratic institutions Stata tools for • biology: genotype and environment SEM • health: difficulty in personal functioning sem gllamm confa • Special cases: gmm • linear regression NHANES daily • confirmatory factor analysis functioning • simultaneous equations Ecology example: • errors-in-variables and instrumental variables observed regression variables References
SEM Origins of SEM Stas Kolenikov U of Missouri Path analysis of Sewall Wright (1918) Introduction Structural ⊗ equation models Formulation Causal modeling of Hubert Blalock (1961) Path diagrams Identification Estimation Stata tools for ⊗ SEM sem gllamm Factor analysis estimation of Karl J¨ oreskog (1969) confa gmm NHANES daily ⊗ functioning Ecology Econometric simultaneous equations of Arthur Goldberger example: observed (1972) variables References
SEM Structural equations model Stas Kolenikov U of Missouri Latent variables: Introduction η = α η + B η + Γ ξ + ζ (1) Structural equation models Formulation Measurement model for observed variables: Path diagrams Identification Estimation y = α y + Λ y η + ε (2) Stata tools for SEM x = α x + Λ x ξ + δ (3) sem gllamm confa ξ , ζ , ε , δ are uncorrelated with one another gmm NHANES daily functioning J¨ oreskog (1973), Bollen (1989), Yuan & Bentler (2007) Ecology example: Other re-expressions: Bentler & Weeks (1980), McArdle & observed variables McDonald (1984). References
SEM Implied moments Stas Kolenikov U of Missouri Denoting Introduction V [ ξ ] = Φ , V [ ζ ] = Ψ , V [ ε ] = Θ ε , V [ δ ] = Θ δ , Structural equation � x � models R = Λ y ( I − B ) − 1 , Formulation z = Path diagrams y Identification Estimation Stata tools for obtain SEM � α y + Λ y R µ ξ � sem � � gllamm µ ( θ ) ≡ E = (4) confa z α x + Λ x µ ξ gmm � Λ x Φ Λ ′ � NHANES � � x + Θ δ Λ x ΦΓ ′ R ′ daily Σ( θ ) ≡ V = (5) functioning z R (ΓΦΓ ′ + Ψ) R ′ + Θ ε R ΓΦ Λ ′ x Ecology example: observed variables References
SEM Path diagrams Stas Kolenikov U of Missouri φ 12 � φ 22 � z 1 � φ 11 � Introduction Structural ξ 1 β 12 equation y 3 ǫ 3 � θ 6 � models λ 6 β 11 Formulation Path diagrams Identification 1 λ 3 λ 5 λ 2 Estimation η 1 y 2 ǫ 2 � θ 5 � Stata tools for SEM x 1 x 2 x 3 sem 1 y 1 gllamm ǫ 1 � θ 4 � � θ 4 � confa gmm NHANES δ 1 δ 2 δ 3 ζ 1 daily � θ 1 � � θ 2 � � θ 3 � � σ 1 � functioning Ecology example: observed variables References
SEM Identification Stas Kolenikov U of Missouri Before proceeding to estimation, the researcher needs to Introduction verify that the SEM is identified : Structural equation models Formulation Pr { X : f ( X , θ ) = f ( X , θ ′ ) ⇒ θ = θ ′ } = 1 I Path diagrams Identification Estimation Stata tools for Different parameter values should give rise to different SEM sem likelihoods/objective functions, either globally, or locally in a gllamm confa neighborhood of a point in a parameter space. gmm NHANES daily functioning Ecology example: observed variables References
SEM Likelihood Stas Kolenikov U of Missouri • Normal data ⇒ likelihood is the function of sufficient Introduction statistic (¯ z , S ) : Structural � � equation + n tr [Σ − 1 ( θ ) S ] − 2 log L ( θ, Y , X ) ∼ n ln det Σ( θ ) models Formulation z − µ ( θ )) ′ Σ − 1 ( θ )(¯ Path diagrams + n (¯ z − µ ( θ )) → min (6) Identification θ Estimation Stata tools for • Generalized latent variable approach for mixed SEM sem response (normal, binomial, Poisson, ordinal, within the gllamm confa same model): gmm NHANES � daily n � functioning − 2 log L ( θ, Y , X ) ∼ f ( y i , x i | ξ, ζ ; θ ) d F ( ξ, ζ | θ ) (7) ln Ecology example: i = 1 observed variables Bartholomew & Knott (1999), Skrondal & References Rabe-Hesketh (2004)
SEM Estimation methods Stas Kolenikov U of Missouri • Normal theory MLE Introduction • Weighted least squares: Structural equation models s = vech S , σ ( θ ) = vech Σ( θ ) Formulation Path diagrams F = ( s − σ ( θ )) ′ V n ( s − σ ( θ )) → min (8) Identification θ Estimation Stata tools for where V n is weighting matrix: SEM sem V ( 1 ) • Optimal ˆ = ˆ gllamm V [ s − σ ( θ )] (Browne 1984) n confa • Simplistic: least squares V ( 2 ) gmm = I n NHANES V ( 3 ) • Diagonally weighted least squares: ˆ = diag ˆ V [ s − σ ] daily n functioning • Model-implied instrumental variables limited information Ecology estimator (Bollen 1996) example: observed variables • Bounded influence/outlier-robust methods (Yuan, References Bentler & Chan 2004, Moustaki & Victoria-Feser 2006) • Empirical likelihood
SEM Goodness of fit Stas Kolenikov U of Missouri • The estimated model Σ(ˆ θ ) is often related to the Introduction “saturated” model Σ ≡ S and/or independence model Structural Σ 0 = diag S equation models • Likelihood formulation ⇒ LRT test, asymptotically χ 2 Formulation k • Non-normal data: LRT statistic ∼ � Path diagrams j w j χ 2 1 , can be Identification Estimation Satterthwaite-adjusted towards the mean and variance Stata tools for of the appropriate χ 2 SEM k (Satorra & Bentler 1994, Yuan & sem Bentler 1997) gllamm confa • Analogies with regression R 2 attempted, about three gmm NHANES dozen fit indices available (Marsh, Balla & Hau 1996) daily functioning • Reliability of indicators: R 2 in regression of an indicator Ecology example: on its latent variable observed variables • Signs and magnitudes of coefficient estimates References
SEM Now, some tools Stas Kolenikov U of Missouri 1 Introduction Introduction Structural equation models 1 Structural equation Formulation models Path diagrams Formulation Path diagrams Identification Identification Estimation Estimation Stata tools for SEM 2 Stata tools for SEM sem gllamm sem confa gmm gllamm NHANES daily confa functioning gmm Ecology example: observed NHANES daily functioning 3 variables References Ecology example: observed variables 4 References 5
SEM sem Stas Kolenikov U of Missouri Introduction sem ? Structural equation models Formulation As announced earlier this week, Stata 12 will be released Path diagrams Identification on 25 July 2011 and will have a full-fledge sem estimation Estimation Stata tools for routine. SEM sem gllamm confa gmm NHANES daily functioning Ecology example: observed variables References
SEM sem Stas Kolenikov U of Missouri Introduction Structural equation models Formulation Path diagrams Identification Estimation Stata tools for SEM sem gllamm confa gmm NHANES daily functioning Ecology example: observed variables References
SEM sem Stas Kolenikov U of Missouri Introduction Structural equation models Formulation Path diagrams Identification Estimation Stata tools for SEM sem gllamm confa gmm NHANES daily functioning Ecology example: observed variables References
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