Introduction to Latent Variable Models A comparison of
Introduction to Latent Variable Models
A comparison of models Model A Model B ξ 1 Y 1 X 2 X 3 X 1 X 2 X 3 δ 1 δ 2 δ 3
The Fundamental Hypothesis of SEM = ( ) Population = Implied Where is the variance-covariance matrix of the entire model and Where is a vector (list) of elements that are matrices: Λ Θδ Θε Φ Γ Β Ψ
Implied Covariance Matrix: Observed Model n For an observed model, the implied matrix is the relationships among all the x and y variables X X xx Y xy Y yx yy It can be decomposed into three pieces: – – – the covariance matrix of y The covariance matrix of x with y
Model A: Observed model Model A Y Y 1 X 1 δ 1 X 2 δ 2 X 3 δ 3 X 1 X 2 X 3 Y YY X 1 YX 1 X 1 X 1 X 2 YX 2 X 1 X 2 X 2 X 2 X 3 YX 3 X 1 X 3 X 2 X 3 X 3 X 3
Covariance Matrix of Y Σyy(Θ) = E(yy’) = (I – -1 B) (ΓΦΓ’ + Ψ) (I – -1’ B)
Covariance Matrix of X Σxx(Θ) = E(xx’) = Φ
Covariance Matrix of XY Σxy(Θ) = E(xy’) = -1 ΦΓ’(I – B)
Put that all together and get: (I – B)-1(ΓΦΓ’ + Ψ) (I – B)-1’ ( ) = ΦΓ’(I – B)-1 Φ
Population vs. Implied Covariance Matrices in Model A
So, the matrices for Model A are: Elements of Θ = Λ Θδ Θε Φ Γ Β Ψ Β=0 Θε = 0 Φ=0 Γ=0 λ 1 n λ 2 Λ= λ 3 Φ = φy Ψ= δ 12 δ 13 0? δ 23 0? 0? δ 3 Θδ = 0
Identification n 4 variables = (4)(5)/2 = 10 n There are 10 parameters we could estimate: – 3 λ (the path coefficients) – 1 ψ (error variance of Y) – 3 δ (error variances of each X) – 3 δ (Covariances among the 3 X errors)
Model A: Observed model Y 1 ζ Model A λ 1 X 1 δ 1 λ 2 λ 3 X 2 X 3 δ 2 δ 3
Covariance Matrix X 1 X 2 X 3 Y 1 X 1 2. 062 0. 783 0. 798 X 2 X 3 Y 1 1. 519 0. 498 1. 558 1. 054 0. 734 0. 783 1. 008
Lisrel Syntax for Model A Three indicator Model A Observed VAriables: Y X 1 X 2 X 3 Covariance Matrix: 2. 062 0. 783 1. 519 0. 798 0. 498 1. 558 1. 054 0. 734 0. 783 1. 008 Sample Size: 1000 Relationships: X 1 = Y X 2 = Y X 3 = Y Let X 1 -X 3 Correlate Path Diagram Print Residuals Lisrel Output: SS SC EF SE VA MR FS PC PT End of problem
Model B: Measurement model (Now Y is ξ) Model B Y ξ 1 X 1 δ 1 X 2 δ 2 X 3 δ 3 X 1 X 2 X 3 Y YY X 1 YX 1 X 1 X 1 X 2 YX 2 X 1 X 2 X 2 X 2 X 3 YX 3 X 1 X 3 X 2 X 3 X 3 X 3
Fundamental Hypothesis = ( ) But now we only have the variancecovariance matrix of X, so: ( ) = E(xx’) = Λx Φ Λx’ + Θδ
So, all info in this model is in Λx Φ and Θδ Θε=0 Γ=0 Β=0 Ψ=0 X 1 δ 1 Φ = E(ξ ξ’)= Var(ξ) = 1 X 2 δ 2 X 3 X= δ= δ 3 λ 1 Var(δ 1) 0 λ 2 Λ x = λ 3 Θδ = 0 0 Var(δ 2) 0 0 0 Var(δ 3)
Restating the model λ 1 λ 2 Var(δ 1) λ 3 0 0 ( ) = E(xx’) = Λx 0Φ Λx’Var + (δΘ) δ =0 2 0 λ 3 (1) + 0 Var(δ 3)
Identification n 3 variables = (3)(4)/2 = 6 n There are 6 parameters we could estimate: – 3 λ (the path coefficients) – 3 δ (error variances of each X)
Model B ξ 1 λ 3 λ 2 X 1 δ 1 X 2 X 3 δ 2 δ 3
Lisrel Syntax for Model B Three indicator Model A Observed VAriables: X 1 X 2 X 3 Covariance Matrix: 2. 062 0. 783 1. 519 0. 798 0. 498 1. 558 Latent Variable: Y Sample Size: 1000 Relationships: X 1 = Y X 2 = Y X 3 = Y Path Diagram Print Residuals Lisrel Output: SS SC EF SE VA MR FS PC PT End of problem
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