Introduction to Latent Variable Models

A comparison of models X1X1 X2X2 X3X3 Y1Y1 δ1δ1 δ2δ2 δ3δ3 Model AModel B ξ1ξ1 X1X1 X2X2 X3X3 δ1δ1 δ2δ2 δ3δ3

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 Θ δ Θ ε Φ Γ Β Ψ

Implied Covariance Matrix: Observed Model For an observed model, the implied matrix is the relationships among all the x and y variables For an observed model, the implied matrix is the relationships among all the x and y variables XY XY X xxyx Y xy yy It can be decomposed into three pieces: –the covariance matrix of y –The covariance matrix of x –The covariance matrix of x with y

Model A: Observed model X1X1 X2X2 X3X3 Y1Y1 δ1δ1 δ2δ2 δ3δ3 Model A Y X1X1X1X1 X2X2X2X2 X3X3X3X3 YYY X1X1X1X1 YX 1 X1X1X1X1X1X1X1X1 X2X2X2X2 YX 2 X1X2X1X2X1X2X1X2 X2X2X2X2X2X2X2X2 X3X3X3X3 YX 3 X1X3X1X3X1X3X1X3 X2X3X2X3X2X3X2X3 X3X3X3X3X3X3X3X3

Covariance Matrix of Y Σ yy (Θ) = E(yy’) = (I – B) -1 (ΓΦΓ’ + Ψ) (I – B) -1’

Covariance Matrix of X Σ xx (Θ) = E(xx’) = Φ

Covariance Matrix of XY Σ xy (Θ) = E(xy’) = ΦΓ’(I – B) -1

Put that all together and get:  (  ) = (I – B)-1(ΓΦΓ’ + Ψ) (I – B)-1’ Φ ΦΓ’(I – B)-1

Population vs. Implied Covariance Matrices in Model A

So, the matrices for Model A are: Elements of Θ = Λ Θ δ Θ ε Φ Γ Β Ψ Elements of Θ = Λ Θ δ Θ ε Φ Γ Β Ψ Β = 0Θ ε = 0 Φ = 0 Γ = 0 Θ δ = 0 Λ = Φ = φ y Ψ = Ψ = λ1λ1λ1λ1 λ2λ2λ2λ2 λ3λ3λ3λ3 δ1δ1δ1δ10?0? δ 12 δ2δ2δ2δ20? δ 13 δ 23 δ3δ3δ3δ3

Identification 4 variables = (4)(5)/2 = 10 4 variables = (4)(5)/2 = 10 There are 10 parameters we could estimate: 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 X1X1 X2X2 X3X3 Y1Y1 δ1δ1 δ2δ2 δ3δ3 Model A λ1λ1λ1λ1 λ2λ2λ2λ2 λ3λ3λ3λ3 ζ

Covariance Matrix X 1 X 2 X 3 Y 1 X X X Y

Lisrel Syntax for Model A Three indicator Model A Observed VAriables: Y X1 X2 X3 Covariance Matrix: Sample Size: 1000 Relationships: X1 = Y X2 = Y X3 = Y Let X1-X3 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 X1X1X1X1 X2X2X2X2 X3X3X3X3 YYY X1X1X1X1 YX 1 X1X1X1X1X1X1X1X1 X2X2X2X2 YX 2 X1X2X1X2X1X2X1X2 X2X2X2X2X2X2X2X2 X3X3X3X3 YX 3 X1X3X1X3X1X3X1X3 X2X3X2X3X2X3X2X3 X3X3X3X3X3X3X3X3 ξ1ξ1 X1X1 X2X2 X3X3 δ1δ1 δ2δ2 δ3δ3

Fundamental Hypothesis  =  (  )  =  (  ) But now we only have the variance- covariance matrix of X, so:  (  ) = E(xx’) = Λ x Φ Λ x ’ + Θ δ

So, all info in this model is in Λ x Φ and Θ δ Θ ε =0 Γ=0 Β=0 Ψ=0 Φ = E( ξ ξ’)= Var(ξ) = 1 X = δ = Λ x = Θ δ = λ1λ1λ1λ1 λ2λ2λ2λ2 λ3λ3λ3λ3 Var(δ 1 ) 000 Var(δ 2 ) 0 00 Var(δ 3 ) X1X1X1X1 X2X2X2X2 X3X3X3X3 δ1δ1δ1δ1 δ2δ2δ2δ2 δ 3δ 3δ 3δ 3

Restating the model  (  ) = E(xx’) = Λ x Φ Λ x ’ + Θ δ = (1) + (1) + λ1λ1λ1λ1 λ2λ2λ2λ2 λ3λ3λ3λ3 Var(δ 1 ) 000 Var(δ 2 ) 0 00 Var(δ 3 ) λ 1λ 1λ 1λ 1 λ 2λ 2λ 2λ 2 λ 3λ 3λ 3λ 3

Identification 3 variables = (3)(4)/2 = 6 3 variables = (3)(4)/2 = 6 There are 6 parameters we could estimate: There are 6 parameters we could estimate: –3 λ (the path coefficients) –3 δ (error variances of each X)

Model B ξ1ξ1 X1X1 X2X2 X3X3 δ1δ1 δ2δ2 δ3δ3 λ1λ1λ1λ1 λ2λ2λ2λ2 λ3λ3λ3λ3

Lisrel Syntax for Model B Three indicator Model A Observed VAriables: X1 X2 X3 Covariance Matrix: Latent Variable: Y Sample Size: 1000 Relationships: X1 = Y X2 = Y X3 = Y Path Diagram Print Residuals Lisrel Output: SS SC EF SE VA MR FS PC PT End of problem