Introduction to Latent
Variable Models
A comparison of models
Model A
Model B
ξ1
Y1
X1
X2
X3
X1
X2
X3
δ1
δ2
δ3
δ1
δ2
δ3
The Fundamental Hypothesis of
SEM
 = ()
Population = Implied
Where
 is the variance-covariance matrix of the
Where
entire model
and
 is a vector (list) of elements that are
matrices: Λ Θδ Θε 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
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
– The covariance matrix of x with y
Model A: Observed model
Model A
Y
Y1
X1
δ1
X2
δ2
X3
δ3
X1
X2
X3
Y
YY
X1
YX1
X1X1
X2
YX2
X1X2 X2X2
X3
YX3
X1X3 X2X3 X3X3
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
Cov(Y 1 , X 1) Cov(Y 1 , X 2) Cov(Y 1 , X 3) 
 Var (Y 1)


Var ( X 1)
Cov( X 1 , X 2) Cov( X 1 , X 3) 
 Cov(Y 1 , X 1)
Σ
Cov(Y 1 , X 2) Cov( X 1 , X 2)
Var ( X 2)
Cov( X 2 , X 3) 


 Cov( , ) Cov( , ) Cov( , )
Var ( X 3) 
Y1 X 3
X1 X 3
X2 X3

 Σyy
 
 Σxy
Σyx 

Σxx 
 E (y y T) E (x y T) 

 
T
T 
 E (x y ) E (x x ) 
 Σyy (θ) Σyx (θ)   E (y y T) E (x y T) 

  
Σ(θ)  
 E ( T y ) E (x T) 
(
θ
)
(
θ
)
Σ
Σ
xy
xx
x 

  x
 (I  Β)1 (ΓΦ ΓT  Ψ ) [(I  Β)1 T (I  Β)1 ΓΦ 
]



T


T
1
Φ
Φ
Γ [(I  Β) ]


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

λ1
Λ=
λ2
Φ = φy
λ3
Ψ=
δ1
δ12
0?
δ2
0?
0?
δ13
δ23
δ3
Θδ = 0
Identification
4 variables = (4)(5)/2 = 10
 There are 10 parameters we could
estimate:

–3
–1
–3
–3
λ (the path coefficients)
ψ (error variance of Y)
δ (error variances of each X)
δ (Covariances among the 3 X errors)
Model A: Observed model
Y1
ζ
Model A
λ1
X1
δ1
λ2
λ3
X2
X3
δ2
δ3
Covariance Matrix
X1
X2
X3
Y1
X1
X2
X3
Y1
2.062
0.783 1.519
0.798 0.498 1.558
1.054 0.734 0.783 1.008
Lisrel Syntax for Model A
Three indicator Model A
Observed VAriables: Y X1 X2 X3
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:
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
ξ1
X1
δ1
X2
δ2
X3
δ3
X1
X2
X3
Y
YY
X1
YX1
X1X1
X2
YX2
X1X2 X2X2
X3
YX3
X1X3 X2X3 X3X3
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
Φ = E(ξ ξ’)= Var(ξ) = 1
X1
δ1
X = X2
δ= δ
2
X3
δ
3
λ1
Λx = λ2
λ3
Var(δ1) 0
Θδ = 0
0
0
Var(δ2) 0
0
Var(δ3)
Restating the model
() = E(xx’) = Λx Φ Λx’ + Θδ =
λ1
λ2 (1) λ
λ3
1
λ
2
λ
3
+
Var(δ1)
0
0
0
Var(δ2)
0
0
0
Var(δ3)
Identification
3 variables = (3)(4)/2 = 6
 There are 6 parameters we could
estimate:

– 3 λ (the path coefficients)
– 3 δ (error variances of each X)
Model B
ξ1
λ1
λ3
λ2
X1
δ1
X2
X3
δ2
δ3
Lisrel Syntax for Model B
Three indicator Model A
Observed VAriables: X1 X2 X3
Covariance Matrix:
2.062
0.783 1.519
0.798 0.498 1.558
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