Ch. 1: Structural Equation Modeling
I. Hayduk’s paper
II. Estimation of parameters
A. Given a super covariance matrix, Σ, with Xs and Ys
y1 y2 . . yp x1 x2 . . xq
)()( qpXqp ++
Σ
=
2
2121
2
2222122212
112
2112111
21
2
21
222122
22221
12111112
211
....
..........
..........
....
....
....
..........
..........
....
....
xqqxqxqxqypxqyxqy
qxxxypxyxyx
qxxxypxyxyx
ypxqypxypxyppypyp
xqyxyxypyyy
xqyxyxypyyy
σσσσσσ
σσσσσσ
σσσσσσ
σσσσσσ
σσσσσσ
σσσσσσ
=
ΣΣ
ΣΣ
xxxy
yxyy
where,(parametization)
Σyy = Λy(I-B)-1(ΓΦΓ’ + Ψ)(I-B’)-1Λy’ + θε,
Σyx = Λy(I-B)-1ΓΦΛx’,
Σxy = ΛxΦΓ’(I-B’)-1Λy’, and
Σxx = ΛxΦΛ x’ + θδ.
B. Estimation of all parameters (B, Γ, Λx, Λy) can be done by
the Newton-Raphson method or other appropriate methods.
III. Goodness-of-fit
A. Given a model-implied covariance matrix, Σ, and a sample
covariance matrix, S, both F and χ2-test can test the
goodness-of-fit between Σ and S.
B. F = ln|Σ| + tr(SΣ-1) – ln|S| - (p+q)
C. χ2 = n[tr(SΣ-1) + ln|Σ| - ln|S| - (p+q)]
with df = .5[(p+q)(p+q+1)] – t
where t = the number of free parameters in the model.
IV. Caution
A. Non-identifiable parameters.
B. Possibility of equivalent models.
C. Theory-driven models.
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