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PSY 6550: Structural Equation Modeling
Ch. 1: Introduction
I. Definition: Structural Equation Modeling (SEM) is a family of statistical methods which
allows confirmatory decisions for sets of relationships between one or more independent
variables and one or more dependent variables. It may be said that SEM is a combination
of exploratory factor analysis (EFA) and multiple regression (MR).
II. Terminology
A. Manifest variables: Observed variables, indicators, or measured variables,
represented by squares or rectangles.
B. Latent variables: unobserved hidden variables, also called constructs or factors,
represented by circles or ovals.
C. Exogenous concepts (Exogenous latent variables): Latent variables which cause
fluctuations in the values of other latent variables in the model (cf. IV).
D. Endogenous concepts (Endogenous latent variables): Latent variables which are
influenced by the exogenous concepts (cf. DV).
E. Measurement model: A structural model which includes both latent and manifest
variables.
F. Latent Model: A structural model with only latent variables.
III. Mathematical (Structural) models
A. Two measurement models
- x = Λ (^) X ξ + δ
where
x = a (qx1) vector of observed exogenous indicators,
Λ X = a (qxn) matrix of structural coefficients,
ξ = an (nx1) vector of exogenous concepts,
δ = a (qx1) vector of errors in the measurement model,
q = the number of x-variables, and
n = the number of ξ -variables.
- The covariances among these errors make a (qxq) matrix, Θ δ.
2. y = Λ y η + ε
where
y = a (px1) vector of observed endogenous indicators,
Λ y = a (pxm) matrix of structural coefficients,
η = an (mx1) vector of endogenous concepts,
ε = a (px1) vector of errors in the measurement model,
p = the number of y-variables, and
m = the number of η -variables.
- The covariances among these errors make a (pxp) matrix, Θ ε.
B. A latent model
where
η = an (mx1) vector of endogenous concepts,
Β = an (mxm) matrix of structural coefficients,
Γ = an (mxn) matrix of structural coefficients,
ξ = an (nx1) vector of exogenous concepts, and
ζ = an (mx1) vector of errors in the latent model.
* The covariances among these exogenous variables ( ξ ) make an (nxn) Φ matrix.
* The covariances among these errors ( ζ ) in the latent model make an (mxm)
Ψ (^) matrix.
IV. Basic Indices
A. Mean
1. E(X) = ∑.
i
xi p ( xi )
Since p ( xi ) = 1 / N , E(X) = ∑
i
xi p ( xi )= ∑ ( 1 / ) = ∑xi/.
i i
xi N N
- For a grouped data set, p^ (^ xi ) =^ fi / N , thus,
E(X) = ∑ = ∑
i
i i i
xi ( fi / N ) ( xf )/ N
- The mean is the center of gravity of the data set.
B. Variance
- Var(X) =
2 2
σ x = E ( X − E ( X ))
i i
i i i i N
f x E X px x E X
2 2 ( ( )) ( ) ( ( ))
= x E X N. i
( (^) i ( )) /
2
- Variance is an index of average squared distance (area concept).
C. Covariance
- Cov(XY) = E[(X – E(X))(Y - E(Y))]
i j
i j i j ( x E ( X ))( y E ( Y )) p ( xy )
i j
( xi μ x )( yj μ y )) p ( xiyj )
= X E X Y EY N
i j
∑∑ ( −^ ( ))( − ( ))^ /.
- Covariance is an index of co-change between X and Y.
- Variance can be said a special case of covariance in that variance is an
index of change of the same variable.
VI. Standardization
A.
x
x
x
x
s
X X M
z
B. Standardization makes the origin (mean) zero (0) and the unit (SD) 1.
C. Standardization does not change the shape of the original distribution.
D. If you standardize any two variables, say X and Y, or X 1 and X 2 ,
then the covariance between the two variables is the same as the correlation.
rxy = SDX SD Y
COV ( XY )
x y
xy
Æ if both σ x and σ y are 1, then rxy = σ xy.
VII. Structural Equation Modeling with simple equations
A. Y = a + bX, regression equation or structural equation where a and b are structural
coefficients.
B. Structural equation and covariance/correlation
- Let X 2 = b 21 X 1 + e 2 (1) and
X 3 = b 31 X 1 + e 3 (2).
Then, the structural coefficient for X 1 predicting X 2 is b 21 and the
structural coefficient for e 2 is 1 for equation (1), and the same logic applies
to equation (2).
- If b 21 and b 31 are near 1 and e 2 and e 3 are random fluctuations, then we can
say that X 2 and X 3 are correlated due to their dependence on the common
cause of X 1 (figure 1.12) Æ Covariance/correlation.
C. Quantification of the covariance and correlation between X 2 and X 3.
- Cov(X 2 X 3 ) = E [( X 2 − μ (^) x 2 )( X 3 − μ x 3 )]
= ( 2 2 )( 3 x 3 ) ( 2 i 3 j ) i j
∑∑ xi −μ^ x x j −^ μ px x
i
[( b (^) 21 x 1 i e 2 i ) ( b 21 μ x 1 μ e 2 )]*[( b 31 x 1 i e 3 i ) ( b 31 μ x 1 μ e 3 )]( 1 / N )
N
[ 21 31 ( 1 1 )( 1 1 ) 21 ( 1 i x 1 )( 3 i e 3 ) i
∑ b b xi −μ^ x xi −μ x + b x −μ e −^ μ
- b 31 ( x 1 i − μ (^) x 1 )( e 2 i − μ e 2 )+( e 2 i − μ e 2 )( e 3 i − μ e 3 )]
N
[ 21 31 ( 1 1 )( 1 1 ) 21 ( 1 i x 1 )( 3 i e 3 ) i i
∑ b b xi −μ^ x xi −μ x +∑ b x −μ e −^ μ
i
i x i e i e i e i
b (^) 31 ( x 1 μ 1 )( e 2 μ 2 ) ( e 2 μ 2 )( e 3 μ 3 )]
= b 21 b 31 Var(X 1 ) + b 21 Cov(X 1 e 3 ) + b 31 Cov(X 1 e 2 ) + Cov(e 2 e 3 ).
- Assuming both e 2 and e 3 are random error, we can rewrite
Cov(X 2 X 3 ) = b 21 b 31 Var(X 1 ).
- If we standardize X 1 , X 2 , and X 3 , then Cov(X 2 X 3 ) = r 23 , and Var(X 1 ) = 1.
Thus, r 23 = β 21 β 31 where β 21 and β 31 are standardized coefficients of b 21
and b 31 , respectively.
- The major task of structural equation modeling is to estimate the
structural coefficients based on a theory about psychological or
behavioral
phenomena.
