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Exploratory Structural Equation Modeling
1
Overview
• Brief overview of EFA, CFA, and SEM
• New approach to structural equation modeling
• Examples
2
Factor Analysis
And Structural Equation Modeling
• Exploratory factor analysis (EFA) is the most frequently used
multivariate analysis technique in statistics
• 1966 Jennrich solved a significant EFA rotation problem by
deriving the direct quartimin rotation
• Jennrich was the first to develop standard errors for rotated
solutions although these have still not made their way into
most statistical software programs
• 1969 development of confirmatory factor analysis (CFA) by
Joreskog
• Joreskog developed CFA further into structural equation
modeling (SEM) in LISREL where CFA was used for the
measurement part of the model
3
Structural Equation Model
(1) Yi = v + Λ ηi + K X i + ε i
(2) ηi = α + Bηi + Γ X i + ξi
Λ is typically specified as having "simple structure"
4
CFA Simple Structure Λ
X 0
X 0
X 0
Λ= 0 X where X is a factor loading parameter to be
estimated
0 X
0 X
• CFA simple structure is often too restrictive in practice
5
Quote From Browne (2001)
"Confirmatory factor analysis procedures are often used for
exploratory purposes. Frequently a confirmatory factor
analysis, with pre-specified loadings, is rejected and a
sequence of modifications of the model is carried out in an
attempt to improve fit. The procedure then becomes
exploratory rather than confirmatory --- In this situation the
use of exploratory factor analysis, with rotation of the factor
matrix, appears preferable. --- The discovery of misspecified
loadings ... is more direct through rotation of the factor matrix
than through the examination of model modification indices."
6
A New Approach: Exploratory SEM
• Allow EFA measurement model parts (EFA sets)
• Integrated with CFA measurement parts
• Allowing EFA sets access to other SEM parameters, such as
– Correlated residuals
– Regressions on covariates
– Regressions between factors of different EFA sets
– Regressions between factors of EFA and CFA sets
– Multiple groups
– EFA loading matrix equalities across time or group
– Mean structures
7
Factor Indeterminacy And Rotations
• Λ Ψ ΛT + Θ
• Λ is p x m, so m2 indeterminacies
• Ψ = I fixes m (m +1)/2 indeterminacies
• Λ ΛT + Θ = Λ * Λ * T +Θ
for Λ * = Λ H-1, where H is orthogonal
• A starting Λ* can be rotated using a rotation criterion
function that favors simple structure in Λ :
( ) (
f Λ* = f Λ H −1 ) (2a)
p m m
f (Λ ) = ∑ ∑ ∑ λij λik
2 2
(2b)
i =1 j =1k ≠ j
• Common rotation: Quartimin
• Good alternative: Geomin rotation
8
Transformation Of SEM Parameters Based
On Rotated Λ
(1) Yi = v + Λ ηi + K X i + ε i (2) ηi = α + Bηi + Γ X i + ξi
Transformations:
(6) v* = v (10) α* = H α
(7) Λ* = Λ(H * )−1 (11) B* = H* B (H*)-1
(8) K* = K (12) Γ* = H* Γ
(9) θ* = θ (13) Ψ* = (H*)T Ψ H*
9
Maximum-Likelihood Estimation And Testing
• ML estimation in several steps
– Compute the unstandardized starting values for Λ, Ψ, and Θ
with identifying restrictions
– Use the Δ method to estimate the asymptotic distribution of
the standardized starting value for Λ
– Find the asymptotic distribution of the rotated standardized
solution (cf Jennrich, 2003)
• Standard errors for rotated solution of the full SEM
• Pre-specified testing sequence: EFA followed by CFA
10
Simulated Data Example: EFA With Covariates
.8 .0
.8 .0
.8 .0 Math:
.8 .25 η1 = 0.5* x + ξ1
Λ= .8 .25 Reading:
.0 .8 η2 = 1.0* x + ξ2
.0 .8
.0 .8 where Cov(ξ1, ξ2) = 0.5
.0 .8
.0 .8
• Classic EFA rule: Ignore loadings < 0.3 (SEs typically not
available/used)
• CFA-SEM: Simple structure Λ gives good fit in terms of
common SEM measures (e.g. CFI)
11
Simulated Data: CFA-SEM, EFA-SEM Results
CFA-SEM Results EFA-SEM Results
95% 95%
Population Average Cover Average Cover
ETA1 BY
Y1 0.8000 0.7535 0.5400 0.8108 0.9000
Y2 0.8000 0.7460 0.4800 0.8045 0.9700
Y3 0.8000 0.7478 0.4800 0.8069 0.9500
Y4 0.8000 0.9905 0.0000 0.8074 0.9500
Y5 0.8000 0.9926 0.0000 0.8133 0.9500
Y6 0.0000 0.0000 1.0000 0.0011 1.0000
Y7 0.0000 0.0000 1.0000 0.0033 0.9700
Y8 0.0000 0.0000 1.0000 -0.0008 1.0000
Y9 0.0000 0.0000 1.0000 0.0108 1.0000
Y10 0.0000 0.0000 1.0000 0.0010 0.9700
12
Simulated Data: CFA-SEM, EFA-SEM Results
(Continued)
CFA-SEM Results EFA-SEM Results
95% 95%
Population Average Cover Average Cover
ETA2 BY
Y1 0.0000 0.0000 1.0000 -0.0108 0.9500
Y2 0.0000 0.0000 1.0000 -0.0122 0.9600
Y3 0.0000 0.0000 1.0000 -0.0130 0.9700
Y4 0.2500 0.0000 1.0000 0.2375 0.9100
Y5 0.2500 0.0000 1.0000 0.2335 0.8900
Y6 0.8000 0.8019 0.9900 0.8014 0.9800
Y7 0.8000 0.8806 0.9900 0.7989 0.9700
Y8 0.8000 0.8020 0.9400 0.8025 0.9600
Y9 0.8000 0.8004 0.9500 0.7994 0.9900
Y10 0.8000 0.8019 0.9400 0.8016 0.9700
13
Simulated Data: CFA-SEM, EFA-SEM Results
(Continued)
CFA-SEM Results EFA-SEM Results
95% 95%
Population Average Cover Average Cover
ETA1 ON
X 0.5000 0.6093 0.1300 0.5169 0.9400
ETA2 ON
X 1.0000 1.0004 0.9600 1.0003 0.9700
ETA1 WITH
ETA2 0.5000 0.6067 0.0100 0.5131 0.9300
14
EFA With Covariates And Direct Effects
y1 ε1
y2 ε2
ζ1
x1 f1 y3 ε3
y4 ε4
y5 ε5
ζ2
x2 f2 y6 ε6
y7 ε7
Model: f1-f2 by y1-y8(*mimic); y8 ε8
f1-f2 on x1-x2;
y1 on x1; y8 on x2;
15
ESEM
y7 y8 y9 y10 y11 y12
y1
y2 f1
y3
f3 f4
y4
y5 f2
y6
Model: ! combination of CFA and EFA measurement parts
f4 by y10-y12;
f3 by y7-y9;
f1-f2 by y1-y6 (*exog);
f4 on f3; f3 on f1-f2; 16
A Test-Retest EFA Model
y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12
f1 f2 f3 f4
Model: !loadings not equal over time Model: !loadings equal across time
f1-f2 by y1-y6(*t1); f1-f2 by y1-y6(*t1 1);
f3-f4 by y7-y12(*t2); f3-f4 by y7-y12(*t2 1);
y1-y6 pwith y7-y12; y1-y6 pwith y7-y12;
f3-f4 with f1-f2; f3-f4 with f1-f2;
17
Multiple-Group EFA
Female Variable: names = y1-y5 grp;
y1 y2 y3 y4 y5 grouping = grp (1=g1 2=g2);
Model: ! group-invariant loadings
! Psi and Theta vary across groups
f1-f2 by y1-y5 (*meas);
[f1-f2@0];
f1 f2 Model g2: [y1-y5]; ! relaxing invariant
!intercepts
Male Factor mean and variance-covariance
y1 y2 y3 y4 y5 differences across groups can be tested
f1 f2
18
Rotation Methods
Choice of rotation important when not relying on CFA
measurement structure:
• With factor complexity > 1 (“cross-loadings”) Geomin is better
than conventional methods such as varimax, promax, quartimin
• Target rotation
19
Target Rotation
Target rotation:
• In between of mechanical rotation and CFA: Rotation guided
by human judgment
• Choose your own rotation by specifying target loading values
(typically zero)
• Target values not fixed as in CFA – zero targets can come out
big if misspecified
• m – 1 zeros in each loading column gives EFA (m = # factors)
• Mplus language:
f1 by y1-y10 y1~0 (*t);
f2 by y1-y10 y5~0 (*t);
References: Browne (1972 a, b; Tucker, 1944)
20
Mplus Output
• Example 1:
http://www.ats.ucla.edu/stat/mplus/seminars/whatsnew_in_mp
lus5_1/example1.out.txt
• Example 2 :
http://www.ats.ucla.edu/stat/mplus/seminars/whatsnew_in_mp
lus5_1/example2.out.txt
• Example 3
– Example 3a :
www.ats.ucla.edu/stat/mplus/seminars/whatsnew_in_mplus
5_1/example3a.out
– Example 3b :
www.ats.ucla.edu/stat/mplus/seminars/whatsnew_in_mplus
5_1/example3b.out
21
Mplus Output (Continued)
• Example 4
– Example 4a :
www.ats.ucla.edu/stat/mplus/seminars/whatsnew_in_mplus
5_1/example4.out
– Example 4b :
www.ats.ucla.edu/stat/mplus/seminars/whatsnew_in_mplus
5_1/example4b.out
– Example 4c :
http://www.ats.ucla.edu/stat/mplus/seminars/whatsnew_in_
mplus5_1/example4c.out.txt
– Example 4d :
www.ats.ucla.edu/stat/mplus/seminars/whatsnew_in_mplus
5_1/example4d.out
22
Mplus Output (Continued)
– Example 4e :
www.ats.ucla.edu/stat/mplus/seminars/whatsnew_in_mplus
5_1/example4e.out
• Example 5 :
www.ats.ucla.edu/stat/mplus/seminars/whatsnew_in_mplus5_
1/example5.out
23
How Was The Monte Carlo Simulation Done?
Slides 11 -14
• Mplus (CFA model) output
• Mplus (EFA model) output
24
References
Asparouhov & Muthen (2008). Exploratory structural equation
modeling. Technical report. www.statmodel.com
25
