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Exploratory Structural Equation Modeling

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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

				
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