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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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