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Bollen, Ray, Zavisca ASA, April 22 , 2005 - slide #1
Bayes Factorsin
Structural Equation Models (SEMs):Schwarz BIC and Other Approximations
.
Kenneth A. BollenUniversity of North Carolina, Chapel Hill
Surajit RaySAMSI and University of North Carolina, Chapel Hill
Jane ZaviscaSAMSI and University of Arizona
Bayes Factor in SEM
Bayes factor in SEM
Model Fit in SEM’s
Approximating Bayes Factor
Simulation
Results
Conclusion
Bollen, Ray, Zavisca ASA, April 22 , 2005 - slide #2
Bayes factor in SEM
■ Model Fit in SEMs
■ Bayes Factor
■ Approximating Bayes Factor
■ Simulation
■ Results
■ Conclusions
Bayes Factor in SEM
Bayes factor in SEM
Model Fit in SEM’s
SEM
Hypothesis testing
Fit Indices
Model Comparisons
Approximating Bayes Factor
Simulation
Results
Conclusion
Bollen, Ray, Zavisca ASA, April 22 , 2005 - slide #3
SEM
Latent Variable Model
η = Bη + Γξ + ζ
Measurement Model
y = Λyη + �
x = Λxξ + δ
Bayes Factor in SEM
Bayes factor in SEM
Model Fit in SEM’s
SEM
Hypothesis testing
Fit Indices
Model Comparisons
Approximating Bayes Factor
Simulation
Results
Conclusion
Bollen, Ray, Zavisca ASA, April 22 , 2005 - slide #4
Hypothesis testing
H0 : Σ = Σ(θ)
Chi square Test Statistic
T = (N − 1)FML ∼ χ2 in large samples
■ excess power when big N
■ excess kurtosis influence on T
■ exact H0, approximate model
Bayes Factor in SEM
Bayes factor in SEM
Model Fit in SEM’s
SEM
Hypothesis testing
Fit Indices
Model Comparisons
Approximating Bayes Factor
Simulation
Results
Conclusion
Bollen, Ray, Zavisca ASA, April 22 , 2005 - slide #5
Fit Indices
RMSEA, CFI, TLI, IFI, Etc.
■ Cutoff values?
■ Nonnested models?
■ Small N issues?
■ Behavior across estimators?
Bayes Factor in SEM
Bayes factor in SEM
Model Fit in SEM’s
SEM
Hypothesis testing
Fit Indices
Model Comparisons
Approximating Bayes Factor
Simulation
Results
Conclusion
Bollen, Ray, Zavisca ASA, April 22 , 2005 - slide #6
Model Comparisons
■ Chi square difference (LR) tests◆ Power, excess kurtosis, N issues◆ Nested Models Only.
■ Fit indices differences◆ Cutoff values for differences◆ Behavior across estimators◆ Properties across N
Bayes Factor in SEM
Bayes factor in SEM
Model Fit in SEM’s
Approximating Bayes Factor
Previous Work
Bayes Factor (BF)
Posterior Odds and BF
Marginal Likelihood
Laplace Approximation
Laplace Approximation - MLE
substitution
BIC
ABF 1
ABF 1 and BIC
ABF 1 and BIC
Simulation
Results
Conclusion
Bollen, Ray, Zavisca ASA, April 22 , 2005 - slide #7
Previous Work
■ Bayesian work small part of SEMs
■ Bayes factor, largely discussed via BIC in SEM literature◆ Cudeck and Browne (1983)◆ Bollen (1989)◆ Raftery (1993, 1995)◆ Haughton, Oud, and Jansen (1997)
Bayes Factor in SEM
Bayes factor in SEM
Model Fit in SEM’s
Approximating Bayes Factor
Previous Work
Bayes Factor (BF)
Posterior Odds and BF
Marginal Likelihood
Laplace Approximation
Laplace Approximation - MLE
substitution
BIC
ABF 1
ABF 1 and BIC
ABF 1 and BIC
Simulation
Results
Conclusion
Bollen, Ray, Zavisca ASA, April 22 , 2005 - slide #8
Bayes Factor (BF)
■ Y : Data
■ Mk : Model
■ Bayes Theorem
P (M1|Y ) =P (Y |M1)P (M1)
P (Y |M1)P (M1) + P (Y |M2)P (M2)
■ Comparing Model M1 and M2Choose the model with higher posterior probability.
P (M1|Y )
P (M2|Y )=
P (Y |M1)
P (Y |M2)
P (M1)
P (M2)
= Bayes Factor × prior odds
Bayes Factor in SEM
Bayes factor in SEM
Model Fit in SEM’s
Approximating Bayes Factor
Previous Work
Bayes Factor (BF)
Posterior Odds and BF
Marginal Likelihood
Laplace Approximation
Laplace Approximation - MLE
substitution
BIC
ABF 1
ABF 1 and BIC
ABF 1 and BIC
Simulation
Results
Conclusion
Bollen, Ray, Zavisca ASA, April 22 , 2005 - slide #9
Posterior Odds and BF
P (M1|Y )
P (M2|Y )=
P (Y |M1)
P (Y |M2)
P (M1)
P (M2)
If P (M1) = P (M2) then prior odds=1
=⇒ Posterior Odds = BF
Define Bayes Factor as
BF12 =P (Y |M1)
P (Y |M2)
Bayes Factor in SEM
Bayes factor in SEM
Model Fit in SEM’s
Approximating Bayes Factor
Previous Work
Bayes Factor (BF)
Posterior Odds and BF
Marginal Likelihood
Laplace Approximation
Laplace Approximation - MLE
substitution
BIC
ABF 1
ABF 1 and BIC
ABF 1 and BIC
Simulation
Results
Conclusion
Bollen, Ray, Zavisca ASA, April 22 , 2005 - slide #10
Marginal Likelihood
■ P (θ|Mk): Prior Distribution of θ given the models Mk.
■ dk dimension of θk■ In general the Marginal Likelihood is
P (Y |Mk) =
∫
θk
P (Y |Mk, θk)P (θk|Mk)dθk
■ If the pdf P (Y |Mk)’s are completely specified ( no free parameters).—– Bayes Factor= Likelihood ratio.
■ Sensitivity to prior is more critical in calculation in BF than in otherBayesian Analysis. ( Raftery 1993)
Bayes Factor in SEM
Bayes factor in SEM
Model Fit in SEM’s
Approximating Bayes Factor
Previous Work
Bayes Factor (BF)
Posterior Odds and BF
Marginal Likelihood
Laplace Approximation
Laplace Approximation - MLE
substitution
BIC
ABF 1
ABF 1 and BIC
ABF 1 and BIC
Simulation
Results
Conclusion
Bollen, Ray, Zavisca ASA, April 22 , 2005 - slide #11
Laplace Approximation
P (Y |Mk) =
∫
θk
P (Y |Mk, θk)P (θk|Mk)︸ ︷︷ ︸
dθk
■ Laplace Approximation on Likelihood × Prior
P (Y |Mk) ≈ (2π)dk/2|Ĩ(θ̃k)|
− 12 P (y|θ̃k, Mk)P (θ̃k)
■ Error of approximation : O( 1n )
Ĩ(θ̃k) =d2
dθdθ′log(P (y|θ, Mk)P (θ))
∣∣∣θ=θ̃k
=d2
dθdθ′l̃(θ)
∣∣∣θ=θ̃k
■ θ̃ not readily available from software outputs
Bayes Factor in SEM
Bayes factor in SEM
Model Fit in SEM’s
Approximating Bayes Factor
Previous Work
Bayes Factor (BF)
Posterior Odds and BF
Marginal Likelihood
Laplace Approximation
Laplace Approximation - MLE
substitution
BIC
ABF 1
ABF 1 and BIC
ABF 1 and BIC
Simulation
Results
Conclusion
Bollen, Ray, Zavisca ASA, April 22 , 2005 - slide #12
Laplace Approximation - MLE substitution
■ Replacing θ̃ by θ̂ (MLE)
P (Y |Mk) ≈ (2π)dk/2|I(θ̂k)|
− 12 P (y|θ̂k, Mk)P (θ̂k)
I(θ̂k) =d2
dθdθ′log(P (y|θ, Mk))
∣∣∣θk=θ̂k
=d2
dθdθ′l(θ)
∣∣∣θk=θ̂k
◆ Error: O( 1n )
◆ Less accurate than using θ̃
◆ Using E(I(θ̂k)) in place of I(θ̂k) has error: O(1
n12)
Bayes Factor in SEM
Bayes factor in SEM
Model Fit in SEM’s
Approximating Bayes Factor
Previous Work
Bayes Factor (BF)
Posterior Odds and BF
Marginal Likelihood
Laplace Approximation
Laplace Approximation - MLE
substitution
BIC
ABF 1
ABF 1 and BIC
ABF 1 and BIC
Simulation
Results
Conclusion
Bollen, Ray, Zavisca ASA, April 22 , 2005 - slide #13
BIC
P (Y |Mk) ≈ (2π)dk/2|I(θ̂k)|
− 12 P (y|θ̂k, Mk)P (θ̂k)
■ Choosing Unit Information prior i.e.
P (θk) ∼ N
θok ,
[
I(θ̂k)
n
]−1
=⇒ P (Y |Mk) ≈ el(θ̂k|y)(n)−dk/2
=⇒ 2 log P (Y |Mk) ≈ 2l(θ̂k|y) −dk
log(n)
2 log BF = BIC
Bayes Factor in SEM
Bayes factor in SEM
Model Fit in SEM’s
Approximating Bayes Factor
Previous Work
Bayes Factor (BF)
Posterior Odds and BF
Marginal Likelihood
Laplace Approximation
Laplace Approximation - MLE
substitution
BIC
ABF 1
ABF 1 and BIC
ABF 1 and BIC
Simulation
Results
Conclusion
Bollen, Ray, Zavisca ASA, April 22 , 2005 - slide #14
ABF 1
P (Y |Mk) =
∫
θk
P (Y |Mk, θk)︸ ︷︷ ︸
P (θk|Mk)︸ ︷︷ ︸
dθk
Likelihood Prior
■ Laplace Approximation only on P (y|θ̂k, Mk)
■ Prior
P (θk|Mk) ∼ N
θ0k,
[
c
I(θ̂k)
n
]−1
Using the c to maximize P (Y |Mk)
Bayes Factor in SEM
Bayes factor in SEM
Model Fit in SEM’s
Approximating Bayes Factor
Previous Work
Bayes Factor (BF)
Posterior Odds and BF
Marginal Likelihood
Laplace Approximation
Laplace Approximation - MLE
substitution
BIC
ABF 1
ABF 1 and BIC
ABF 1 and BIC
Simulation
Results
Conclusion
Bollen, Ray, Zavisca ASA, April 22 , 2005 - slide #15
ABF 1 and BIC
Using the c
2 log P (Y |Mk) ≈ 2l(θk) − dk
(
1 + log
[
dk
θ̂kTI(θ̂k)θ̂k
])
=⇒ ABF 1 = BIC − d1 log
[
d1
θ̂1T I(θ̂1)
nθ̂1
]
+ d2 log
[
d2
θ̂2T I(θ̂2)
nθ̂2
]
Bayes Factor in SEM
Bayes factor in SEM
Model Fit in SEM’s
Approximating Bayes Factor
Previous Work
Bayes Factor (BF)
Posterior Odds and BF
Marginal Likelihood
Laplace Approximation
Laplace Approximation - MLE
substitution
BIC
ABF 1
ABF 1 and BIC
ABF 1 and BIC
Simulation
Results
Conclusion
Bollen, Ray, Zavisca ASA, April 22 , 2005 - slide #16
ABF 1 and BIC
BIC ABF 1
Prior implicit Yes Yes: has more flexibility thanthe unit information prior
Uses standard software output Yes Yes
Need for defining n. a Yes No: Enters through
θ̂kTI(θ̂k)θ̂k
aSee Raftery 1993 and 1995 on uncertainty about n in SEM’s
Bayes Factor in SEM
Bayes factor in SEM
Model Fit in SEM’s
Approximating Bayes Factor
Simulation
Simulation Study
True Model (MT)
Missing Crossloading (M1)
Missing Crossloading (M2)
Missing Crossloading (M3)
Correlated errors Replace
Crossloadings (M4)
Over-specified Model (M5)
Over-specified Model (M6)
Over-specified Model (M7)
Extra Latent Variable (M8)
Missing Latent Variable
Wrong Structural Model
Wrong Structural Model
Wrong Structural Model
Missing Indicator
Switched Loadings
Results
Conclusion
Bollen, Ray, Zavisca ASA, April 22 , 2005 - slide #17
Simulation Study
■ We tested model selection properties on a simulated data set.
■ 20 replicate data sets were created at each of 4 sample sizes (N=100,N=250, N=500, N=1000). Full scale Monte Carlo simulation study stillneeds to be done.
■ We fit a variety of under- and over-specified models.
For further details on the simulated data, see Paxton et “Monte CarloExperiments” Structural Equation Modeling, 2001.
Bayes Factor in SEM
Bayes factor in SEM
Model Fit in SEM’s
Approximating Bayes Factor
Simulation
Simulation Study
True Model (MT)
Missing Crossloading (M1)
Missing Crossloading (M2)
Missing Crossloading (M3)
Correlated errors Replace
Crossloadings (M4)
Over-specified Model (M5)
Over-specified Model (M6)
Over-specified Model (M7)
Extra Latent Variable (M8)
Missing Latent Variable
Wrong Structural Model
Wrong Structural Model
Wrong Structural Model
Missing Indicator
Switched Loadings
Results
Conclusion
Bollen, Ray, Zavisca ASA, April 22 , 2005 - slide #18
True Model (MT)
Bayes Factor in SEM
Bayes factor in SEM
Model Fit in SEM’s
Approximating Bayes Factor
Simulation
Simulation Study
True Model (MT)
Missing Crossloading (M1)
Missing Crossloading (M2)
Missing Crossloading (M3)
Correlated errors Replace
Crossloadings (M4)
Over-specified Model (M5)
Over-specified Model (M6)
Over-specified Model (M7)
Extra Latent Variable (M8)
Missing Latent Variable
Wrong Structural Model
Wrong Structural Model
Wrong Structural Model
Missing Indicator
Switched Loadings
Results
Conclusion
Bollen, Ray, Zavisca ASA, April 22 , 2005 - slide #19
Missing Crossloading (M1)
Bayes Factor in SEM
Bayes factor in SEM
Model Fit in SEM’s
Approximating Bayes Factor
Simulation
Simulation Study
True Model (MT)
Missing Crossloading (M1)
Missing Crossloading (M2)
Missing Crossloading (M3)
Correlated errors Replace
Crossloadings (M4)
Over-specified Model (M5)
Over-specified Model (M6)
Over-specified Model (M7)
Extra Latent Variable (M8)
Missing Latent Variable
Wrong Structural Model
Wrong Structural Model
Wrong Structural Model
Missing Indicator
Switched Loadings
Results
Conclusion
Bollen, Ray, Zavisca ASA, April 22 , 2005 - slide #20
Missing Crossloading (M2)
Bayes Factor in SEM
Bayes factor in SEM
Model Fit in SEM’s
Approximating Bayes Factor
Simulation
Simulation Study
True Model (MT)
Missing Crossloading (M1)
Missing Crossloading (M2)
Missing Crossloading (M3)
Correlated errors Replace
Crossloadings (M4)
Over-specified Model (M5)
Over-specified Model (M6)
Over-specified Model (M7)
Extra Latent Variable (M8)
Missing Latent Variable
Wrong Structural Model
Wrong Structural Model
Wrong Structural Model
Missing Indicator
Switched Loadings
Results
Conclusion
Bollen, Ray, Zavisca ASA, April 22 , 2005 - slide #21
Missing Crossloading (M3)
Bayes Factor in SEM
Bayes factor in SEM
Model Fit in SEM’s
Approximating Bayes Factor
Simulation
Simulation Study
True Model (MT)
Missing Crossloading (M1)
Missing Crossloading (M2)
Missing Crossloading (M3)
Correlated errors Replace
Crossloadings (M4)
Over-specified Model (M5)
Over-specified Model (M6)
Over-specified Model (M7)
Extra Latent Variable (M8)
Missing Latent Variable
Wrong Structural Model
Wrong Structural Model
Wrong Structural Model
Missing Indicator
Switched Loadings
Results
Conclusion
Bollen, Ray, Zavisca ASA, April 22 , 2005 - slide #22
Correlated errors Replace Crossloadings (M4)
Bayes Factor in SEM
Bayes factor in SEM
Model Fit in SEM’s
Approximating Bayes Factor
Simulation
Simulation Study
True Model (MT)
Missing Crossloading (M1)
Missing Crossloading (M2)
Missing Crossloading (M3)
Correlated errors Replace
Crossloadings (M4)
Over-specified Model (M5)
Over-specified Model (M6)
Over-specified Model (M7)
Extra Latent Variable (M8)
Missing Latent Variable
Wrong Structural Model
Wrong Structural Model
Wrong Structural Model
Missing Indicator
Switched Loadings
Results
Conclusion
Bollen, Ray, Zavisca ASA, April 22 , 2005 - slide #23
Over-specified Model (M5)
Bayes Factor in SEM
Bayes factor in SEM
Model Fit in SEM’s
Approximating Bayes Factor
Simulation
Simulation Study
True Model (MT)
Missing Crossloading (M1)
Missing Crossloading (M2)
Missing Crossloading (M3)
Correlated errors Replace
Crossloadings (M4)
Over-specified Model (M5)
Over-specified Model (M6)
Over-specified Model (M7)
Extra Latent Variable (M8)
Missing Latent Variable
Wrong Structural Model
Wrong Structural Model
Wrong Structural Model
Missing Indicator
Switched Loadings
Results
Conclusion
Bollen, Ray, Zavisca ASA, April 22 , 2005 - slide #24
Over-specified Model (M6)
Bayes Factor in SEM
Bayes factor in SEM
Model Fit in SEM’s
Approximating Bayes Factor
Simulation
Simulation Study
True Model (MT)
Missing Crossloading (M1)
Missing Crossloading (M2)
Missing Crossloading (M3)
Correlated errors Replace
Crossloadings (M4)
Over-specified Model (M5)
Over-specified Model (M6)
Over-specified Model (M7)
Extra Latent Variable (M8)
Missing Latent Variable
Wrong Structural Model
Wrong Structural Model
Wrong Structural Model
Missing Indicator
Switched Loadings
Results
Conclusion
Bollen, Ray, Zavisca ASA, April 22 , 2005 - slide #25
Over-specified Model (M7)
Bayes Factor in SEM
Bayes factor in SEM
Model Fit in SEM’s
Approximating Bayes Factor
Simulation
Simulation Study
True Model (MT)
Missing Crossloading (M1)
Missing Crossloading (M2)
Missing Crossloading (M3)
Correlated errors Replace
Crossloadings (M4)
Over-specified Model (M5)
Over-specified Model (M6)
Over-specified Model (M7)
Extra Latent Variable (M8)
Missing Latent Variable
Wrong Structural Model
Wrong Structural Model
Wrong Structural Model
Missing Indicator
Switched Loadings
Results
Conclusion
Bollen, Ray, Zavisca ASA, April 22 , 2005 - slide #26
Extra Latent Variable (M8)
Bayes Factor in SEM
Bayes factor in SEM
Model Fit in SEM’s
Approximating Bayes Factor
Simulation
Simulation Study
True Model (MT)
Missing Crossloading (M1)
Missing Crossloading (M2)
Missing Crossloading (M3)
Correlated errors Replace
Crossloadings (M4)
Over-specified Model (M5)
Over-specified Model (M6)
Over-specified Model (M7)
Extra Latent Variable (M8)
Missing Latent Variable
Wrong Structural Model
Wrong Structural Model
Wrong Structural Model
Missing Indicator
Switched Loadings
Results
Conclusion
Bollen, Ray, Zavisca ASA, April 22 , 2005 - slide #27
Missing Latent Variable
Bayes Factor in SEM
Bayes factor in SEM
Model Fit in SEM’s
Approximating Bayes Factor
Simulation
Simulation Study
True Model (MT)
Missing Crossloading (M1)
Missing Crossloading (M2)
Missing Crossloading (M3)
Correlated errors Replace
Crossloadings (M4)
Over-specified Model (M5)
Over-specified Model (M6)
Over-specified Model (M7)
Extra Latent Variable (M8)
Missing Latent Variable
Wrong Structural Model
Wrong Structural Model
Wrong Structural Model
Missing Indicator
Switched Loadings
Results
Conclusion
Bollen, Ray, Zavisca ASA, April 22 , 2005 - slide #28
Wrong Structural Model
Bayes Factor in SEM
Bayes factor in SEM
Model Fit in SEM’s
Approximating Bayes Factor
Simulation
Simulation Study
True Model (MT)
Missing Crossloading (M1)
Missing Crossloading (M2)
Missing Crossloading (M3)
Correlated errors Replace
Crossloadings (M4)
Over-specified Model (M5)
Over-specified Model (M6)
Over-specified Model (M7)
Extra Latent Variable (M8)
Missing Latent Variable
Wrong Structural Model
Wrong Structural Model
Wrong Structural Model
Missing Indicator
Switched Loadings
Results
Conclusion
Bollen, Ray, Zavisca ASA, April 22 , 2005 - slide #29
Wrong Structural Model
Bayes Factor in SEM
Bayes factor in SEM
Model Fit in SEM’s
Approximating Bayes Factor
Simulation
Simulation Study
True Model (MT)
Missing Crossloading (M1)
Missing Crossloading (M2)
Missing Crossloading (M3)
Correlated errors Replace
Crossloadings (M4)
Over-specified Model (M5)
Over-specified Model (M6)
Over-specified Model (M7)
Extra Latent Variable (M8)
Missing Latent Variable
Wrong Structural Model
Wrong Structural Model
Wrong Structural Model
Missing Indicator
Switched Loadings
Results
Conclusion
Bollen, Ray, Zavisca ASA, April 22 , 2005 - slide #30
Wrong Structural Model
Bayes Factor in SEM
Bayes factor in SEM
Model Fit in SEM’s
Approximating Bayes Factor
Simulation
Simulation Study
True Model (MT)
Missing Crossloading (M1)
Missing Crossloading (M2)
Missing Crossloading (M3)
Correlated errors Replace
Crossloadings (M4)
Over-specified Model (M5)
Over-specified Model (M6)
Over-specified Model (M7)
Extra Latent Variable (M8)
Missing Latent Variable
Wrong Structural Model
Wrong Structural Model
Wrong Structural Model
Missing Indicator
Switched Loadings
Results
Conclusion
Bollen, Ray, Zavisca ASA, April 22 , 2005 - slide #31
Missing Indicator
Bayes Factor in SEM
Bayes factor in SEM
Model Fit in SEM’s
Approximating Bayes Factor
Simulation
Simulation Study
True Model (MT)
Missing Crossloading (M1)
Missing Crossloading (M2)
Missing Crossloading (M3)
Correlated errors Replace
Crossloadings (M4)
Over-specified Model (M5)
Over-specified Model (M6)
Over-specified Model (M7)
Extra Latent Variable (M8)
Missing Latent Variable
Wrong Structural Model
Wrong Structural Model
Wrong Structural Model
Missing Indicator
Switched Loadings
Results
Conclusion
Bollen, Ray, Zavisca ASA, April 22 , 2005 - slide #32
Switched Loadings
Bayes Factor in SEM
Bayes factor in SEM
Model Fit in SEM’s
Approximating Bayes Factor
Simulation
Results
Distn. (%) of Selected Models
Distn. (%) of Selected Models
% of Samples Selecting Over vs Under
Specified Models
Ranking of Models 1-3
Conclusion
Bollen, Ray, Zavisca ASA, April 22 , 2005 - slide #33
Distn. (%) of Selected Models
Bayes Factor in SEM
Bayes factor in SEM
Model Fit in SEM’s
Approximating Bayes Factor
Simulation
Results
Distn. (%) of Selected Models
Distn. (%) of Selected Models
% of Samples Selecting Over vs Under
Specified Models
Ranking of Models 1-3
Conclusion
Bollen, Ray, Zavisca ASA, April 22 , 2005 - slide #34
Distn. (%) of Selected Models
Bayes Factor in SEM
Bayes factor in SEM
Model Fit in SEM’s
Approximating Bayes Factor
Simulation
Results
Distn. (%) of Selected Models
Distn. (%) of Selected Models
% of Samples Selecting Over vs Under
Specified Models
Ranking of Models 1-3
Conclusion
Bollen, Ray, Zavisca ASA, April 22 , 2005 - slide #35
% of Samples Selecting Over vs Under Specified Models
Bayes Factor in SEM
Bayes factor in SEM
Model Fit in SEM’s
Approximating Bayes Factor
Simulation
Results
Distn. (%) of Selected Models
Distn. (%) of Selected Models
% of Samples Selecting Over vs Under
Specified Models
Ranking of Models 1-3
Conclusion
Bollen, Ray, Zavisca ASA, April 22 , 2005 - slide #36
Ranking of Models 1-3
Bayes Factor in SEM
Bayes factor in SEM
Model Fit in SEM’s
Approximating Bayes Factor
Simulation
Results
Conclusion
Concluding Remarks
Future Direction
Bollen, Ray, Zavisca ASA, April 22 , 2005 - slide #37
Concluding Remarks
■ BIC 6= ABF 6= Bayes Factor.
■ Other approximations are possible◆ ABF2, GBIC, Houghton’s BICR
■ PERFORMANCE: Small-scale simulation suggests◆ ABF performs better than BIC in small samples◆ ABF performs better than PVAL based conclusion in large samples◆ ABF performs as good or better than the best performance of BIC or
PVAL for all sample size.
Bayes Factor in SEM
Bayes factor in SEM
Model Fit in SEM’s
Approximating Bayes Factor
Simulation
Results
Conclusion
Concluding Remarks
Future Direction
Bollen, Ray, Zavisca ASA, April 22 , 2005 - slide #38
Future Direction
■ Study the sensitivity of Bayes Factors to priors.
■ Study the consistency property of ABF.
■ Choosing more flexible priors. (ABF 2, GBIC)
■ Do a large simulation study
Bayes Factor in SEM
Bayes factor in SEM
Model Fit in SEM’s
Approximating Bayes Factor
Simulation
Results
Conclusion
Acknowledgement
Acknowledgement
Bollen, Ray, Zavisca ASA, April 22 , 2005 - slide #39
Acknowledgement
We thank theSAMSI Model Averaging Group
. Slides Prepared by LATEX HA-prosper Package.
A Statisticians OverviewBayes factor in SEM MFBSEM Hypothesis testing Fit Indices Model Comparisons
ABFPrevious Work Bayes Factor (BF) Posterior Odds and BF Marginal Likelihood Laplace Approximation Laplace Approximation - MLE substitution BIC ABF 1 ABF 1 and BIC ABF 1 and BIC
SIMSimulation Study True Model (MT) Missing Crossloading (M1) Missing Crossloading (M2) Missing Crossloading (M3) Correlated errors Replace Crossloadings (M4) Over-specified Model (M5) Over-specified Model (M6) Over-specified Model (M7) Extra Latent Variable (M8) Missing Latent Variable Wrong Structural Model Wrong Structural Model Wrong Structural Model Missing Indicator Switched Loadings
RESDistn. (%) of Selected Models Distn. (%) of Selected Models % of Samples Selecting Over vs Under Specified Models Ranking of Models 1-3
CONCConcluding Remarks Future Direction
ACKAcknowledgement