Bayes Factors. Structural Equation Models (SEMs): Schwarz BIC and Other Approximations
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1 Bayes Factors in Structural Equation Models (SEMs): Schwarz BIC and Other Approximations Kenneth A. Bollen University of North Carolina, Chapel Hill Surajit Ray SAMSI and University of North Carolina, Chapel Hill Jane Zavisca SAMSI and University of Arizona Bollen, Ray, Zavisca SAMSI, May slide #1
2 Model Fit in SEMs Bayes Factor s Bollen, Ray, Zavisca SAMSI, May slide #2
3 SEM Latent Variable Model Measurement Model η = Bη + Γξ + ζ y = Λ y η + ε SEM Hypothesis testing Fit Indices Model Comparisons x = Λ x ξ + δ Bollen, Ray, Zavisca SAMSI, May slide #3
4 Hypothesis testing Chi square Test Statistic H 0 : Σ = Σ(θ) T = (N 1)F ML χ 2 in large samples excess power when big N excess kurtosis influence on T exact H 0, approximate model SEM Hypothesis testing Fit Indices Model Comparisons Bollen, Ray, Zavisca SAMSI, May slide #4
5 Fit Indices RMSEA, CFI, TLI, IFI, Etc. Cutoff values? Nonnested models? Small N issues? Behavior across estimators? SEM Hypothesis testing Fit Indices Model Comparisons Bollen, Ray, Zavisca SAMSI, May slide #5
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 SEM Hypothesis testing Fit Indices Model Comparisons Bollen, Ray, Zavisca SAMSI, May slide #6
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) Previous Work Bayes Factor (BF) Posterior Odds and BF Marginal Likelihood Laplace Approximation Laplace Approximation - MLE substitution BIC ABF 1 Bollen, Ray, Zavisca SAMSI, May slide #7
8 Bayes Factor (BF) Y : Data M k : Model Bayes Theorem P(M 1 Y ) = P(Y M 1 )P(M 1 ) P(Y M 1 )P(M 1 ) + P(Y M 2 )P(M 2 ) Comparing Model M 1 and M 2 Choose the model with higher posterior probability. Previous Work Bayes Factor (BF) Posterior Odds and BF Marginal Likelihood Laplace Approximation Laplace Approximation - MLE substitution BIC ABF 1 P(M 1 Y ) P(M 2 Y ) = P(Y M 1) P(M 1 ) P(Y M 2 ) P(M 2 ) = Bayes Factor prior odds Bollen, Ray, Zavisca SAMSI, May slide #8
9 Posterior Odds and BF P(M 1 Y ) P(M 2 Y ) = P(Y M 1) P(Y M 2 ) If P(M 1 ) = P(M 2 ) then prior odds=1 Define Bayes Factor as P(M 1 ) P(M 2 ) = Posterior Odds = BF Previous Work Bayes Factor (BF) Posterior Odds and BF Marginal Likelihood Laplace Approximation Laplace Approximation - MLE substitution BIC ABF 1 BF 12 = P(Y M 1) P(Y M 2 ) Bollen, Ray, Zavisca SAMSI, May slide #9
10 Marginal Likelihood P(θ M k ): Prior Distribution of θ given the models M k. d k dimension of θ k In general the Marginal Likelihood is Z P(Y M k ) = θ k P(Y M k,θ k )P(θ k M k )dθ k If the pdf P(Y M k ) s are completely specified ( no free parameters). Bayes Factor= Likelihood ratio. Sensitivity to prior is more critical in calculation in BF than in other Bayesian Analysis. ( Raftery 1993) Previous Work Bayes Factor (BF) Posterior Odds and BF Marginal Likelihood Laplace Approximation Laplace Approximation - MLE substitution BIC ABF 1 Bollen, Ray, Zavisca SAMSI, May slide #10
11 Laplace Approximation P(Y M k ) = Z P(Y M k,θ k )P(θ k M k ) θ k }{{} dθ k Laplace Approximation on Likelihood Prior P(Y M k ) (2π) d k/2 Ĩ( θ k ) 1 2 P(y θ k,m k )P( θ k ) Error of approximation : O( 1 n ) Ĩ( θ k ) = d2 dθdθ log(p(y θ,m k)p(θ)) = θ= θ k d2 dθdθ l(θ) θ= θ k Previous Work Bayes Factor (BF) Posterior Odds and BF Marginal Likelihood Laplace Approximation Laplace Approximation - MLE substitution BIC ABF 1 θ not readily available from software outputs Bollen, Ray, Zavisca SAMSI, May slide #11
12 Laplace Approximation - MLE substitution Replacing θ by ˆθ (MLE) I(ˆθ k ) = Error: O( 1 n ) P(Y M k ) (2π) d k/2 I(ˆθ k ) 1 2 P(y ˆθ k,m k )P(ˆθ k ) d2 dθdθ log(p(y θ,m k)) = θk = θˆ k Less accurate than using θ Using E(I(ˆθ k )) in place of I(ˆθ k ) has error: O( 1 d2 dθdθ l(θ) θk = ˆ θ k n 1 2 ) Previous Work Bayes Factor (BF) Posterior Odds and BF Marginal Likelihood Laplace Approximation Laplace Approximation - MLE substitution BIC ABF 1 Bollen, Ray, Zavisca SAMSI, May slide #12
13 BIC P(Y M k ) (2π) d k/2 I(ˆθ k ) 1 2 P(y ˆθ k,m k )P(ˆθ k ) Previous Work Choosing Unit Information prior i.e. P(θ k ) N θ ok, [ ] 1 I(ˆθ k ) n Bayes Factor (BF) Posterior Odds and BF Marginal Likelihood Laplace Approximation Laplace Approximation - MLE substitution BIC ABF 1 = P(Y M k ) e l(ˆθ k y) (n) d k/2 = 2logP(Y M k ) 2l(ˆθ k y) d k log (n) 2logBF = BIC Bollen, Ray, Zavisca SAMSI, May slide #13
14 ABF 1 P(Y M k ) = Z P(Y M k,θ k ) θ k }{{} Likelihood Laplace Approximation only on P(y ˆθ k,m k ) Prior P(θ k M k ) N Using the c to maximize P(Y M k ) P(θ k M k ) }{{} dθ k Prior [ ] 1 θ 0 k, c I(ˆθ k ) n Previous Work Bayes Factor (BF) Posterior Odds and BF Marginal Likelihood Laplace Approximation Laplace Approximation - MLE substitution BIC ABF 1 Bollen, Ray, Zavisca SAMSI, May slide #14
15 Using the c 2logP(Y M k ) 2l(θ k ) d k (1 + log [ = ABF 1 = BIC d 1 log [ d 1 θˆ T I( θˆ 1 ) 1 ˆ n θ 1 d k T θˆ k I( θˆ k ) θˆ k ] [ + d 2 log ]) d 2 θˆ T I( θˆ 2 ) 2 ˆ n θ 2 ] Previous Work Bayes Factor (BF) Posterior Odds and BF Marginal Likelihood Laplace Approximation Laplace Approximation - MLE substitution BIC ABF 1 Bollen, Ray, Zavisca SAMSI, May slide #15
16 BIC ABF 1 Prior implicit Yes Yes: has more flexibility than the unit information prior Uses standard software output Yes Yes Need for defining n. a Yes No: Enters through a See Raftery 1993 and 1995 on uncertainty about n in SEM s T θˆ k I( θˆ k ) θˆ k Previous Work Bayes Factor (BF) Posterior Odds and BF Marginal Likelihood Laplace Approximation Laplace Approximation - MLE substitution BIC ABF 1 Bollen, Ray, Zavisca SAMSI, May slide #16
17 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 still needs to be done. We fit a variety of under- and over-specified models. For further details on the simulated data, see Paxton et Monte Carlo Experiments Structural Equation Modeling, 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 Missing Indicator Switched Loadings Bollen, Ray, Zavisca SAMSI, May slide #17
18 True Model (MT) 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 Missing Indicator Switched Loadings Bollen, Ray, Zavisca SAMSI, May slide #18
19 Missing Crossloading (M1) 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 Missing Indicator Switched Loadings Bollen, Ray, Zavisca SAMSI, May slide #19
20 Missing Crossloading (M2) 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 Missing Indicator Switched Loadings Bollen, Ray, Zavisca SAMSI, May slide #20
21 Missing Crossloading (M3) 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 Missing Indicator Switched Loadings Bollen, Ray, Zavisca SAMSI, May slide #21
22 Correlated errors Replace Crossloadings (M4) 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 Missing Indicator Switched Loadings Bollen, Ray, Zavisca SAMSI, May slide #22
23 Over-specified Model (M5) 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 Missing Indicator Switched Loadings Bollen, Ray, Zavisca SAMSI, May slide #23
24 Over-specified Model (M6) 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 Missing Indicator Switched Loadings Bollen, Ray, Zavisca SAMSI, May slide #24
25 Over-specified Model (M7) 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 Missing Indicator Switched Loadings Bollen, Ray, Zavisca SAMSI, May slide #25
26 Extra Latent Variable (M8) 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 Missing Indicator Switched Loadings Bollen, Ray, Zavisca SAMSI, May slide #26
27 Missing Latent Variable 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 Missing Indicator Switched Loadings Bollen, Ray, Zavisca SAMSI, May slide #27
28 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 Missing Indicator Switched Loadings Bollen, Ray, Zavisca SAMSI, May slide #28
29 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 Missing Indicator Switched Loadings Bollen, Ray, Zavisca SAMSI, May slide #29
30 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 Missing Indicator Switched Loadings Bollen, Ray, Zavisca SAMSI, May slide #30
31 Missing Indicator 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 Missing Indicator Switched Loadings Bollen, Ray, Zavisca SAMSI, May slide #31
32 Switched Loadings 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 Missing Indicator Switched Loadings Bollen, Ray, Zavisca SAMSI, May slide #32
33 Distn. (%) of Selected Models Distn. (%) of Selected Models Distn. (%) of Selected Models % of Samples Selecting Over vs Under Specified Models Ranking of Models 1-3 Bollen, Ray, Zavisca SAMSI, May slide #33
34 Distn. (%) of Selected Models Distn. (%) of Selected Models Distn. (%) of Selected Models % of Samples Selecting Over vs Under Specified Models Ranking of Models 1-3 Bollen, Ray, Zavisca SAMSI, May slide #34
35 % of Samples Selecting Over vs Under Specified Models Distn. (%) of Selected Models Distn. (%) of Selected Models % of Samples Selecting Over vs Under Specified Models Ranking of Models 1-3 Bollen, Ray, Zavisca SAMSI, May slide #35
36 Ranking of Models 1-3 Distn. (%) of Selected Models Distn. (%) of Selected Models % of Samples Selecting Over vs Under Specified Models Ranking of Models 1-3 Bollen, Ray, Zavisca SAMSI, May slide #36
37 Concluding Remarks BIC ABF 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. Concluding Remarks Future Direction Bollen, Ray, Zavisca SAMSI, May slide #37
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 Concluding Remarks Future Direction Bollen, Ray, Zavisca SAMSI, May slide #38
39 Acknowledgement We thank the SAMSI Model Uncertainty Group Acknowledgement Acknowledgement. Slides Prepared by L A T E X HA-prosper Package. Bollen, Ray, Zavisca SAMSI, May slide #39
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