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Dear list, I have a question about the BSEM approach to CFA that was outlined in Muthén, B. O., & Asparouhov, T. (2012). Bayesian structural equation modeling: A more flexible representation of substantive theory. Psychological Methods, 17(3), 313–335. doi:10.1037/a0026802 Having thought a bit about this approach, it seems to me that there are two uses to it; 1) to test the hypothesis that parameters (e.g. crossloadings or residual covariances) are close to zero rather than exactly zero, and 2) to provide estimates for these parameters that can be used similarly to modification indices in ML estimation. My question regards the first (1) issue: If one wants to conduct a stringent test for this hypothesis, wouldn't one want to specify the model so that no parameter estimate is allowed outside the boundary of the specified range that is thought to be trivial? For example, if one thinks residual correlations + .20 are trivial, then one would want to estimate the model so that no parameter is estimated outside this range. Otherwise it seems hard to interpret model fit information? Best, Fredrik Falkenström 


If I understand your question correctly ... you are considering a prior distribution that doesn't have tails. This would be more readily accomplished for crossloadings possibly with uniform priors, however, I don't really see the problem with interpretation even with normal prior that essentially has the range of .2 to .2. If the posterior is out of that range the interpretation is clear  the parameter is nontrivial and the model does not fit. If the posterior is within that range you can interpret it as a trivial parameter that is approximately zero and is not a source of misfit. You can of course also use a uniform prior with a fixed range that does not allow the data to drive the parameter away and get a PPP value to determine the fit. 


Thanks! Fredrik 


Hello, I am running a BSEM which includes 2 latent factors and 12 continuous indicators. Small crossloadings and residual variance. Following the syntax. MODEL: T1 BY F8 F9 F12; T1 BY F1F7 F10 F11(O1O9); T2 BY F1 f2F6 F7 F10 F11; T2 BY F8F9 F12 (O10O12); f1f12(pp1pp12); f1f12 with f1f12 (y1y66); MODEL PRIOR: O1O12~N(0,0.01); pp1pp12~IW(1,27); y1y66~IW(0,27); I am not sure I specified the Inverse Wishart correctly, I'd appreciate your feedback. Thanks. 


Advice on how to use the Inverse Wishart is given in the paper on our website: Asparouhov, T., Muthén, B. & Morin, A. J. S. (2015). Bayesian structural equation modeling with crossloadings and residual covariances: Comments on Stromeyer et al. Journal of Management, 41, 15611577. 

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