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Sample size 300.I have totally 5 latent variables with multiple indicators (continuous): 3 independent and 2 dependent. Both the IVs and DVs are answered by the same respondent, however with regards to three different firms i.e.) All the five variables are scales requesting the respondent to answer about Firm A, firm B and firm C. The survey looks like:
 Firm A  Firm B  Firm B  Factor 1     1. Rate satisfaction of service with (1 to 5):     2     3…     Factor 2     1. Rate you ability … (1 to 5)     2..     3..     Factor 5     1.. , 2… , 3….  For CFA, to account for the correlated residuals I used Bayesian SEM (BSEM) and have got robust fit indices and factor lodgings, Scales show very good fit. Can I check measurement invariance using BSEM webnote 17; however I don't have a grouping variable, so I can't use Type=mixture & knownclass? Please advice how to use BSEM Measurement invariance for single group. 


BSEM measurement invariance for a single group is like the longitudinal example in web note 17. 


Dear Prof. Muthen. Thank you very much. I am using the same setup as in the longitudinal example. however the model is not converging. These are the steps I am following: 1.) I am using correlated CFA model as per the BSEM 2012 article. The model worked very well. 2.) I am testing BSEM measurement invariance using approximate invariance. The model is not converging for iterations 50000 and 100000. Please advice. 


Dear Prof. Muthen: I am in touch with you on this through email. Kindly ignore the above doubt here. 


For my analysis (point 2 in my above post) I am still getting posterior probability to be 0. I first tried with DIFF variance of 0.1, then .01, and then .001. Still PPP is 0. I having three indicators with * in the difference output though. 


PPP of zero can have many causes. We need to see more details to give advice  please send input, output, data and license number to support. 


Hello. Aside from considering the benefit and meaning: I am wondering if it is possible to specify approximate measurement invariance across time and groups simultaneously in Mplus Bayesian CFA? I haven't found a way to expand the label assigning feature together with type=mixture, knownclass and the do/diff options to include both types of invariance. Is there any? 


I'll email an example to you. 


Dear Drs. Muthen, I have a question about prior distributions in testing approximate measurement invariance. In Muthen & Asparouhov article (2013), prior distributions for DIFFERENCES between parameters were used to test approximate measurement invariance. Loading1Loading2 ~ N(0,0.01) Because variance of 0.01 may represent different magnitude of variability of differences depending on scales of factors’ indicators, instead of priors for differences between parameters, I think that using priors for PROPORTIONS of two parameters can be used. So we may assign priors such as Loading1/Loading2 ~N(1, small variance) I tried to use this in Mplus, but Mplus gives error message. “Unknown parameter label: Loading1/Loading2” Q1. What do you think about using priors for proportions, not for differences in tests of approximate measurement invariance? Q2. I don’t know why Mplus recognizes "Loading1/Loading2" as a new label although it recognizes "Loading1Loading2" well. I also created new parameter for proportion (L1= Loading1/Loading2) under the model constraint and then assign a prior to new parameter (L1), but Mplus still gives error message (Unknown parameter label). Is there any way to use priors for proportions in this case? Thanks a lot in advance!!! Yoonjeong 


The scale of the variables does influence the prior variance choices as you say. The size of a loading is related to the SD of the variable. Also, different variables may have very different SDs. But if you are concerned about this, I think you could transform your variables to be on a more similar scale and then check the sensitivity to prior variances. If you want to work with ratios, you have to declare those parameters as "NEW" parameters in Model Constraint before applying priors to them. 


Dear Dr. Muthen, Thank you so much for your clarification! Yes. It would be one option to transform variables to be on a similar scale. I got it! To work with ratios, I followed your advice. I declared parameters as "New" parameters in model constraint and apply priors to those new parameters but it didn't work. Could you let me know if there is anything wrong in my code?(for simplicity, I didn't include priors for intercept terms) MODEL: %OVERALL% F1 BY X1X4; X1X4*; F1*;[F1*]; %CG#1% F1 BY X1X4(CG1X1CG1X4); [X1X4*](CG1IX1CG1IX4); X1X4*; F1@1;[F1@0]; %CG#2% F1 BY X1X4(CG2X1CG2X4); [X1X4*](CG2IX1CG2IX4); X1X4*; F1*;[F1*]; MODEL CONSTRAINT: NEW(LX1 LX2 LX3 LX4); LX1=CG1X1/CG2X1; LX2=CG1X2/CG2X2; LX3=CG1X3/CG2X3; LX4=CG1X4/CG2X4; MODEL PRIORS: LX1 ~ N(1,0.001); LX2 ~ N(1,0.001); LX3 ~ N(1,0.001); LX4 ~ N(1,0.001); *** ERROR in MODEL PRIORS command Unknown parameter label: LX1 


I forgot that we don't yet have the option of giving priors for NEW parameters. One approach is to fix one lambda to 1 and let others have mean 1 and smallvariance priors. 


Dear Dr. Muthen, When I took a look at the code in technical report regarding approximate measurement invariance, I realized that only a variance and mean of the reference group is fixed to 1 and 0. Variances and means for the other groups were freely estimated and also all parameters were freely estimated across groups. With Maximum likelihood, this is not sufficient for model identification. In multiple group analysis, we additionally need to constrain at least one factor loading and intercept to be equal across group. Q1. I wonder how the model is identified with Bayesian, particularly approximate measurement invariance case. My guess is that assigning strong prior distributions to the differences between factor loadings and intercepts with Bayesian works similarly to ML identification method (constraining at least one lambda and intercept). Could you let me know whether this is correct reasoning? Q2. If my guess is correct in Q1, with a two group CFA model with Bayesian estimation, I think that a model would be identified when (1) a variance and mean of the reference group is fixed to 1 and 0 , (2) and strong prior distributions are assigned to difference of only one factor loading and intercept (preferably of reference indicator). What do you think? 


Q1. Yes. If the prior is strong enough, you essentially have the case of exact invariance which we know leads to identification. Q2. Right. Partial invariance like this is possible in BSEM as well. 


Hi, Mplus team, Could you please point me at the example input file for the approximate measurement invariance test for categorical indicators? I would greatly appreciate your input. Thank you, Dmitriy 


Mplus Web Note 17 has that input. 


Hi I recently asked Linda M. for an advice on how to define "approximate" in approximate measurement invariance. I use Bayesian estimations for the alignment. Linda forwarded Tihomir's response:  I would define approximate invariance as: noninvariance which is not statistically significant or is too small to be of practical importance, i.e., the differences between the parameters are either not statistically significant or too small in magnitude to be of practical significance. Based on my experience with models using approximate measurement invariance, I still find it difficult to have a clear understanding of what is estimated to be "too smal in magnitude to be of practical significance" (as opposed to loadings that are estimated not to be approximately equal). Also, I don't think the reviewer of my paper would be satisfied with this explanation. Do Tihomir or Bengt have an explanation that is short/clear enough to be used in a paper? Kind regards, Christopher 


Christopher The full technical definition is given in the "INVARIANCE ANALYSIS" section, page 5, in http://statmodel.com/download/webnotes/webnote18.pdf Due to multiple testing and uncertainty about what constitutes practical significance (which of course is subjective and can vary from person to person and application to application), we have used a complicated definition based on multiple tests and low pvalues. Note however  that this is all a by product of the alignment estimation, and the estimation does not depend on that definition. You can choose to define it and use it in a different way. Our intent however was to define it as exactly that "noninvariance which is not statistically significant or is too small to be of practical importance". This discussion is very similar in nature to the discussion of "approximate fit" in SEM. On the other hand, it is conceivable that we need some precise cutoff values in standardized metric (something similar to how EFA cross loadings of less than 0.3 are often considered of lesser practical importance). That would be a good research paper. I don't have a value I can recommend at this point. Tihomir 


Tihomir, thanks a lot for the detailed answer! Keep up the good work. Best, Chris 

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