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I have a question regarding PPP when using the Bayesian estimator. In Muthén, B. & Asparouhov, T. (2011), it says low PPP indicates poor fit but also says a PPP around 0.5 indicates an excellentfitting model. However, in my analysis, I had a PPP about 0.99. Did this indicate a poor fit model or a good fit model? Thanks. 


Good fit. 

Rohini Sen posted on Thursday, May 03, 2012  7:20 am



If PPP is around 0.36, is that a very poor fit? 


This would indicate good fit. You are looking for values greater than .05. A value of .5 would indicate excellent fit. 

Phil Wood posted on Tuesday, November 27, 2012  8:35 am



Is it ever acceptable to use the ppp from a Bayesian analysis to compute a Bayes Factor between two models, or is it always preferable to use the BIC? (Assuming you have enough draws to believe it as a point estimator). I read a recent article by Meng in the Annals of Statistics which frowned on doing so, but it seems that argument is based on "using the same data twice," which we also did when using the BIC. Any thoughts from anyone? 


Phil I haven't seen any methodology on computing BF using PPP. PPP definition typically involves just 1 model while BF involves 2 models. Note also that PPP can be defined in many ways and the way it is defined in Mplus has nothing to do with what is in Meng's article. Mplus uses SEM style chi2 fit function. If you want to compare PPP and BF testing power and quality I would recommend looking at a simulation study. Tihomir 

Phil Wood posted on Tuesday, December 04, 2012  7:25 am



I had just meant dividing the PPP from one model by the PPP from another model. Looking at just a few calculations, it doesn't sem to work very well in practice relative to using, say, the BIC. Thanks for clearing up my confusion on Meng's article, though! 


Hello. I am running a secondorder factor model using both ML estimation and Bayesian estimation. I have 16 subscale scores and 16 subscale scores load on four factors and the four factors load on one higher factor. I found that Model fits from ML estimation seem to be good based on CFI(0.973) and RMSEA (0.044). However PPP from Bayesian estimation was very low. It was PPP=0.00. I don't understand why two estimation methods provide very different results in terms of model fit. Do you have any idea why this happens? (For the reference, I used default priors in bayesian estimation.) 


PPP is more related to chisquare than to CFI. What is your sample size, chisquare value, df, and pvalue? 


Dear Dr. Muthen, Sorry for the late response. I have a sample size of 2,200. I got a chisquare value of 689.058, df of 101, and p value of 0.000 using ML. Thanks a lot in advance for your help.



So the pvalue for the chisquare and the PPP agree as is expected. These fit statistics are less forgiving than CFI. So it isn't a matter of Bayes vs ML but a matter of which fit statistics you deem best. 


Dear Dr. Muthen, Thanks a lot for your answer. I have followingup questions. I prefer using other fit indices such as CFI and RMSEA than the chisquare when assessing the model fit. It's because the literature has demonstrated that the chisquare is sensitive to sample size. I wonder whether PPP is also sensitive to sample size or not. If so, I wouldn't want to use PPP to assess model fit. Then my another question is that is there any other model fit indices to assess model fit in Bayesian SEM? Thanks a lot for your help in advance.



I don't think one should be so quick to abandon chisquare (or PPP). Instead, try to investigate why the chisquare is not good enough. What's good about the Bayes approach is that you get an interval for the fit and when you relax the model you can see how the lower limit decreases, getting closer to zero (and into the negative). PPP is based on chisquare and therefore also has its power increase with increasing sample size. Mplus does not give any other Bayes fit measures. Note, again however that I would think you want to relax some restrictions in your model as discussed in Muthén, B. & Asparouhov, T. (2012). Bayesian SEM: A more flexible representation of substantive theory. Psychological Methods, 17, 313335. If you do so, some of your key results might change. 

anonymous Z posted on Tuesday, February 24, 2015  1:56 pm



Hi Dr. Muthen, I just started to use Bayes estimation, and I have two questions. 1. The output showed "THE MODEL ESTIMATION TERMINATED NORMALLY USE THE FBITERATIONS OPTION TO INCREASE THE NUMBER OF ITERATIONS BY A FACTOR OF AT LEAST TWO TO CHECK CONVERGENCE AND THAT THE PSR VALUE DOES NOT INCREASE." Is this a warning/error message? 2. model fit: I got a model fit as below.I assume it is good since p=0.515. But I don't understand the confidence interval info. Is it good or bad? 95% Confidence Interval for the Difference Between the Observed and the Replicated ChiSquare Values 18.234 17.604 Posterior Predictive PValue 0.515 Thank you very much! 


1. No, it's a suggestion for good analysis practice. 2. Read my into Bayes paper on our website. 


Hi, I've got a question concerning the interpretation of the output on PPP values in Monte Carlo Simulations using Bayesian estimation. The information given on the PPP value is a table with expected and observed proportions, however no percentiles. I understand that the PPP does not follow a known distribution, hence no percentiles? I was wondering how to interpret the expected proportions then. Is it correct if I assume that the expected proportions listed for the PPP are just the possible values of the PPP (and the observed column then gives me the cumulative frequency of values observed in the respective range) ? Thanks! 


The description of how to interpret that output is given on page 412 of the current user's guide. This is in Chapter 12 under the title Monte Carol Output. 


Unfortunately the info in the user guide doesn't exactly answer my question. On page 412 it says that "The column labeled Proportions Expected (column 1) should be understood in conjunction with the column labeled Percentiles Expected (column 3)." But there is no column with Percentiles in case of PPP values. Thanks! 


Please send the output and your license number to support@statmodel.com and I will look into this. 


Dear Linda and Bengt, Please can you help me understand why I get two identical PPpvalues and 95% C.I's in two separate models where the IV's are the same but the DV is different? There are no latent variables in either model and each model has the same number of parameters. Thanks for your time, Freya 


Please send the two outputs to Support along with your license number. 


Hi Bengt, I am using the universities student license which doesn't cover support. Is there any more information I can give to help you answer the question on this discussion page? Thank you, Freya 


This can happen if your DV versions are linear transformations of each other. Beyond that, we can only tell from the output and data. 


Can you provide references for the PPP as indicating good fit at 0.5? Also, references to better understand and interpret convergence of chisquare with the Bayes Predictive Probability chisquare? 


I found a brief discussion of PPPvalues in: Muthén, Bengt and Tihomir Asparouhov. 2012. “Bayesian Structural Equation Modeling: A More Flexible Representation of Substantive Theory.” Psychological Methods 17(3):313–35. However, more references, especially to the latter question, would be appreciated! 


The best ref is the 2014 Gelman et al Bayesian Data Analysis book, 3rd ed. 


Dear Muthen and Muthen, I have a 3 factor bayes cfa that is producing a PPP of .7 95% C.I = 87, 32 My interpretation is that a PPP of .7 would indicate good fit and would be equivalent to a PPP of .3 (if say the C.I was reversed to 32, 87) Would you agree? Thanks for your time, Freya 


Freya I would not quite agree with reversing the confidence limits. Either way the PPP indicates a good model fit and there is no evidence to reject the model. 


Great, Thanks Tihomir. 


Sorry Tihomir I just want to follow that up with another question because I think my confidence limit reversal example was a bad way to frame it... I have run models that have had a PPP of .5 where the confidence limits centre around zero. When I have had PPP's of approx. .1 the upper limits (e.g. 90) have usually been further from zero than the lower limit (e.g. 10) and when I have had PPP's of approx. .9 the lower limit has been much further from zero (e.g. 120) than the upper limit (e.g. 20)... So, if a PPP of >.05 indicates good fit and .5 excellent fit would it be correct to think that when the PPP increases past .5 fit starts to worsen (as reflected in the confidence limits)? If so does this mean a PPP >.05 and <.95 indicate good fit? Thanks again, Freya 


I would disagree. PPP should be interpreted similarly to how Pvalue is interpreted. If the SEM test of model fit Pvalue is >0.95 it is an unusually good fit and most likely a more restricted model will also fit well but I would not call this a poor fit. Note this is very specific to this PPP value that Mplus computes for SEM models using this particular discrepancy function which is the regular SEM chisquare. Almost all other PPP values used in the statistical literature that are based on different discrepancy function obey the logic that you describe  but not this one. Also consider this: asymptotically  with large samples and normal data the PPP value is almost identical to the pvalue. 


This now makes a lot of sense... I have imposed stricter priors on both crossloadings and residual covariance's and as you said, the model still fits very well. Thanks for this Tihomir! Freya 

Chi Hang Au posted on Sunday, August 06, 2017  1:27 pm



Hello, Is it possible to save out the posteriorpredictive distribution used to compute a ppp value as a separate file? Thank you! Chi 


No. 


Hi, I am estimating highly complex twolevel models with multiple mediators, some of which are categorical (binary and ordinal) variables. All mediation occurs at the first level. I use Bayesian estimation. I first estimated all paths (first and second stage of mediation) in separate models, which converged and had excellent to acceptable fit. When I estimated the full mediation model, it converged but had a terrible fit, and estimated effects deviated from those from the separate models. I included all covariances between the predictors at each level and all covariances between the mediators, so I cannot see the source of misfit. Model parameters: 444 free parameters, 95% Confidence Interval for the Difference Between the Observed and the Replicated ChiSquare Values 4137.377 4518.237, ppp < .001. I have also tried to estimate the second stage of mediation separately in order to use an online mediation calculator that multiplies the paths and uses bootstrap. To get categorical mediators to be treated as latent variables as at the first stage, I specified them as categorical variables. (However, they are actually predictors in this model as only one equation  second stage  is being estimated.) These models don't converge, no matter what I try, not even a simple covariance matrix. I am just wondering whether I miss some essential tip on how to estimate such models. Thanks in advance! 


We need to see the separate and full model outputs  send to Support along with your license number. 

Hazel B posted on Friday, April 20, 2018  4:12 pm



Dr. Muthen, I read some of your articles, but I am still wondering if the model fit can be determined based on the number of free parameters and 95% confidence interval. (Like when we use ML estimator, we sometimes conclude that the model fits the data well based on other values (e.g., CFI, TLI...), even if the pvalue is less than .001.) Number of free parameters: 119 (119 * 2 = 338) 95% confidence interval: 35.028  144.421 PPP: 0.001 What do you think about this model fit information? 

Hazel B posted on Saturday, April 21, 2018  1:29 pm



Nevermind. I understand that the positive lower limit indicates a poor fit. 


That is correct. If you are interested in establishing concepts such as "approximate fit" (which is what CFI, TLI do) you should look into BSEM and these three articles http://www.statmodel.com/download/BSEMFINAL10212011.pdf http://www.statmodel.com/download/BSEMRejoinder.pdf http://www.statmodel.com/download/PPPP.pdf 

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