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Greetings, I have two questions on ESEM. (1) In ESEM (EFA) tests of invariances (MG or longitudinal), is it possible to separately tests the invariance of factor variances and covariances ? I believe not. (2) I do not understand why factor variancecovariances cannot be constrained to equality in tests of longitudinal ESEM (EFA) invariance since it can be done in multigroup tests... It seems to have something to do with the defaults... In the baseline model, all factor variances are fixed to 1 and loadings are "free". Then, when loadings are constrained to equality, Time 2 variances are freed as the default. I tried to get the variances back to 1 after fixing the loadings to equality and it did not happen... Could it be a program bug ? Thank you very much 


Alexandre (1) You are correct. The factor variance/covariance is obtained from the optimal rotation which can be either entirely the same or entirely different only  no partial equality is available at this time. (2) Can you please send your example to support@statmodel.com and we will check for errors. Here is how to code such a model. I am modifying example 5 from the UG addendum MODEL: f1 BY y1y5*.8 y6y10*0 (*1 1); f2 BY y1y5*0 y6y10*.8 (*1 1); f1f2@1; f1 WITH f2*.5 (2); y1y10*1; [y1y10*1]; [f1f2@0]; MODEL g2: f1 BY y1y5*.8 y6y10*0 (*1 1); f2 BY y1y5*0 y6y10*.8 (*1 1); f1f2@1; f1 WITH f2*.5 (2); y1y10*2; [f1*.5 f2*.8]; 


Greetings Tihomir, Thanks for this answer. Yes, it does work in multigroup tests. It is in the context of "single group" tests of longitudinal invariance that I had problems but I guess that was due to the fact that I tried them separately. It now works. Thanks. Input below for those who are wondering. f1f2 BY SE1_1SE1_10 (*t1 1); f3f4 BY SE2_1SE2_10 (*t2 1); [SE1_1SE1_10] (1120); [SE2_1SE2_10] (1120); SE1_1SE1_10 (2130); SE2_1SE2_10 (2130); [f3 f4]; f3@1; f4@1; f1 WITH f2 (40); f3 WITH f4 (40); SE1_1SE1_10 PWITH SE2_1SE2_10; 


Greetings, If, as a logical extension of ESEM, I want to do a latent profile analysis but with indicators measured with an EFA model. This is close to a factor mixture analysis but I dont want the EFA factor model to vary across classes. I want only the factor means and variances (if possible) to vary from class to class. Would that be possible ? Would one or more classes be forced to have factor means fixed at 0 and factor variances fixed at 1 ? How would you specify such a model ? Thank you very much in advance. 


ESEM is not available for Mixture models yet so you cannot do this directly. You can do a twostage approach which serves as an approximate estimation for this model. In step 1 you would do the ESEM model and save the factor scores. In step 2 you can model these factor scores with a mixture model. 


Too bad, and thanks for the answer (will save me a lot of time)! But EFA factor mixture is avalaible, is it not ? But, if I understand correctly, only if the factor model varies across classes ? Best regards and have a nice week! 


Yes, mixture EFA gives classvarying factor loading matrices. You can do a mixture model where you specify an "EFA within a CFA". How to do an EFA within CFA is shown in the handout for The Mplus Short Course Topic 1 on our web site. 


Thank you ! Let me try a recap (just to make sure). (1) ESEM and Mixture models have not YET been combined (is it upcoming ?) (2) EFA mixture models are available but only when the full factor model vary across classes. (3) To do a "fully latent" latent profile model in which the latent mixture indicators are estimated by way of an EFA factor model that does not vary from one class to the other I would have to rely on the "EFA within a CFA" method and specifying that items loadings, intercepts and uniquenesses are class invariant. Up to this point it's alright. I guess that if the intercepts are fixed to equality, the factor means will be able to vary from one class to the other (will the means from one class have to be fixed to 0)? What about the factor variances ? Will they have to remain = to 1 in all classes ? or will I be able to free some of them ? In ESEM, variances and covariances are either simultanesouly freed or constrained to equality. Does this mean that if I want to estimate classvarying factor variances I will also have to estimate classvarying covariances even if one of the goal of latent profile is to "eliminate" these correlations ? Thanks again. 


1. Right (no comment; we like to produce pleasant surprises) 2. Yes, so a true EFA, exploring the factor loading pattern. 3. Answering your questions in turn: Right. Yes, right. You can let the factor variances also vary across classes, just make sure that the metric of the factors is set  so e.g. have no loadings fixed at 1 and have one class have the factor variances fixed at 1 and the other classes having them free. Regarding the last question, if you consider mixture EFA, you will by definition have withinclass variancecovariance, so you should let the covariances be free  and possibly different across classes. So the model is more a "Factor Mixture Model" than an LPA model. 


Thank you, And I just love pleasant surprises ! 


Hello, I am intrigued by what distinguishes EFA within CFA and Exploratory SEM. I'm only starting to read about these analyses, and one thing I came across is EFA within CFA seems to use 'anchor items' (no crossloading), whereas ESEM doesn't seem to. 1. Would you mind stating the main differences bw the two? 2. Is ESEM making EFA within CFA obsolete? Thank you very much in advance. 


Here are some preliminary answers and please let it not stop the Mplus team to answer (and to correct me). 1) ESEM incorporate EFA within an SEM framework. So you can do SEM (test relations between constructs, multigroup analyses) with factors identified on the basis of EFA. I believe the EFA part of ESEM is similar to traditional EFA. In ESEM you can test the invariance of the EFA factor model and you can also incorporate EFA and CFA factors at the same time (and even with the same items: bifactors/method factors). 2) I believe that the answer is a preliminary yes. It depends on how fast will the ESEM component of Mplus connect with its other components... My question (see previous posts here) was on fully latent mixture models with a fully invariant EFA factor model. I will let someone else answer the part about the anchor items. 


Greetings Dr Muthen, Let me get back to the last question of my [March 23, 2009  10:39 am ]. I'll be more precise. Suppose I have 40 items (i1i40) that I load on 5 factors (F1F5) through an EFA ("EFA within CFA" for the moment) method. And then that I load these factors on "C" (latent categorical) with an invariant EFA model (loadings, intercepts, uniquenesses). Normally, mixture models attempt to "explain" or "reproduce" the correlations between the mixture indicators (here f1f5). If I do an "EFA within a CFA", this is alright since after all it is a CFA. So I can freely (and separately) estimate class specific means (with 1 class fix to 0) and variances (with the same class fix to 1). This is possible since variances can be estimated separately from covariances in CFA (and then the mixture part will attempt to explain the covariance). But lets suppose that ESEM gets someday connected with mixture models. In EFA, variancescovariances are either free or constrained together (due to rotational issues). Do you believe that it will be possible to reconcile that with what I'm saying in the previous paragraph (about estimating class varying factor variances) ? 


Mixture modeling can be thought of as latent multiplegroup analysis, so therefore the multiplegroup features ESEM has now could be made available in mixtures as well. This includes studying how factor covariance matrices differ across groups/classes. 


Dorothee, regarding your question 2., I think "EFA in CFA" is now obsolete thanks to ESEM unless you are doing 2level or mixture EFA for which ESEM is not yet available. 


So I guess that with an orthogonal rotation, the factor correlations will all be "absorbed" by C and it will become possible to "disconnect" factor variances from covariances ? Am I clear ? I'm asking that since in MG ESEM, factor variances and covariances either vary or not together (across groups)(I'm refering to Tihomir post up there on MG ESEM). If my mixture indicators ARE factors (with an invariant measurement model), my goal would be to explain the correlations/covariances between the factors with C. In other words, I would like to really do a latent profile analysis (the classical type) with class varying indicators variances (and means) but with mixture indicators that are factors. It is now possible with the "EFA within CFA" (since in MG CFA, variances and covariances are separate parameters). From what you said, I'm not sure this will be possible trough future "mixture ESEM" exept maybe with orthogonal rotations... 


The factor covariance matrix pertains to withinclass variances nad covariances  that is, the varcovs left to explain beyond what C explains. With C influencing the factors there will be less factor variance and covariance to explain than without C, but there may still be variancecovariance in which case an oblique rotation may still be useful. 


Thank you! 


Greetings, I started to play with the "EFA within CFA" method and have two questions. (1) (a) In the context of a multigroup (lets say 5 groups) model, do I have to keep the factor variances fixed to 1 in all groups when the loadings are constrained to equality ? I believe I do. (b) However, do you have another idea on how to identify the multigroup model when the loadings are constrained to equality if I need to free the variances in the groups (exept the referent group)? (2) (a) This method ("EFA within CFA") seems to provide results equivalent to those from a rotated EFA model. Is that it ? (b) and if this is the case, to which form of rotation will it be equivalent ? 


See the Version 5.1 Examples and Language Addenda on the website with the user's guide and the topic Exploratory SEM on the website. 1. When factor loadings are equal, factor variances are fixed to one in one group and free in the others. 2. It is EFA. The default rotation is GEOMIN. The ROTATION option can be used to select other rotations. 


Hi Linda, I am not talking about EFA or ESEM here (I may have chosen the wrong post, sorry for that). I am talking about doing an "EFAWITHINCFA", as described in handout number 1 from slide 132 on. Bengt suggest this to me earlier in this same post to circumvent some of the current limitations of ESEM. 


With EFA in CFA, you get the same model fit as the EFA but the rotation will not be exactly the same. It depends on the choice of anchor items. When factor loadings are constrained, factor variances are fixed to one in one group and free in the others. 


Dear Mplus Team, I recently used EFA in a CFA framework ("E/CFA")to reorganise and revise a 29 item instrument into a 12 item instrument with a simple structure relating to 3 factors and was delighted at the ability to easily identify significant crossloadings. I used the latest version of Mplus which I recently purchased and didn't realise that use of E/CFA was no longer necessary due to the ESEM capability of Mplus. Am I right in thinking that anchor items are no longer necessary as the oblique rotations (Geomin/Quartimin) used provide the additional constraints required? Also, if you are only using the CFA/EFA part of the model (as in E/CFA) is that still strictly "ESEM" as there may not be a structural part as such or is it better thought of as E/CFA but using rotations rather than anchor items to provide the necessary constraints to allow model identification? Lastly, does the team consider the Bonferroni correction too stringent when assessing significance of Est/SEs? Surely each test of significance is unlikely to be truley independent. Can anyone suggest a rational way of "relaxing" the correction a little? I am keen to hear any thoughts on these issues. 


Greetings Paul, ESEM is nicely covered in the 2009 3rd issue of Structural Equation Modeling in two papers. Yes, I think ESEM can now replace "E/CFA" in most contexts. And yes, ESEM does not require anchor items. The Marsh et al. team (who wrote one of the ESEM 2009 papers) are currently playing at streching the capabilities of ESEM which may not be the most efficient method in ALL contexts (but still it is pretty generalisable). For instance, I do not beleive that ESEM and multilevel of mixture models have been merged yet. In those cases, "ECFA" is still better to work with crossloadings. Working with only the factor part of the model is not strickly ESEM, it is EFA. But, ESEM allow you to do multigroup EFA (and to conduct tests of invariances). For the predecessor to ESEM multigroup EFA, see the Dolan paper in issue 2 of 2009 Structural Equation Modeling. I prefer to let others answer the last part (bonferroni) of your questions. 


Thanks Alexandre. I have just read the two excellent papers I think you were referring to ESEM does look very promising as an approach. Interestingly I reworked my data using the new method (EFA in an "ESEM context"). As I was working with 3 and 4 factor models I tried using the Target rotation, rather than the Geomin/Quartimin. The 3 factor model suited the instrument design better than the four (the 4th factor had only one "pure item"). The result was I achieved a better fitting model (CFI/TLI both .99) and the output had suggested more MIs/revisions than when I had done the E/CFA. However, the resulting instrument was slightly different than when guided by the earlier method, and one subscale only ended up with two items. As such it didn't seem quite as practical as the instrument revision guided by the E/CFA, even if the fit was slightly poorer (CFI.98 etc). I did wonder whether E/CFA still has a place, particularly when looking at data where there are likely to be numerous crossloadings and if you don't need to bring in structural parts of a model. No doubt as people try out the ESEM approach more we will understand the advantages and limitations. I would be interested to hear if anyone else has had experience of using both approaches to evaluate a measurement model and which they preferred? 


Hi, In my experiences, ECFA and ESEM factor extraction (particularly target) alwasy yielded similar results (are you sure you used a similar specification both methods, with oblique or orthogonal rotation). And, when you look solely at the factor model, ESEM is only a plain old EFA. And, as in regular EFA, extraction and rotation method may produce different results. Just to make sure we are saying the same thing, by "ECFA" you are refering to the method described in EFAWITHINCFA", as described in the Mplus handout number 1 from slide 132 on ? The main difference between both approaches is that in ECFA you arbitrarily decide on anchor items with no crossloadings. In EFAESEM, you freely estimate all loadings. This may result in differing results. Personally, when I need to use ECFA to go where ESEM does not go yet, I always start from a ESEMEFA and then use to solution to decide which anchor items to use in the ECFA model. I believe you can also fix the ECFA anchor items cross loadings to their "real" values (as identified in the EFAESEM model) rather than to 0. This should give you identical results for both approaches. 


ESEM essentially replaces EFAwithinaCFA framework when a standard EFA model is considered. EFAwithinaCFA framework was designed to provide 2 things that conventional EFA didn't: SEs for rotated loadings and modification indices for residual covariances. Both of those features are now available in an ESEM EFA. In addition, ESEM EFA avoids the somewhat tedious work in EFAwithinaCFA framework of finding anchor items, doing the right msquare fixing of parameters, and setting starting values. Bonferroni adjustments can be valuable and are discussed in the excellent article on the benefits of using SEs for rotated loadings: Cudeck, R. & O’Dell, L.L. (1994). Applications of standard error estimates in unrestricted factor analysis: Significance tests for factor loadings and correlations. Psychological Bulletin, 115, 475487. EFAwithinaCFA essentially forces you to choose a certain rotation by your choice of where the zero factor loadings should be. So in a sense you have more direct influence on the rotation. But if you want to have such influence, probably a better way is to use the ESEM "target rotation" approach. 


Thanks Alexandre and Bengt, Yes, I was referring to efa in a cfa framework. The results differed very little and I think could easily have been explained by the different rotations. I will have a look at the recommended paper on corrections. I also liked the suggestions about using an esem approach to select and fix anchor items. I haven't come across any papers relating to selecting starting values. Does clear guidance exist? I certainly agree with prof muthen that the esem approach is quicker! I am a fairly new user of mplus but have been incredibly impressed by how much can be achieved with such little syntax and I think the esem capability will prove a real advance. Thanks and keep up the great work! 


EFAwithinCFA requires starting values and guidance is given in our Topic 1 handout at: http://www.statmodel.com/newhandouts.shtml ESEM EFA does not require starting values. The Mplus team is appreciative for the encouragement. 


Greetings, I am doing a multiple group (based on gender) ESEM model with two different ESEM sets of factors (lets suppose 30 variables load on 3 ESEM factors and another set of 30 variables load on three other ESEM factors) and using the first set of ESEM factors (F1F3) to predict the second set (F4F6). I want to test the invariance of the prediction and the models are already specified as having strong invariance at the measurment level so I know that F1 in group 1 is identically defined than F1 in group 2. The question: to I need to constrain ALL regression paths to invariance simultaneously (F4 on F1; F5 on F1; F6 on F1; F4 on F2; etc.) or can I constrain them to invariance one at a time. I know that in ESEM the full variancecovariance matrix needs to be free or constrained at the same time but I am not sure in this case given the fact that these are different ESEM sets. Thanks 


I think all coefficients for a set of factors need to work in tandem because of the rotation. Try it and you will find out for sure. 

nanda mooij posted on Friday, October 08, 2010  7:38 am



Dear Drs. Muthen, I'm wondering if an ESEM analysis is right for my problem. I have a 3factor model and I want to see if a morefactor model fits better. So I want to do an EFA on the residual matrix (difference between basemodel and 3factor model) of the 3factor model to see if the residuals show high factor loadings on other than the 3 factors. But I can only do an EFA on a correlation matrix, so I thought maybe I can do ESEM? So I will define the 3 factor model as it is(SEM), and then I let all the items load on factor 4 and 5 (EFA). Does this makes sense? Thanks in regard, Nanda 


I would start with a regular EFA for 3, 4, and 5 factors asking for modification indices. I would see how these look for residual correlations and use this information to determine how to specify the ESEM model, for example, which residual correlations to include. 

fritz posted on Tuesday, March 01, 2011  9:55 am



Dear all, this is kind of a basic question. Would you recommend ESEM to decide on the correct number of factors for example by testing a 5 factor solution against a 6 factor solution? I'm particularly interested in the comparison of a 5 vs. 6 factor model. Usually I would have used CFA for this question, but as you state in your ESEM papers, the CFA fit is not very well and modification indices suggest many crossloadings. So I thought, ESEM might be a more appropriate way of testing my models. Does it make sense to use ESEM in such a case? Thanks in advance! 


Using ESEM alone is the same as EFA and can be used for this purpose. 

Daniel posted on Wednesday, March 02, 2011  6:10 am



Greetings, I am new to Mplus and ESEM. I am trying to apply ESEM to distinguish among sociological constructs and test measurement invariance across countries. For the analysis, I am replicating the approach used in Marsh et al (2009) integrating CFA and EFA. The authors appear to use EFA to obtain the goodness of fit for the total ESEM (Appendix B). My question is: why do I obtain different goodness of fit indices with EFA and ESEM? Loadings and factor correlations are the same. My models are: EFA Variables: Usevariables q1q14; Analysis: type= efa 3 3; ESEM Variables: Usevariables q1q14; Model: f1f3 BY q1q14 (*1); Perhaps you can refer me to related literature. Thank you very much in advance. 


You should get identical results including fit statistics. Perhaps you are not using the same estimator for both analyses. If you want further help on this, please send the two outputs and your license number to support@statmodel.com. You may find the following paper which Marsh et al. based their paper on of interest: Asparouhov, T. & Muthén, B. (2009). Exploratory structural equation modeling. Structural Equation Modeling, 16, 397438. It is available on the website. 

Jing Jin posted on Monday, March 07, 2011  6:15 am



Hi, I have a question about fixing item loading in ESEM: in my analysis, one item has a factor loading > 1.00, I want to know how can I fix this problem? For example, what would be the code for constraining this particular item loading to .999? Thanks! 


Ask for the standardized solution. You will then be able to see if you have a small negative residual variance. If so, you can use MODEL CONSTRAINT to force the parameter to be greater than zero. 

Jing Jin posted on Monday, March 07, 2011  5:27 pm



Hi, I did find negative residual variance for that parameter. So I tried model constraint: Model: X (veX); MODEL CONSTRAINT: veX > 0; But then the model could not be identified. When I tried to make veX equal to a small positive number, the factor loading still couldn't be pulled down below 1. What might be the reason? Thanks! 


Send input, output, data, and license number to support. 

mari posted on Wednesday, May 04, 2011  8:35 am



Hello, I am running ESEM with 2 factors with 20 binary indicators. I have three binary covariates and one binary distal outcome. I have two questions. Q1. In my preliminary EFA analyses, 2 factor solution was supported. Also, in the ESEM output, indicators have high loadings on one factor and very little loadings on the other factors. But the factor correlation is very high (0.817). I am wondering how this can happen. Can I use 2factor solution even though they are highly correlated? My colleague mentioned a secondorder factor. Would that be a solution for my case? Q2. I have two types of standardized estimates (StdYX and STd). I have seen many posts about this, but still not sure about which estimate should be used for my paper. Would you please give me some suggestions? I appreciate your help. 


Q1. You could try a bifactor model with one general factor influencing all items and one specific factor (uncorrelated with the general) that influences only the items that loaded high on one factor. Q2. Look at the UG regarding standardization  that gives you the answers. 

mari posted on Wednesday, May 04, 2011  11:20 am



Thank you so much for your quick response!! In my model, 7 indicators highly loaded on the first factor and 13 indicators highly loaded on the second factor. Then, did you mean a model with one general factors for 20 indicators, one specific factor for 7 indicators, and one specific factor for 13 indicators? Actually, I have not seen a bifactor model. Do you have any good reference? 


Answers to your 2 paragraphs: That might work, but you may only need one general for all 20 and one specific for the first 7 (or for the last 13). See our Topic 1 handout. 


Hello. I want to test invariance of measure with categorical items across four groups using ESEM. For configural invariance (no constraints), I think I should: (1) Repeat the BY statement for all groups to relax the default equality constraint on factor loading matrices (2) Fix factor variances to one for all groups (default per my output) (3) Use bracket statements to relax the default equality constraint of intercepts / thresholds in groups 24 (4) Set scale factors for categorical variables to one in groups 24 when thresholds and factor loadings are estimated (5) Fix factor means to zero across all groups to override default of factor means being fixed at zero in group 1 and being free in groups 24. (Note: I did this and model fit was poor when compared with a no group ESEM I ran. Mod indices suggested freeing means, so I removed the "constraints to zero" command and model fit jumped back where expected.) Questions: (A) Is 15 correct for assessing basic configural similarity across groups (i.e., no acrossgroup invariance)? If not, what would be correct? (B) What would be next step for assessing invariance of factor loadings aside from removing groups 24 BY statements? 


(A) 15 look correct. The mod indices >0 that you mention in (5) should not happen, so something is amiss. If you can't see it, send to support with the output showing the nonzero MIs. (B) Here you don't want to fix factor variances to one for all groups. By default, the variances are fixed to one in the first group and the remaining groups have a free factor covariance matrix. 


Thank you, Bengt, for addressing my (A) and (B) questions. Linda was able to help me correct my syntax for thresholds to fix my issue; many thanks to her. If I may confirm next steps for testing invariance w categorial indicators ... Step 2: Factor Loading Invariance (1) Remove BY statements in remaining groups (2) Free factor variances in remaining groups Step 3: Factor Variance/Covariance Invariance (1) Fix factor covariances to equality across groups (2) Fix factor variances to one across groups Step 4: Test Invariance of Intercepts (1) Remove bracket statements from remaining groups (2) Free scale factors in remaining groups (3) Free factor means in remaining groups Step 5: Factor Mean Invariance (1) Fix factor means to zero across all groups. Sorry if I need to restate what is obvious to others to be sure I understand ... Many thanks. 


The models we recommend to test measurement invariance of categorical indicators are shown on pages 433434 of the Version 6 User's Guide. The inputs are shown under multiple group analysis in the Topic 2 course handout. Testing of structural parameters is shown under multiple group analysis of the Topic 1 course handout. This is the same for continuous or categorical indicators. 


Thank you, Linda. Is the above all true for ESEM? I am perceiving differences between examples provided above and 5.27, to which I was originally told to refer. 433434 suggests freeing / fixing factor loadings and thresholds in tandem other examples indicate intercepts being fixed after loadings. I have mostly categorical variables (and, therefore, thresholds) but I do have a few continuous variables (and, therefore, intercepts). I could really use a clear, single, stepbystep statement of what to do for multiple group ESEM with some categorical and some continuous indicators. Thanks for your patience. 


Example 5.27 can be generalized to the categorical case by referring to thresholds and scale factors for categorical variables in the first step rather than intercepts. If you have a combination of categorical and continuous, you use thresholds and scale factors for categorical and intercept for continuous. So, for example, in Example 5.27 you have the statement [y1y10]: For binary categorical, it would be replaced by [y1$1y10$1]; {y1y10@1}; 


Yes, thank you. So, do I then fix thresholds to equality at the same time I fix intercepts to equality when testing invariance? 


Yes. 


Hi, In Asparouhov & Muthén (2009) it is argued: "Furthermore, misspecification of zero loadings in CFA tends to give distorted factors. When nonzero crossloadings are specified as zero, the correlation between factor indicators representing different factors is forced to go through their main factors only, usually leading to overestimated factor correlations and subsequent distorted structural relations." I think I have an example of the opposite: higher factor correlations in SEM than in CFA. I compared a 5factor model (using WLSMV) for ordered categorical items. My interpretation would be that when in reality the hypothesized factors are substantially correlated, then the derived factors in SEM are a more realistic representation of the 'true' factors (thanks to the allowance of crossloadings). In this case, the crossloadings cause the factors to resemble each other more than in CFA. Would that be a correct interpretation, or am I missing something? thanks for your advice! Wim 


I don't think this is the case. If you want to send an example that shows this, we can take a look at it. 


Quick question....I ran both ECFA and ESEM. In the latter I specified Oblimin rotation. In the former, I chose anchors based on a Promax rotation EFA in SPSS (those with the highest pattern coefficient). What is somewhat odd is that the ESEM approach I specified provides significant estimates for the conceptually relevant items on their factors...while in the ECFA approach, most of the items I expect to provide significant estimates on a given factor do not do so. I am wondering if this is because, despite the choice of anchors in the CFA, the rotation is not optimized to provide the best simple structure? 


Just a note to disregard the abovethe results are basically the same [ECFA/ESEM]...I detected an incorrect anchor specification in a line of my syntax (C20@0 instead of C21@0)...this made everything better Apologies. 


Actuallyone conceptual question that I am not clear on: In the ECFA approach, how does choosing the anchor items actually affect the rotation process? 


In EFA within CFA your choice of anchor items and fixed zero loadings define a certain rotation  no further rotation is done because the m^2 restrictions have been determined (a rotation imposes its own m^2 restrictions). 


Thanks. So would it be correct to say that a drawback of the ECFA approach is by having to choose specific anchors you can not have as much control over the rotation method as you would otherwise normally would? When I compare the results of my ECFA with my ESEM, it does appear that the same crossloaders show up...so that is consistent across the approaches... 


You have more control in ECFA in a way because you can come up with a lot of ways of applying the EFA m^2 restrictions  but these ways may be inferior to those provided by the various rotation methods, since they automate the implementation of good criteria for simplicity. But I would typically use EFA/ESEM over ECFA. Apart from some special uses, I think ECFA was mostly an intermediate step in the evolution, getting SEs and MIs. 


So in this sense, would setting the anchor items in ECFA based on the highest loading from a Promax rotation be equivalent to setting that rotation (in the ECFA) to Promax? 


It isn't the anchor item choice that is the important one  that merely has to do with convergence ease  but instead where you put the loadings fixed at zero. That's what determines the "rotation". So the closer to zero the suitable loadings of Promax are, the more ECFA will agree with that rotation. 

EFried posted on Wednesday, September 12, 2012  1:33 pm



Is it possible to change rotation method and estimator in ESEM  like this (ys are categorical with 3 thresholds) : USEVAR = s1s16; CATEGORICAL = s1s16; ANALYSIS: !estimator = MLR; !rotation = promax; MODEL: F1F4 by s1s16 (*1); When I try rotation=promax I receive the error that this is only available for type=EFA (I thought the (*) after the ESEM specification is telling MPLUS that this is a EFA). When I try estimator=MLR I receive the message that " EFA factors are not allowed with ALGORITHM = INTEGRATION. EFA factors are declared with (*label)." Thanks 


Yes, you can change the rotation method in ESEM. The PROMAX and VARIMAX rotations are not allowed with ESEM. The other rotations are. ESEM is not available with maximum likelihood and categorical factor indicators. 

EFried posted on Thursday, September 13, 2012  12:10 pm



Thank you Linda. 


Hi, I have 40 items and 90 observations, I would identify the dimensionality of the instrument; is it correct to perform a multigroup ESEM where the groups are the data of the same subjects gathered at two time point? thank you very much 


No, you don't have independence across time points. You should instead use the Version 7 UG ex 5.27. 


Hi, I followed your instructions, and of course it worked! I am here to ask: of the 40 items 20 are positive emotions and 20 negative. I think that a ESEM with 40 items may be demanding for 90 observations. Do you think it could be feasable to conduct 2 esems (longitudinal) the first to test the dimensions of negative emotions (just including th 20 items) and the second to test the dimensions of positive emotions? 


This seems reasonable. 


Thank you very much, Andrea 

Xu, Man posted on Thursday, March 14, 2013  8:06 am



Hi Just to follow up Alex' post in 2009, I wondered whether it is now possible to run factor mixture analysis with EFA (or ESEM), and with predictors? I guess what I am asking is whether I can run the 3step analysis with a EFA in the LCA/LPA part of the model. Thanks! 


EFA mixture analysis is available, but not mixture ESEM so no predictors. 3step with EFA is not possible since it would involve mixture ESEM. 

Xu, Man posted on Thursday, March 14, 2013  12:04 pm



Thanks. I had a look at ex4.4 for efa mixture analysis. It seems that if I have no idea about the number of classes in the first place, I would need to get a few model runs changing number of classes each time, then choose the best fitting one from these combinations of models with different number of classes and number of factors. I am interested in using an external variable to predict class membership obtained from factor mixture model (maybe here one would need to either using CFA approach or EFAinCFA approach to maintain the results from factor mixture analysis). But factor ladings in EFA mixture analysis are not the same across classes  hence, I think this might indicate measurement noninvariance of the factors across classes. This is probably the whole point of factor mixture analysis, but in this case, does it make sense to look at the profiles of the classes in relation the factors, and on top of that, look at these class memberships in relation to an external predictor? 


When there is not measurement invariance across classes you don't want to make comparisons with respect to the factors in different classes. 


Dear Dr. Muthen, I am new with ESEM, but very interested in this method. Am I right that there is (so far) no option to set part of the factor loadings equal (by putting a defining number/statement in brackets)? For example: f1 by var1var5 (1) var6var10 (*1 2) Or is this option implemented in Mplus 7? Thanks in advance for your reply! Best regards, Isabel Thielmann 


No, this is not possible. 


Dear Dr Muthens, I would like to perform configural, metric and strong measurement invariance with two sets of ESEM factors simultaneously: e.g. f1f5 by i1i10(*1) f6f10 by i20i30(*2) The total model works fine but I receive an error that the latent variable covariance matrix is not positively definite when I try to perform any kind of MI with two sets of factors. I am familiar with multiple group ESEM with one set of factors and it has worked well for me in the past but I can't get rid of the errors for two set of factors. Kindly could you help? 


Please send the output and your license number to support@statmodel.com. 


I would like to combine results from an ESEM analysis (i.e. complex structure in factor loadings of 5 factors) with a separate latent interaction model (Note: other factors than those used in the ESEM part are supposed to be used in the interaction). To my knowledge the two issues – ESEM and latent interactions by using ML estimation – cannot be combined, so far. Would it be appropriate to, first, run the simpler model (without the interaction) and, in a second step, restrict factor loadings according to the unstandardized ESEM solution (complex structure) in order to finally add the latent interaction part (then using TYPE=RANDOM, ALGORITHM=INTEGRATION)? Thanks in advance for your reply! 


It would be a rough approximation with underestimated SEs. 

Eiko Fried posted on Tuesday, January 06, 2015  10:24 am



I am looking for example syntax for longitudinal ESEM measurement invariance in MPLUS, but haven't been able to find any so far (in this thread here, or in other sources such as the videos). Is anybody aware of tutorials or example syntax? Thank you very much. 


See Example 5.26. 

Eiko Fried posted on Wednesday, January 07, 2015  7:54 am



Thank you Linda. I have cat indicators (0,1,2,3), and adapted the 5.26 example using the information provided on p. 485 on measurement invariance for cat indicators (WLSMV; delta) that states thresholds and factor loadings should be contrained / relaxed in tandem. Does that mean there is no partial measurement invariance for such models because either both are relaxed or contrained? Is the syntax below correct: Unconstrained: F1F3 by H1_t1H17_t1 (*t1 1); F4F6 by H1_t2H17_t2 (*t2 2); H1_t1H17_t1 PWITH H1_t2H17_t2; Constrained : F1F3 by H1_t1H17_t1 (*t1 1); F4F6 by H1_t2H17_t2 (*t2 1); H1_t1H17_t1 PWITH H1_t2H17_t2; [H1_t1$1 H1_t1$2 H1_t1$3] (a); [H1_t2$1 H1_t2$2 H1_t2$3] (a); [H2_t1$1 H2_t1$2 H2_t1$3] (b); [H2_t2$1 H2_t2$2 H2_t2$3] (b); ! .. constrain all other thresholds here .. 


In regular CFA invariance settings, the binary case does not offer a chance to separately test threshold and loading invariance while at the same time allowing noninvariant residual variances. But in the polytomous case it is possible using a specialized setup  see the Millsap book Statistical Approaches to Measurement Invariance. But I am not sure if and how that translates to the EFA setting of ESEM. Note that your setups treat the residual variances as invariant and fixed to 1 since you don't mention them (Mplus default in a singlegroup setting). So in that case your "Unconstrained" case should work (be identified). Note that your unconstrained case still has loading invariance. You also want to try out your constrained setup. 

Eiko Fried posted on Monday, January 19, 2015  5:30 am



Thank you Bengt. 1. You say our unconstrained model has loading invariance, but (*t1 1) and (*t1 2) make Mplus estimate different loadings, no? Our output shows different loadings across time. 2. My second question pertains to the interpretation of the chisquare statistic using WSLVM for orderedcategorical variables in ESEM. Comparing invariant loadings to unconstrained baseline in one case decreases the chisquare from 2420 (df 439) to 2330 (df 481). How can that be? Also, we find a dramatic increase in chisquare when comparing loading to threshold invariance that looks enormous (baseline 5000 (df 1341), loadings invariant 5700 (df 1416), thresholds invariant 22000 (df 1493). Specification of the models are fine (Mplus support looked over them)  can such results be trusted or is the increase too enormous? 


1. You are right. 2. Please send relevant outputs and data to Support. Make sure you are using version 7.3. 

Eiko Fried posted on Monday, April 20, 2015  6:19 am



I have a brief question on longitudinal measurement invariance testing in Mplus, using the ESEM framework. We have 3 factors and 2 measurement points: F1F3 by H1_1H17_1 (*t1 1); F4F6 by H1_2H17_2 (*t2 2); H1_1H17_1 PWITH H1_2H17_2; This is the first model M1. Model 2 constrains loading to be equal, model 3 the thresholds. I cannot find the factor means in the output; how do I get them? Thank you 

Eiko Fried posted on Monday, April 20, 2015  6:30 am



I found the means in Tech4, apologies. My follow up question: Why are all factor means 0 in the above ESEM model? Thank you 


If you hold the factor loadings and intercepts equal across time, you can free the factor means at one time point. Otherwise, the factor means are fixed at zero for model identification. 

Eiko Fried posted on Monday, April 20, 2015  10:49 am



Thank you Linda. Do I understand you correctly that the Mplus default for ESEM models like the one described above is fixing factor means to zero, even in case loadings and intercepts/thresholds are _not_ constrained to be equal across time/groups, and that said factor means have to be freed manually? Because in the example above both loadings and thresholds are freely estimated and not constrained. 


If intercepts and loadings are not constrained to be equal, factor means must be zero. So if you want to free a factor mean, you must constrain the intercepts and factor loadings. This must be done manually. It is not possible for the program to know whether the factors are repeated measures of the same factor or three different factors so the default is for a crosssectional model. 

Eiko Fried posted on Tuesday, April 21, 2015  7:33 am



Thank you Linda. To make sure I got this right (categorical indicators, so we have thresholds and no intercepts): (1) In M1 (all free) and M2 (loadings invariant) factor means all must be 0 (and variances 1). (2) In M3 (threshold invariance), factor means (and with that, variances) of one time point can be freed (either t0 or t1) (3) What cannot be done is freeing all factor means (identification), or subsets of factors per timepoint (e.g., 2 of the 3 factors at baseline) 


See the Version 7.1 language addendum which is on the website with the user's guide. This document contains detailed descriptions about the models to use for testing measurement invariance in many different situations. Although this talks about groups, the same rules apply to time. 

Eiko Fried posted on Wednesday, April 22, 2015  3:39 am



Thank you. Last question: our scalar invariance model converges with 4 time points and 3 factors, with means@0 and variances@1 at baseline and freed at the 3 other time points, but we receive 2 identical Theta warnings (for same variable). Theta matrix looks ok, no negative residual variances in standardized rsquare output either. What could this be due to? 


Please send the output and your license number to support@statmodel.com. 


Hello! I would like to specify the MGI9 model from the Marsh2009 paper (http://www.statmodel.com/download/ESEM%20SETs%20Final.pdf). It imposes invariant factor loadings and intercepts, invariant item uniquenesses and equal factor (co)variances across groups. I tried to impose equality of factor variances, factor covariances and residual variances but I will get an error message saying "Improper parameter constraint for efa measurement specification". I would be very pleased if anyone could explain to me why I get this message. Thankyou very much! 


Please send the output and your license number to support@statmodel.com. 


ESEM analysis provides a number of estimates which show relationships between factors and are reported using "with" in the output. Two questions: 1 My understanding is that because the factors are standardized in ESEM, these should be considered factor correlations not factor covariances. Is my understanding true? 2 A factor correlation of .3 in esem means that the two factors have .3*.3 = 9% shared variance? many thanks in advance 


Point 1 is correct as long as the factors are not regressed on other variables. For 2  I would say the shared variance is 30%. If V(f1)=V(f2)=1 and Cov(f1,f2)=0.3 then f1=f0+e1 f2=f0+e2 where e1 e2 and f0 are uncorrelated and V(e1)=V(e2)=0.7, V(f0)=0.3. 

Cheng posted on Sunday, March 13, 2016  11:24 pm



In ESEM model, it is logical to introduce the convariances among the items' residuals (errors) if it is theoretically sound and its improve the fit indices? 


Sure. 


Hi, I'm estimating a bifactor ESEM model and one item has a negative residual variance and a standardized factor loading larger than 1.0. I've been trying to constrain the residual variance to be 0 or a positive value using the MODEL CONSTRAINT command but still get a negative residual variance. Is there any other way that I can try to deal with this issue? Or do I merely conclude that this is an inadmissible solution and respecify my model? Cheers, Andreas 


I would respecify the model. 


Greeting, Mrs. Bengt., I¡¯m a Chinese master student who is quite interested in ESEM. As you previously mentioned, the ESEM proceeds at first similarly to the traditional EFA, and then test the measurement invariance and structural invariance. The factor variances and covariances should be restricted at the same time. But in a guidebook about using m plus, in a multiple group analysis within a CFA framework, the factor variance invariance(reflect the item reliability) and the factor covariance invariance(reflect the factor correlation) can be tested separately. Question1: what¡¯s the difference of the two cases? Question2:The ESEM uses traditional EFA and test invariances, it is finally an EFA or CFA, or combination of the two? In the official m plus user¡¯s guidebook, the ESEM is classified in the CFA chapters. I need your confirmation. In my research, I want to use the parallel analysis, the MAP test and multiple group ESEM to decide on the number of latent variables. At this time, Question3:do I need to proceed the multiple group analysis again? However, after the previous steps, I want to know the differences on the factors within the 3 student groups, whether a LPA is recommended?(observed variables are continuous, with 4 points Likert scale) Any response is really appreciated! Thank you! MR.WEN 


Q1: With ESEM you can't test them separately. Q2. It's not classified as either EFA or CFA. Q3. Parallel analysis is available only for EFA. You can run them one group at a time. 


Thank you very much! 


Greetings, Mr. Tihomir, In your paper published in 2009, it is said that in ESEM, the loading matrix rotation gives a transformation of both measurement and structural coefficients. Extending the work summarized in Jennrich (2007), ESEM provides standard errors for all rotated parameters. I want to ask you, what does this transformation mean? In my opinion, I think it means not only the factor loadings are rotated as it was done in traditional EFA, but all the other parameters are also rotated. Therefore, ESEM can provides SE for all parameters. Please give me some light on this question. Thank you very much! 


Yes, the rotation transformation matrix for the loadings is applied to the other parameters as well. 


Thank you very much! 


Hello, MR. Tihomir and MR. Bengt! I have some questions about the details of using MG ESEM. In the paper ESEM published in 2009, you provided an empirical example and used the multiple group EFA to test the gender invariance. After examining the configural model and the model which restricted the factor loadings,the p value was not significant at the third step, which restricted the item intercepts. You freed the highest MI and went to the covariancevariance invariance step and obtained also a strong nonsignificant p value. I want to ask you, the output in Table 1 resulted from which step? From the factor loading step,the intercept step or the covariancevariance step? In my study, at the factor loading invariance testing step, I freed a high MI and tested with success the factor loading and item intercept invariance, but failed to reach the covariancevariance invariance. Which output I should use and present in a table? And another question. In my study, when testing the factor loading invariance, the output indicated that in group 3, the MI of Y2 WITH Y1(Y2,Y1 are observed variables) was 115. When I free this parameter, We should say "free the covariance between Y2 and Y1" or "free the correlation between Y2 and Y1"? In my opinion, when we standardize the items¡¯ covariance, it becomes the items¡¯ correlation. Thank you very much! 


This is from the paper “The Geominrotated solution for the model with invariant loadings, invariant intercepts except for break rules, and noninvariant factor covariance matrix is presented in Table 1.” You should present whatever model is most parsimonious and fits the data. if you free Y1 with Y2 calling it either covariance or correlation is fine and has the same meaning but in the output we present the covariance 


Hi again, For the first question,the reason why I ask this question is because when I restricted the variance and covariance invariance, the means are 0, variances are 1,we should use the results from previous step,but I don't know which step.The difference does exist between the results of the factor loading restriction step and intercept restriction step. In my study,the structures of factor loadings of different steps are similar, but they are not identical in values. Therefore, according to the "most parsimonious and fits the data"rule, the factor loading restriction step has the highest CFI, lowest RMSEA,SRMR,and less restricted structure. Should I use the output of the factor loading invariance step? Thank you! 


That seems reasonable. 

Kathy Xiao posted on Friday, February 03, 2017  3:17 am



I am quite new to ESEM, after I read the papers post in the Special Mplus Model. I am confused whether it is necessary for me to redo my multiple group analysis. I have a 12item scale, it was in a questionnaire as a scale, but there is no reference for its validity or whether there is subdimensions within this scale. Also, it has not been tested measurement invariance among racial groups. Previously I tested its measurement invariance followed the suggestions in the website, from groupspecific baseline model > configural > metric > scalar model, and I got my results. But it seems I haven't test the model fit for onefactor/twofactor/threefacotr model of this scale. Then I saw the application of ESEM on multiple group analysis. In this case, would you recommend me doing the ESEM to test the dimentionality (1/2/3factor model) and then doing multiple group ESEM? Thanks! 


I would do an EFA on each group separately to see if you find that the same number of factors are appropriate for each group. I would then proceed as described in the Topic 1 course video and handout on the website. 

Herb Marsh posted on Monday, July 03, 2017  5:43 pm



I am exploring ESEM with Target rotation with different values of "~.x" other than the default ~.0. With simulated data, sometimes when I specify ~.1 one of the factors is reversed (i.e., major factor loadings are negative). When this happens, the factor loading for the negatively oriented factor is always negatively biased (even after reversing the direction) and all the other factor loadings are positively biased. Is there any way to force all the ESEM factors to be positively oriented? 


One way would be to use 0.1 but that will work only for a single data set. In a simulation study you might have to save and split the data sets in two and use 0.1 for one and 0.1 in the other. Adding some additional targets would work as well but that will change the rotation. Using nonzero targets means that the negative orientation is not and equivalent solution, so it is not as simple as just keeping a positive orientation  the negative orientation simply gets a better target fit. 


Hi drs!, i have a question, can i use ESEM with interactions and use fit (test CFI, TLI, etc). My model have direct and interaction effects and is multilevel (only controlling by team). In ESEM model without interactions, my CFI, TLI, RMSEA are good but when i use path model normaly, aggregating the observable variables, my cfi is 0,00 (i think this is because i aggregate the oobs. var. and not multiply by the factor loading). can you help me? 


Send the 2 outputs to Support along with yor license number. 


Hi, I'm trying to test ESEM multigroup measurement invariance using binary outcome variables but I'm getting the following error message for Model M1 (according to the Topic 2 handout), the model with loading and intercept invariance: WARNING: THE RESIDUAL COVARIANCE MATRIX (THETA) IN GROUP G1 IS NOT POSITIVE DEFINITE. THIS COULD INDICATE A NEGATIVE VARIANCE/RESIDUAL VARIANCE FOR AN OBSERVED VARIABLE, A CORRELATION GREATER OR EQUAL TO ONE BETWEEN TWO OBSERVED VARIABLES, OR A LINEAR DEPENDENCY AMONG MORE THAN TWO OBSERVED VARIABLES. CHECK THE RESULTS SECTION FOR MORE INFORMATION. PROBLEM INVOLVING VARIABLE SCORE15. I am able to run the ESEM model on each group separately and get similar fit statistics. Is there a problem with my code? GROUPING IS group (1 = g1 2 = g2); Analysis: Estimator = WLSMV; Rotation = Geomin(Oblique, .5); MODEL: f1f5 BY score3score76 (*1); [f1f5@0]; MODEL g2: f1f5 BY score3score76(*1); [score3$1score76$1]; {score3score76@1}; MODEL: f1f5 BY score3score76 (*1); OUTPUT: TECH1 SAMPSTAT MODINDICES STANDARDIZED; Thanks, Steve 


The models are fine. The message is regarding a Heywood case which is not very unusual for these circumstances. You can modify the model to avoid it (if it appears important to you, most of the time these are not significant anyway). You can switching to the theta parameterization where you have more direct control of the residual variances. 


Thank you. 


Dear Tihomir / Linda /Bengt, my question is a little bit off topics, but your advice would be of help for me. In a ESEM bifactor model, I suppose that ALL estimated factor loadings of specific factors have to be used in calculating Omega, hierarchical Omega and ECV (explained common variance), not only those targeted in (target) rotation. Eg. when there are 26 items, there are all 26 factor loadings for any specific factor which have to be used. Thanks a lot in advance Martin 


That's right. 


Just another question: in fitting twotier model (2 general factors instead of 1 as in bifactor model), should these 2 general factors have multiple loadings and orthogonal (target) rotation as well? Or each item should have only 1 loading on its respective general factor as in CFA? Thanks a lot. 


The 2 general factors should have multiple items loading on them. They don't need to be orthogonal as long as they have some items that load on only one of these 2 factors. 

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