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 Alexandre Morin posted on Friday, March 13, 2009 - 6:29 am

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 variance-covariances 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
 Tihomir Asparouhov posted on Friday, March 13, 2009 - 10:21 am

(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 and we will check for errors. Here is how to code such a model. I am modifying example 5 from the UG addendum

f1 BY y1-y5*.8 y6-y10*0 (*1 1);
f2 BY y1-y5*0 y6-y10*.8 (*1 1);
f1 WITH f2*.5 (2);
[y1-y10*1]; [f1-f2@0];

f1 BY y1-y5*.8 y6-y10*0 (*1 1);
f2 BY y1-y5*0 y6-y10*.8 (*1 1);
f1 WITH f2*.5 (2);
[f1*.5 f2*.8];
 Alexandre Morin posted on Friday, March 13, 2009 - 11:55 am
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.

f1-f2 BY SE1_1-SE1_10 (*t1 1);
f3-f4 BY SE2_1-SE2_10 (*t2 1);
[SE1_1-SE1_10] (11-20);
[SE2_1-SE2_10] (11-20);
SE1_1-SE1_10 (21-30);
SE2_1-SE2_10 (21-30);
[f3 f4];
f1 WITH f2 (40);
f3 WITH f4 (40);
SE1_1-SE1_10 PWITH SE2_1-SE2_10;
 Alexandre Morin posted on Friday, March 20, 2009 - 1:55 pm

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.
 Tihomir Asparouhov posted on Friday, March 20, 2009 - 3:16 pm
ESEM is not available for Mixture models yet so you cannot do this directly. You can do a two-stage 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.
 Alexandre Morin posted on Monday, March 23, 2009 - 4:40 am
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!
 Bengt O. Muthen posted on Monday, March 23, 2009 - 8:27 am
Yes, mixture EFA gives class-varying 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.
 Alexandre Morin posted on Monday, March 23, 2009 - 10:39 am
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 class-varying factor variances I will also have to estimate class-varying covariances even if one of the goal of latent profile is to "eliminate" these correlations ?

Thanks again.
 Bengt O. Muthen posted on Monday, March 23, 2009 - 4:31 pm
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:


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 within-class variance-covariance, 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.
 Alexandre Morin posted on Tuesday, March 24, 2009 - 4:20 am
Thank you,

And I just love pleasant surprises !
 Dorothee Durpoix posted on Tuesday, March 31, 2009 - 11:45 pm

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 cross-loading), 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.
 Alexandre Morin posted on Wednesday, April 01, 2009 - 4:37 am
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.
 Alexandre Morin posted on Wednesday, April 01, 2009 - 4:56 am
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 (i1-i40) that I load on 5 factors (F1-F5) 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 f1-f5).

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, variances-covariances 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) ?
 Bengt O. Muthen posted on Wednesday, April 01, 2009 - 10:23 am
Mixture modeling can be thought of as latent multiple-group analysis, so therefore the multiple-group features ESEM has now could be made available in mixtures as well. This includes studying how factor covariance matrices differ across groups/classes.
 Bengt O. Muthen posted on Wednesday, April 01, 2009 - 10:54 am
Dorothee, regarding your question 2., I think "EFA in CFA" is now obsolete thanks to ESEM unless you are doing 2-level or mixture EFA for which ESEM is not yet available.
 Alexandre Morin posted on Wednesday, April 01, 2009 - 11:07 am
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...
 Bengt O. Muthen posted on Thursday, April 02, 2009 - 10:34 am
The factor covariance matrix pertains to within-class variances nad covariances - that is, the var-covs 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 variance-covariance in which case an oblique rotation may still be useful.
 Alexandre Morin posted on Thursday, April 02, 2009 - 10:52 am
Thank you!
 Alexandre Morin posted on Wednesday, June 03, 2009 - 5:46 pm

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 ?
 Linda K. Muthen posted on Thursday, June 04, 2009 - 9:40 am
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.
 Alexandre Morin posted on Thursday, June 04, 2009 - 9:58 am
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 "EFA-WITHIN-CFA", 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.
 Linda K. Muthen posted on Thursday, June 04, 2009 - 10:30 am
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.
 Paul A.Tiffin posted on Friday, September 25, 2009 - 2:04 am
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.
 Alexandre Morin posted on Friday, September 25, 2009 - 6:19 am
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, "E-CFA" is still better to work with cross-loadings.

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 multi-group 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.
 Paul A.Tiffin posted on Friday, September 25, 2009 - 7:57 am
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?
 Alexandre Morin posted on Friday, September 25, 2009 - 8:32 am
In my experiences, E-CFA 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 "E-CFA" you are refering to the method described in EFA-WITHIN-CFA", as described in the Mplus handout number 1 from slide 132 on ?

The main difference between both approaches is that in E-CFA you arbitrarily decide on anchor items with no cross-loadings. In EFA-ESEM, you freely estimate all loadings. This may result in differing results. Personally, when I need to use E-CFA to go where ESEM does not go yet, I always start from a ESEM-EFA and then use to solution to decide which anchor items to use in the E-CFA model. I believe you can also fix the E-CFA anchor items cross loadings to their "real" values (as identified in the EFA-ESEM model) rather than to 0. This should give you identical results for both approaches.
 Bengt O. Muthen posted on Friday, September 25, 2009 - 9:17 am
ESEM essentially replaces EFA-within-a-CFA framework when a standard EFA model is considered. EFA-within-a-CFA 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 EFA-within-a-CFA framework of finding anchor items, doing the right m-square 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, 475-487.

EFA-within-a-CFA 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.
 Paul A.Tiffin posted on Friday, September 25, 2009 - 11:43 am
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!
 Bengt O. Muthen posted on Friday, September 25, 2009 - 11:51 am
EFA-within-CFA requires starting values and guidance is given in our Topic 1 handout at:

ESEM EFA does not require starting values.

The Mplus team is appreciative for the encouragement.
 Alexandre Morin posted on Wednesday, April 07, 2010 - 2:31 pm
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 (F1-F3) to predict the second set (F4-F6).
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 variance-covariance 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.
 Linda K. Muthen posted on Wednesday, April 07, 2010 - 4:31 pm
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 3-factor model and I want to see if a more-factor model fits better. So I want to do an EFA on the residual matrix (difference between base-model and 3-factor model) of the 3-factor 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,
 Linda K. Muthen posted on Friday, October 08, 2010 - 8:38 am
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 cross-loadings. 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!
 Linda K. Muthen posted on Tuesday, March 01, 2011 - 11:07 am
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

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:


Usevariables q1-q14;

type= efa 3 3;


Usevariables q1-q14;

f1-f3 BY q1-q14 (*1);

Perhaps you can refer me to related literature. Thank you very much in advance.
 Linda K. Muthen posted on Wednesday, March 02, 2011 - 9:11 am
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

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, 397-438.

It is available on the website.
 Jing Jin posted on Monday, March 07, 2011 - 6:15 am
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?

 Linda K. Muthen posted on Monday, March 07, 2011 - 9:29 am
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
I did find negative residual variance for that parameter. So I tried model constraint:

Model: X (veX);
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?
 Bengt O. Muthen posted on Monday, March 07, 2011 - 6:12 pm
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 2-factor solution even though they are highly correlated?

My colleague mentioned a second-order 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.
 Bengt O. Muthen posted on Wednesday, May 04, 2011 - 8:42 am
Q1. You could try a bi-factor 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 bi-factor model. Do you have any good reference?
 Bengt O. Muthen posted on Wednesday, May 04, 2011 - 8:29 pm
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.
 Stacey Farber posted on Monday, May 30, 2011 - 7:39 am

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 2-4
(4) Set scale factors for categorical variables to one in groups 2-4 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 2-4. (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.)

(A) Is 1-5 correct for assessing basic configural similarity across groups (i.e., no across-group invariance)? If not, what would be correct?

(B) What would be next step for assessing invariance of factor loadings aside from removing groups 2-4 BY statements?
 Bengt O. Muthen posted on Monday, May 30, 2011 - 8:49 am
(A) 1-5 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 non-zero 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.
 Stacey Farber posted on Saturday, June 04, 2011 - 11:50 am
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.
 Linda K. Muthen posted on Saturday, June 04, 2011 - 12:39 pm
The models we recommend to test measurement invariance of categorical indicators are shown on pages 433-434 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.
 Stacey Farber posted on Saturday, June 04, 2011 - 1:28 pm
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. 433-434 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, step-by-step statement of what to do for multiple group ESEM with some categorical and some continuous indicators. Thanks for your patience.
 Linda K. Muthen posted on Saturday, June 04, 2011 - 1:43 pm
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


For binary categorical, it would be replaced by

 Stacey Farber posted on Saturday, June 04, 2011 - 1:49 pm
Yes, thank you. So, do I then fix thresholds to equality at the same time I fix intercepts to equality when testing invariance?
 Linda K. Muthen posted on Saturday, June 04, 2011 - 4:28 pm
 Wim Van den Broeck posted on Saturday, December 24, 2011 - 2:48 am

In Asparouhov & Muthén (2009) it is argued: "Furthermore, misspecification of zero loadings in CFA tends to give distorted factors. When nonzero cross-loadings 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 5-factor 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 cross-loadings). In this case, the cross-loadings 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!
 Linda K. Muthen posted on Monday, December 26, 2011 - 8:38 am
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.
 J.D. Haltigan posted on Thursday, January 19, 2012 - 12:08 pm
Quick question....I ran both E-CFA 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 E-CFA 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?
 J.D. Haltigan posted on Thursday, January 19, 2012 - 1:57 pm
Just a note to disregard the above--the results are basically the same [E-CFA/ESEM]...I detected an incorrect anchor specification in a line of my syntax (C20@0 instead of C21@0)...this made everything better :-) Apologies.
 J.D. Haltigan posted on Thursday, January 19, 2012 - 2:32 pm
Actually--one conceptual question that I am not clear on: In the E-CFA approach, how does choosing the anchor items actually affect the rotation process?
 Bengt O. Muthen posted on Thursday, January 19, 2012 - 8:51 pm
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).
 J.D. Haltigan posted on Friday, January 20, 2012 - 2:13 pm
Thanks. So would it be correct to say that a drawback of the E-CFA 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 E-CFA with my E-SEM, it does appear that the same cross-loaders show that is consistent across the approaches...
 Bengt O. Muthen posted on Friday, January 20, 2012 - 5:26 pm
You have more control in E-CFA 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 E-CFA. Apart from some special uses, I think E-CFA was mostly an intermediate step in the evolution, getting SEs and MIs.
 J.D. Haltigan posted on Monday, January 23, 2012 - 2:32 pm
So in this sense, would setting the anchor items in E-CFA based on the highest loading from a Promax rotation be equivalent to setting that rotation (in the E-CFA) to Promax?
 Bengt O. Muthen posted on Monday, January 23, 2012 - 6:22 pm
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 E-CFA 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 = s1-s16;

!estimator = MLR;
!rotation = promax;

F1-F4 by s1-s16 (*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)."

 Linda K. Muthen posted on Thursday, September 13, 2012 - 10:45 am
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.
 Andrea Norcini Pala posted on Thursday, November 29, 2012 - 1:57 pm

I have 40 items and 90 observations, I would identify the dimensionality of the instrument; is it correct to perform a multigroup E-SEM where the groups are the data of the same subjects gathered at two time point?

thank you very much
 Bengt O. Muthen posted on Thursday, November 29, 2012 - 2:08 pm
No, you don't have independence across time points. You should instead use the Version 7 UG ex 5.27.
 Andrea Norcini Pala posted on Wednesday, December 05, 2012 - 12:56 pm

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?
 Linda K. Muthen posted on Wednesday, December 05, 2012 - 6:27 pm
This seems reasonable.
 Andrea Norcini Pala posted on Thursday, December 06, 2012 - 2:21 pm
Thank you very much,
 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 3-step analysis with a EFA in the LCA/LPA part of the model.

 Bengt O. Muthen posted on Thursday, March 14, 2013 - 8:45 am
EFA mixture analysis is available, but not mixture ESEM so no predictors.

3-step 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 EFA-in-CFA 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 non-invariance 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?
 Bengt O. Muthen posted on Thursday, March 14, 2013 - 4:12 pm
When there is not measurement invariance across classes you don't want to make comparisons with respect to the factors in different classes.
 Isabel Thielmann posted on Thursday, July 04, 2013 - 1:03 am
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 var1-var5 (1) var6-var10 (*1 2)

Or is this option implemented in Mplus 7?

Thanks in advance for your reply!

Best regards,
Isabel Thielmann
 Bengt O. Muthen posted on Thursday, July 04, 2013 - 1:35 pm
No, this is not possible.
 Arielle Bonneville-Roussy posted on Monday, November 11, 2013 - 3:32 am
Dear Dr Muthens,

I would like to perform configural, metric and strong measurement invariance with two sets of ESEM factors simultaneously:


f1-f5 by i1-i10(*1)
f6-f10 by i20-i30(*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?
 Linda K. Muthen posted on Monday, November 11, 2013 - 5:45 am
Please send the output and your license number to
 Julian Aichholzer posted on Wednesday, January 15, 2014 - 5:23 am
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!
 Bengt O. Muthen posted on Wednesday, January 15, 2014 - 10:33 am
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.
 Linda K. Muthen posted on Tuesday, January 06, 2015 - 12:02 pm
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:

F1-F3 by H1_t1-H17_t1 (*t1 1);
F4-F6 by H1_t2-H17_t2 (*t2 2);
H1_t1-H17_t1 PWITH H1_t2-H17_t2;

Constrained :
F1-F3 by H1_t1-H17_t1 (*t1 1);
F4-F6 by H1_t2-H17_t2 (*t2 1);
H1_t1-H17_t1 PWITH H1_t2-H17_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 ..
 Bengt O. Muthen posted on Wednesday, January 07, 2015 - 5:14 pm
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 non-invariant 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 single-group 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 chi-square statistic using WSLVM for ordered-categorical variables in ESEM. Comparing invariant loadings to unconstrained baseline in one case decreases the chi-square from 2420 (df 439) to 2330 (df 481). How can that be? Also, we find a dramatic increase in chi-square 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?
 Bengt O. Muthen posted on Monday, January 19, 2015 - 11:01 am
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:

F1-F3 by H1_1-H17_1 (*t1 1);
F4-F6 by H1_2-H17_2 (*t2 2);
H1_1-H17_1 PWITH H1_2-H17_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
 Linda K. Muthen posted on Monday, April 20, 2015 - 10:34 am
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.
 Linda K. Muthen posted on Monday, April 20, 2015 - 11:19 am
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 cross-sectional 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)
 Linda K. Muthen posted on Tuesday, April 21, 2015 - 12:24 pm
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 r-square output either. What could this be due to?
 Linda K. Muthen posted on Wednesday, April 22, 2015 - 6:01 am
Please send the output and your license number to
 Luisa Wiegand posted on Monday, June 08, 2015 - 8:44 am

I would like to specify the MGI9 model from the Marsh2009 paper ( 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!
 Linda K. Muthen posted on Monday, June 08, 2015 - 9:27 am
Please send the output and your license number to
 Ebrahim Hamedi posted on Monday, October 19, 2015 - 6:15 pm
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
 Tihomir Asparouhov posted on Tuesday, October 20, 2015 - 9:30 am
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
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?
 Bengt O. Muthen posted on Monday, March 14, 2016 - 5:35 pm
 Andreas Stenling posted on Wednesday, April 13, 2016 - 11:47 pm

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?

 Bengt O. Muthen posted on Thursday, April 14, 2016 - 6:17 pm
I would re-specify the model.
 WEN Congcong posted on Saturday, April 16, 2016 - 7:52 am
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!
 Tihomir Asparouhov posted on Monday, April 18, 2016 - 2:05 pm
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.
 WEN Congcong posted on Tuesday, April 19, 2016 - 4:54 am
Thank you very much!
 WEN Congcong posted on Friday, April 29, 2016 - 9:52 pm
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!
 Bengt O. Muthen posted on Saturday, April 30, 2016 - 6:03 am
Yes, the rotation transformation matrix for the loadings is applied to the other parameters as well.
 WEN Congcong posted on Monday, May 02, 2016 - 6:59 am
Thank you very much!
 WEN Congcong posted on Tuesday, May 31, 2016 - 8:21 pm
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 covariance-variance invariance step and obtained also a strong non-significant 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 covariance-variance 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 covariance-variance 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!
 Linda K. Muthen posted on Wednesday, June 01, 2016 - 4:16 pm
This is from the paper

“The Geomin-rotated 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
 WEN Congcong posted on Wednesday, June 01, 2016 - 5:42 pm
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!
 Bengt O. Muthen posted on Friday, June 03, 2016 - 5:37 pm
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 re-do my multiple group analysis.

I have a 12-item scale, it was in a questionnaire as a scale, but there is no reference for its validity or whether there is sub-dimensions 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 group-specific baseline model -> configural -> metric -> scalar model, and I got my results.

But it seems I haven't test the model fit for one-factor/two-factor/three-facotr 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-/3-factor model) and then doing multiple group ESEM?

 Linda K. Muthen posted on Friday, February 03, 2017 - 6:21 am
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?
 Tihomir Asparouhov posted on Tuesday, July 04, 2017 - 10:38 am
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 non-zero 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.
 giuliani coluccio posted on Wednesday, December 13, 2017 - 7:01 pm
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?
 Bengt O. Muthen posted on Thursday, December 14, 2017 - 12:28 pm
Send the 2 outputs to Support along with yor license number.
 Stephen Leach posted on Thursday, May 31, 2018 - 8:16 am

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:


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);

Estimator = WLSMV;
Rotation = Geomin(Oblique, .5);

f1-f5 BY score3-score76 (*1);

f1-f5 BY score3-score76(*1);

f1-f5 BY score3-score76 (*1);


 Tihomir Asparouhov posted on Thursday, May 31, 2018 - 3:22 pm
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.
 Stephen Leach posted on Wednesday, June 06, 2018 - 9:54 am
Thank you.
 Martin Kanovsky posted on Monday, June 11, 2018 - 2:14 am
Dear Tihomir / Linda /Bengt,

my question is a little bit off topics, but your advice would be of help for me. In a ESEM bi-factor 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
 Bengt O. Muthen posted on Monday, June 11, 2018 - 5:38 pm
That's right.
 Martin Kanovsky posted on Sunday, June 17, 2018 - 12:47 pm
Just another question: in fitting two-tier 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.
 Bengt O. Muthen posted on Monday, June 18, 2018 - 9:39 am
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.
 Heiko Breitsohl posted on Thursday, October 25, 2018 - 7:56 am
Dear Mplus Team,

I am trying to set up an exploratory factor model with scalar invariance constraints across ~100 groups.
My goal is to find the optimal factor structure for a given set of indicators that will exhibit scalar invariance.

I tried setting this up as a multilevel ESEM model with with the factor loadings constrained to equality across levels (and the between-level residuals constrained to zero), but Mplus refuses to run the model.

Is there a way (or workaround) of modeling this in Mplus?

Thanks so much!
 Bengt O. Muthen posted on Thursday, October 25, 2018 - 6:20 pm
We have to see exactly how you set it up to be able to tell. Please send your full output to Support along with your license number.
 Jenny posted on Wednesday, November 21, 2018 - 3:15 am

I'm a beginner with ESEM, having only used EFA and CFA before. I have 3 questions regarding ESEM:

1) Can you use ESEM instead of EFA to not only explore the latent structure in a pool of items but to reduce this initial pool at the very first stage of measure development? My pool of items is large (around 80 items) and I'd like this to be reduced as well as a factor structure indicated, as with EFA.

2) If you'd recommend ESEM for this initial stage of measure development, is there a recommended cut-off for item loadings (i.e. retaining items for factors above a certain loading)? Would you then rerun the ESEM with the smaller item pool (above a certain cut off) as you would do with EFA?

3) If you can use ESEM for initial measure development, what would be a good sequence of analyses? Would you recommend conducting ESEM then CFA in an independent sample to confirm the factor structure or would there be no need to conduct the follow up CFA?

Thanks for your help in advance - I haven't been able to find any guidelines on using ESEM in a measure development context.
 Bengt O. Muthen posted on Wednesday, November 21, 2018 - 2:30 pm
Note that ESEM is the same as EFA if you don't have other model parts such as covariates or multiple groups. So general EFA strategies should be used.
 Jenny posted on Thursday, November 22, 2018 - 12:01 am
Thank you Bengt - I'll use general EFA strategies with ESEM.

If I conduct ESEM do you know if there is any need to then confirm the factor structure in an independent sample using CFA (and cross-validate using CFA)? For measure development, would I just need to conduct ESEM?

 Bengt O. Muthen posted on Thursday, November 22, 2018 - 11:12 am
Those are general analysis strategy questions that you may want to get SEMNET input on; different researchers have different opinions.
 Rimantas Vosylis posted on Thursday, July 04, 2019 - 3:13 am
Dear Muthens,
I have a Bi-Factor ESEM model with target rotation that represents the factor structure of my six sbuscales. I want to test the longitudinal measurement invariance of such model.
Is there any example on how I can fix factor loadings equal across time for this solution?
Thank you!
Here is my syntax for the first wave:

ROTATION = TARGET (orthogonal);

fbns1 fbns7 fbns13 fbns19
fbpns20-fbpns24~0 (*1);

fbns2 fbns8 fbns14 fbns20 (*1);

fbns3 fbns9 fbns15 fbns21 (*1);

fbns4 fbns10 fbns16 fbns22 (*1);

fbns5 fbns11 fbns17 fbns23
fbns24~0 (*1);

fbns6 fbns12 fbns18 fbns24 (*1);

fbns1-fbns24 (*1);
 Tihomir Asparouhov posted on Friday, July 05, 2019 - 4:09 pm
Yes. See page 721 in the User's Guide.

f1-f2 BY y1-y5 (*1 1);
f3-f4 BY y6-y10 (*2 1);

holds the loadings equal for the two EFA blocks
 Heiko Breitsohl posted on Thursday, August 29, 2019 - 9:30 am
Dear Muthéns (et al.),

I am working on an ESEM model with longitudinal invariance constraints for two time points.

A few things strike me as odd about the results (or lack thereof):
1. Mplus provides estimates for factor covariances within and across time points, but TECH1 says that none are estimated within Time 1.
2. The same is true for some crossloadings (but not others).
3. With loadings constrained to longitudinal invariance, when I free the factor variances in Time 2, the model does not converge.

Thanks in advance!
 Bengt O. Muthen posted on Saturday, August 31, 2019 - 5:13 pm
That sounds strange. We need to see your full output - send to Support along with your license number.
 Alison Riddle posted on Friday, November 22, 2019 - 2:02 pm
Hello. Is it possible to add a direct effect for only one factor in an ESEM? I am working on an ESEM model that produced 5 factors. The modification indices recommend I add a direct effect between a covariate and one of the factors but I get an error message saying that all factors have to have the same number of regressions. Is there a way I can code it so that the regressions for the remaining factors are set to zero? Many thanks.
 Tihomir Asparouhov posted on Friday, November 22, 2019 - 2:56 pm
The simplest thing to do is to regress all five factors on the covariate. If you want to fix 4 of the regression coefficients to 0 you will have to use the EwC method described here, page 92
 Paolo Iovino posted on Tuesday, August 18, 2020 - 2:24 pm
I recently set up and run an ESEM model with ten indicators and two factors. Above and beyond the fixed loadings on those indicators, I have allowed the others to correlate. However, the MPLUS output gives me the following warning, I cannot work out what the meaning is:

WARNING in MODEL command
Fewer than 2 loadings have been given target values in EFA set *1.
The model may not be identified.

The input is the following:
DATA: file is self-eff.txt;
M5 M6 M7 M8 M9 M10;
M5 M6 M7 M8 M9 M10;
MODEL: illman BY M1* M2 M3 M4 M5 M6~0
M7~0 M8~0 M9~0 M10~0 (*1);
sympt BY M6* M7 M8 M9 M10 M1~0
M2~0 M3~0 M4~0 M5~0(*1);
M7 ON M6;
M2 ON M3;

May I have some information about the meaning of this warning? Also regarding why this model may not be identified? Thanks in advance!
 Tihomir Asparouhov posted on Tuesday, August 18, 2020 - 3:54 pm
I don't get such an error when using your input. You will need to send your input output and data to
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