(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 ?
(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 email@example.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 y1-y5*.8 y6-y10*0 (*1 1); f2 BY y1-y5*0 y6-y10*.8 (*1 1); f1-f2@1; f1 WITH f2*.5 (2); y1-y10*1; [y1-y10*1]; [f1-f2@0];
MODEL g2: f1 BY y1-y5*.8 y6-y10*0 (*1 1); f2 BY y1-y5*0 y6-y10*.8 (*1 1); f1-f2@1; f1 WITH f2*.5 (2); y1-y10*2; [f1*.5 f2*.8];
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]; f3@1; f4@1; f1 WITH f2 (40); f3 WITH f4 (40); SE1_1-SE1_10 PWITH SE2_1-SE2_10;
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 ?
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.
(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 ?
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:
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.
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?
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.
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) ?
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.
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 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.
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 ?
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, "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.
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, 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.
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.
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!
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 (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. 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 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, 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.
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.
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:
Variables: Usevariables q1-q14;
Analysis: type= efa 3 3;
Variables: Usevariables q1-q14;
Model: f1-f3 BY q1-q14 (*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 firstname.lastname@example.org.
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
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?
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?
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?
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.)
Questions: (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?
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 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.
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.
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
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?
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?
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.
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 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 up...so that is consistent across the approaches...
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.
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; CATEGORICAL = s1-s16;
ANALYSIS: !estimator = MLR; !rotation = promax;
MODEL: 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)."
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?
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?
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?
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)?
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: 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 ..
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?
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 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)
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.