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Hi, I recently read the paper "Exploratory Structural Equation Modeling" from Asparouhov & Muthén and I wanted to try the ESEM approach in Mplus (v5.21) to modelise my data. To keep things simple to start with, I considered subset of my data which has 4 factors having 4 indicators each. When I use the "type = EFA 4 4", the model converges and gives interesting results. However, when I want to use a more complex model (e.g. having paths and correlated residuals), I will have to write the syntax in a different way. Something more like : ANALYSIS: type = general; rotation = geomin(0.0001); MODEL: ext BY ext1ext4*1 mi1mi4*0 iden1iden4*0 intro1intro4*0; mi BY ext1ext4*0 mi1mi4*1 iden1iden4*0 intro1intro4*0; iden BY ext1ext4*0 mi1mi4*0 iden1iden4*1 intro1intro4*0; intro BY ext1ext4*0 mi1mi4*0 iden1iden4*0 intro1intro4*1; ext WITH intro iden mi; intro WITH iden mi; iden WITH mi; ...if I understood right from the example given in the paper. Yet, I receive a "THE STANDARD ERRORS OF THE MODEL PARAMETER ESTIMATES COULD NOT BE COMPUTED" error when I run the model. Am I not getting the syntax to do ESEM right? 


The syntax is shown in the Version 5.1 Examples and Language Addendums on the website with the user's guide. It should be: ext mi iden intro BY ext1ext4 mi1mi4 iden1iden4 intro1intro4 (*1); The factors will be correlated with oblique rotations. 


Thank you very much! 

naomi saito posted on Tuesday, July 13, 2010  10:04 am



Hello, After EFA, I am trying to run ESEM using 16 binary variables with 3 factors. I recently updated Mplus to version6. I can run by using default estimator WLSMV, but it returns error when I use “estimator = MLR”: *** ERROR in MODEL command EFA factors are not allowed with ALGORITHM=INTEGRATION. EFA factors are declared with (*label). Can we use MLR for categorical variables? Do I need to use CFA if I want to use MLR? I was hoping use MLR to obtain AIC/BIC to compare models (3 factor model VS latent class models VS factor mixture models). VARIABLE: NAMES = ID STRATUM weight y1y16 psu; USEVAR = y1y16; STRATIFICATION = STRATUM; CLUSTER = psu; WEIGHT = weight; CATEGORICAL = y1y16; ANALYSIS: type = complex; estimator = MLR; MODEL: f1f3 by y1y16 (*con); Thank you for your help. 


You cannot use MLR with categorical outcomes with the ESEM model. You can use it for regular EFA or for CFA. 


I recently ran an EFA with my new version of Mplus (6.1) and was delighted to see fit indices. Are these the result of incorporating an ESEM framework into Mplus? If so, then CFI/TLI and SRMR/RMSEA can be interpreted via convention regarding incremental and absolute fit indices, respectively, correct? thanks for your response... 


EFA (outside ESEM) also has those indices in Mplus. And yes, they are interpreted as usual. 


Drs. Muthen, First, I would like to thank you both for providing the discussion board for users and responding to inquiries. I have personally found it to be an invaluable resource. My question is: Will you please explain why, when perfoming an EFA using ESEM, the output does not contain two sets of factor loading matrices (i.e.  pattern and structure)? Thank you in advance. 


There is no particular reason that we don't give these. 


Thank you for your response. I have read some articles that have argued that interpretation of both the pattern and structure matrices is necessary for determining the most appropriate factor structure when oblique rotation is used because the factor correlation matrix is not an identity matrix. Given that ESEM analyses provide significance tests for each loading, would you mind commenting on how using this information provides an improvement over analysis of the structure matrix when oblique rotation is used? 


I would disagree with those articles that this is necessary. Factor structure is Lambda*Psi, that is the covariance between the indicators and the factors. As you say this is different from Lambda when Psi is not the identity matrix, that is, with oblique rotation. However, I personally don't think that knowing this indicatorfactor covariance is essential or necessary to interpret the factor analysis solution  studying the pattern (Lambda) together with the factor correlations (Psi) is sufficient, even with oblique rotation. This is why we haven't yet added factor structure to ESEM, although we probably will in the future. Providing SEs for loadings is useful for both EFA and ESEM. Two good references are mentioned in our 2009 ESEM article (none uses factor structure): Browne, M. W. (2001). An overview of analytic rotation in exploratory factor analysis. Multivariate Behavioral Research, 36, 111–150. 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. 


Thank you both for your time and input. I'll be sure to read the references you provided. 


Hi, Is there in reason to expect EFA and ESEM to differ in there results when looking only at a measurement model? I know Mplus used the gradient projection algorithm for ESEM, but I haven't seen any situations where ESEM differs from EFA, which is further evidenced by example 4.1 in the Mplus manual. Thanks, Tom 


ESEM and EFA should agree when using only the measurement model in ESEM as long as the rotations are the same. The measurement model in ESEM is EFA. 


Dear Dr Bengt Muthen, I am most interested in this discussion on the interpretation of an EFA/ESEM solution. Referring to your suggestion to Christopher T Allen, on October 21, 2011, 8:39am, I am unclear as to how to interpret the pattern coefficients together with the factor correlations without referring to the structure coefficients. Could you please expand on this, or more specifically, how to interpret the factor structure by considering both the pattern and structure coefficients? One recommendation I have read (referenced below) suggests: “attending to the structure coefficients first then evaluating the pattern coefficients to understand the unique factorvariable relationships”. Bandalos, D. L., & Finney, S. J. (2010). Factor Analysis. In G. R. Hancock & R. O. Mueller (Eds.), The Reviewer's Guide to Quantitative Methods in the Social Sciences. Hoboken: Routledge. Thank you in advance. 


Simply put, the difference between the factor pattern and the factor structure is akin to the difference between regression coefficients and correlations. Both refer to the relationship between factors and items. The factor model is a regression model, regressing the items on the factors. We know how to interpret regression models  an item's loadings on a set of factors are partial regression coefficients just like regressing a y on a set of x's. And in EFA/ESEM all coefficients are standardized so the interpretation is even easier. This is why I think the factor pattern is sufficient for interpretation of a factor model. It is working with the parameters that we are modeling. The factor structure tells you about the correlations between the items and the factors. It could be of interest to see how a certain item correlates with certain factors. These are not parameters in the model, but are consequences of the model which may augment the understanding of the model. So I am not against looking at structure, but I don't think it is (1) necessary, or (2) should precede looking at the pattern. 


Thank you very much for this clarification. It is most helpful. 


Drs Muthen, I collected data on a large pool of items that were designed to capture 6 psych constructs. I am using ESEM for item trimming purposes, and then will crossvalidate on an independent sample. From a statistical standpoint, I’d greatly appreciate your insights into whether it is preferable to focus on the strength of the regression coefficient (i.e., high on one factor only, with low crossloadings) or whether the loading is significant (or both strength and significance) in determining whether to retain an item. For example, there are instances of items with moderate loadings (e.g., >.40) that are not statistically significant. Alternatively, any recommended readings would be greatly appreciated. Regards, Daniel 


You want the loading to be significant  and preferrably also large. You may want to take a look at 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. 


Many thanks  appreciate the prompt response and recommended info/reading. 


Dear the Mplus team, I have two questions about Mplus output for ESEM. Q1. What is the difference in factor loadings between ‘ROTATED LOADINGS’ and ‘FACTOR STRUCTURE’ sections? Q2. Which factor loadings should I report in a paper for publication? Thank you in advance for your consideration. Kind Regards, Masato 


The factor structure is the correlations between the factors and the items. The rotated loadings should be reported. 


Dear Dr Muthen, Thank you very much for your prompt reply! Best regards, Masato 


Dear Mplus team, After reading the ESEMpapers of Marsch et al and the Asparouhov & Muthén manual we are now conducting the superior ESEM method on our dataset. We have to decide on the appropriate number of factors of the ESEM model. Given the distribution of WLSMV, direct chi square comparison is impossible and we have used the Difftest option. However, the type=efa command makes this test impossible: *** WARNING in SAVEDATA command DIFFTEST is not available only TYPE=EFA. Request for DIFFTEST will be ignored. In the Multigroup analysis presented in the Marsch paper, models are compared based on rmsea/cli/tli and srmr. Given that rmsea is regarded of as the best fit index for the WLSMV, together with the mention in the Asparouhov & Muthén manual that the SRMR should decrease >.001; should we use these fit indices only? Or is there another way to compare model quality? Thanks in advance. Mark Patrick Roeling 


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


I am doing ESEM using the taxonomy in Marsh et al. paper. I have my final model but how to i get the correlation for each group. what is the difference between F1 and F1_SE Correlations.lastly what are the possible graphs in ESEM. 


I think you are saying that you see covariances not correlations. Ask for StdY in the OUTPUT command to see the correlations. 


I am doing ESEM using the 13 taxonomy in Marsh et al. paper. Will like to know if the taxonomy is valid for categorical data as well. if not what are the restriction and any advise on how to go about it. 


Yes. 


so i just have to use the WLSMV without any PARAMETRIZATION conditions. e.g Theta If you look at al the articles on the TAXONOMY chiSQUARE diff test was not use. so i can also ignore it and use the CFI, RMSEA, TLI arguments. 


TO correct my post i realized there is a default PARAMETRIZATION of Delta for WLSMV. Can i ignore the chisqaure text and use CFI, RMSEA, TLI arguments.I am just confuse because when i use the difftest and i start to use the modification indexes it does not work when i introduce a second misfit always. I want to send my work but i am using MPLUS on university computer pool. I've treid to the IT center about the license registration so i can add but they dont understand why they should give me the info. Can you explain what i really have to include for verification. 


Support is available to one registered user per license. You must be the registered user of a current license to be eligible for support. 


Drs. Muthen I am a student, and new user of MPlus. I was able to utilize the user guide to produce an output of my analysis, but am not sure how to interpret the results. Could you point me toward resources that might help guide me through interpretation? Thank you! 


See Chapter 18 of the user's guide. The basic output is described there. If the interpretation is due to not being familiar with the method you are using, see the short course handouts and videos on the website. 


Thank you for responding so quickly. I have found the handouts, videos, and user guides. I will be making thorough use of these. 

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