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Mplus Discussion > Confirmatory Factor Analysis >
 Anonymous posted on Thursday, February 12, 2004 - 6:39 am
Dear Bengt/Linda

I'm conducting CFA on 8 categorical items measuring different aspects of "social capital" resulting in a model with 3 latent factors.

However, in this case I would like to allow the error terms of the manifest variables in the model to be correlated. Is this possible in Mplus (as I believe it is for instance in LISREL?)

Clearly, in Mplus you can allow for correlation among manifest variables and between latent factors, but is it possible to specify hypotheses concerning the correlation strucure of the error terms in the model? I have not been able to fnd any information on this subject in the Mplus manual.

Thank you for an excellent piece of software!
 Linda K. Muthen posted on Thursday, February 12, 2004 - 7:41 am
If you want to add residual covariances of the observed factor indicators to the model, you can do so using the WITH option. So if your factor indicators are y1-y4, to specify the residual covariance of y1 and y2, you would say:

y1 WITH y2;

This is described on page 18 of the Mplus User's Guide.
 Tait Medina posted on Thursday, April 03, 2008 - 3:45 pm
Dear Professors.

I have a question regarding Modification Indices for a CFA with categorical factor indicators. I noticed in a prior post that it is necessary to open the matrix for residual covariances in order to obtain suggested WITH statements. Are suggested WITH statements not included by default because you find these to be inappropriate modifications to make when using categorical indicators?

Thank you for your time.
 Bengt O. Muthen posted on Thursday, April 03, 2008 - 3:53 pm
Residual covariances are appropriate with categorical outcomes. We just don't want to open potentially big matrices when we don't need to.
 Ke Anne Zhang posted on Thursday, March 31, 2011 - 9:11 am
Dear Drs. Muthen and Muthen,

I am running a CFA model with 6 observed categorical variables (assuming they're called v1-v6). I specified that they all load onto 1 factor. I also want to allow the error terms of v1 to correlate WITH v2, v3 WITH v4, v5 WITH v6.

However, when I run this model, I get this error for all correlated error terms:

Covariances for categorical, censored, count or nominal variables with other observed variables are not defined. Problem with the statement:
v1 WITH v2

The estimator I'm using is MLR. Is it the case that Mplus will not run maximum likelihood estimation with correlated error terms of categorical variables? Is there any way around this?

Thank you very much for your time!

 Linda K. Muthen posted on Thursday, March 31, 2011 - 9:49 am
With categorical factor indicators, latent variables, and maximum likelihood estimation, each residual covariance requires one dimension of integration. You can use the BY option to specify these, for example,

f1 BY v1@1 v2;
f@1; [f@0];

where the factor loading for v2 will contain the estimate of the residual covariance parameter.

Note that weighted least squares estimation does not have these complication and the WITH option can be used to specify residual covariances.
 ellen posted on Thursday, November 29, 2012 - 11:51 pm
I have 3 continuous indicators (AD1, AD2, AD3) to measure a latent construct (called AD), the Modification index showed that:

AD3 WITH AD1 35.778 (M.I.) 0.123 (E.P.C)

should I allow the error terms of AD3 and AD1 to correlate in my model? I am not sure how to make a decision on this. Do I need to have a theory to allow the two error terms to correlate?

I have another 6 latent constructs (with all continuous indicators) in the model, including RS and BP in the model, and I noticed that when I allowed the error terms of AD3 and AD1 to correlate, RS and BP became significant predictors of AD in the structural model, but if the error terms were not correlated, the paths from RS and BP to AD became non-significant. I am wondering why it is so, and whether I should decide to allow the error terms of AD3 and AD1 to correlate or not.

Could you please let me know? Thanks so much for your time!
 Linda K. Muthen posted on Friday, November 30, 2012 - 10:16 am
Given that a factor with three indicators is just identified, no modification indices would be given if you estimate the model with only that factor. I would not add that residual covariance. In general, one would add a parameter only if substance dictates that this makes sense.
 ellen posted on Friday, November 30, 2012 - 3:39 pm
Hi Linda,

Thanks for your reply to my posting above. I am not sure whether I understand you correctly-- you said "no modification indices would be given if you estimate the model with only that factor." Did you mean that if I have more than 1 factor in my model, it would be fine to utilize modification indices to improve my model fit?

I have 7 latent factors (each has 3 indicators) in my measurement model, and there are 3 X variables (covariates) in the measurement model. When I estimated the model, the modification indices for the correlation between two error terms (AD3 WITH AD1) of a specific latent factor (AD) was 35.778.

Should I allow the two error terms to correlate in my measurement model ? --again, my model has 7 latent factors, and each factor has 3 indicators (all continuous). Besides, there are 3 categorical X variables as covariates of this measurement model.

Your advice on how to deal with the error terms is greatly appreciated.
 Bengt O. Muthen posted on Saturday, December 01, 2012 - 4:50 pm
I think the idea is that when you have only 3 indicators of a factor, misfit cannot be judged when using information from only those 3 indicators - this is because that part of the model is just-identified - so you are in a tricky spot. This is a weakness of your modeling; you really want to be able to investigate fit and the need for residual correlations for each factor by themselves. When you consider all parts of the model you say that this residual correlation has a high modindex. This may be due to many different sources of misfit having to do with misspecified relationships between this factor's indicators and other factors' indicators. So you are in a tricky situation. One guiding principle is - does this residual correlation make strong substantive sense? If yes, free it. Another approach is to test different parts of the model, say this factor together with another factor - so all together analyzing 6 indicators. This will tell you more about why this modindex shows up.
 Amy Gedal Douglass posted on Tuesday, November 18, 2014 - 1:01 pm
Dear Drs. Muthen and Muthen,

I am new to Mplus. I am trying to establish factorial invariance of two latent variables over time measured at 3 waves.

I'm getting slightly different model fits depending on the scale setting technique I use. Is this to be expected? (it appears with the effects coding method I have 2 fewer free parameters than with the marker method). Or would I get identifical results, if I was doing this correctly?

Ultimately, I'd like to use the effects coding method, but am unsure whether I am coding it correct.

 Linda K. Muthen posted on Tuesday, November 18, 2014 - 1:18 pm
Please send the two outputs and your license number to
 Megan Brokenbourgh posted on Monday, February 23, 2015 - 5:47 pm
Hello Drs. Muthen and Muthen,

I am investigating measurement invariance via a MIMIC model with categorical variables and am wondering if allowing error terms to correlate will influence the analysis and interpretation in any way?

For example, I read the example of a MIMIC model with categorical variables on the Mplus Short Courses involving Antisocial Behavior (ASB) Data. Hypothetically, if this measurement model included correlated error terms between the items “seriously threatened” and “intent to injure,” would the analysis and interpretation of population heterogeneity and direct effects (i.e., DIF) change in any way?

Thank you in advance for your time.
 Bengt O. Muthen posted on Tuesday, February 24, 2015 - 7:36 am
Probably not. DIF has to do with x-y relationships, whereas correlated errors have to do with y-y relationships.
 Jane Doe posted on Tuesday, January 12, 2016 - 2:23 am
I had a query about the WITH command. When does it add correlation between the variables, and when does it add correlation between the resudluas. I understand from the discussions so far that this depends on the model being specified.

Lets say my model is:
f1 BY x1 x2 x3;
f2 BY x4 x5 x6;
f3 BY x7 x8 x9;
f3 ON f1 x10; !eq1
f2 ON f1 x11; !eq2
f2 WITH f3;

I want to specify that the residuals of eq1 and eq 2 are correlated. Am I doing it correctly above? OR am I just specifying that f2 and f3 are correlated? Also if I run the model excluding and then including the WITH command the coefficients of eq1 and eq2 change both in size and significance. Is that OK? I am not surprised by the change in the significance (as I expected standard errors to change with the inclusion of the WITH command), but am surprised by the change in size.

Any help would be appreciated.
Thanks in advance.
 Linda K. Muthen posted on Tuesday, January 12, 2016 - 8:17 am
For endogenous variables, the WITH statement covaries the residuals. For exogenous variables, the WITH statement covaries the variables. Note that these are covariances not correlations.
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