Residual correlations
Message/Author
 Kaja posted on Thursday, July 07, 2005 - 10:52 am
Hi there,
We are trying to interpret a situation in which we have huge modification indices for all of the WITH statements, even though we are accounting for what we think is the best factor structure for this data. This suggests to us that our model violates the assumption that the errors are uncorrelated. If we add the residual correlations to the model, the fit improves substantially. We are wondering how people usually interpret this kind of result given that we think we have found the best factor structure for the data.
Kaja
 Linda K. Muthen posted on Thursday, July 07, 2005 - 5:13 pm
Residual covariances can sometimes capture minor factors or methods factors. I would suggest going back and starting with an EFA to see if perhaps some of your items are not behaving as expected. EFA is a good way to find problematic items, that is, items that may load on factors not intended.
 Liu Xiao posted on Wednesday, September 12, 2007 - 5:25 am
Hi, I read the Mplus user's guide, and I read that "residuals are not correlated as default" in Growth Modeling. How to free the residual correlations?
In the following:

i1 s1| y1@0 y2@1 y3@2 y4@3;
i2 s2| x1@0 x2@1 x3@2 x4@3;
y1 with y2;
x1 with y1;
Does "y1 with y2" define the residual correlation or just the correlation between two observation variables --y1 and y2?

 Linda K. Muthen posted on Tuesday, September 18, 2007 - 4:24 am
y1 WITH y2 describes a covariance or a residual covariance depending on whether the variables are exogenous or endogenous variables in the model. In your situation, a residual covariance is specified.
 Ina Prokjev posted on Friday, November 08, 2013 - 2:02 am
Hi Linda,

within a "combined" CFA model I want to correlate the residual variances of the latent first order factors of two second order models (both with a latent second order general-factor at the apex and three latent content specific groupfactors on the first-order level below g). (I´m interested in the interaction of the content-specific first-order factors across batteries)

my question follows the post above.

When I use the "with" statement for the correlation between the first-order factors, can I be sure, that mplus uses the residual variances of the first-order factors instead of factor variances?

Ina
 Linda K. Muthen posted on Friday, November 08, 2013 - 8:57 am
Some but not all of the residual covariances among the first-order factors can be identified. It is the residuals that are covaried.
 Ina Prokjev posted on Friday, November 08, 2013 - 11:37 am

But, one thing: what what do you mean by
"Some but not all of the residual covariances among the first-order factors can be identified."?

Which of the residuals can not be identified? and why?

Thanks again,
Ina
 Linda K. Muthen posted on Friday, November 08, 2013 - 11:38 am
They cannot all be identified. A certain number can be not particular residual covariances. You would need to try adding them and see if they are identified.
 Nate Breznau posted on Thursday, June 16, 2016 - 8:05 am
I have a CFA for individual data across many countries. I am looking at measurement invariance and one of the items has a very large residual variance by country. The item is a question that contains the word 'unemployed' and I suspect that it is picking up something about the unemployment situation in each country. I don't want to model unemployment in the model if I can help it, but just want to look at the correlation of the residual of this item with a measure of unemployment across the 20 countries in my sample. Is this possible (to keep unemployment outside of the model but to investigate the correlation) in MPlus somehow?

I am ever grateful for both your software and responses.
 Bengt O. Muthen posted on Thursday, June 16, 2016 - 4:21 pm
An advanced approach would be to represent the residual as a factor, estimate the factor scores, and then in a separate analysis see what the correlation with unemployment is. How to represent a residual as a factor is described in our FAQ:

Regressing on a residual
 Nate Breznau posted on Wednesday, June 22, 2016 - 3:10 am
Thank you Bengt. I have a problem with generating a residual as a factor and then predicting it with unemployment because the residual factor is individual-level and unemployment is country-level. Thus, unemployment has zero variance within each group.

Again, this is a multigroup CFA with 20 countries (groups). I suspect that the residual variance is greatly different in each country for an item that has a second meaning linked to unemployment in that country.
 Bengt O. Muthen posted on Wednesday, June 22, 2016 - 5:02 pm
I can't get further into your topic without it getting too close too statistical consulting.