How to deal with Heywood case with negative variance

[Juliette van Acker]

Hi all,

This is the first time that I am using the WLSMV to perform my analysis. So far I run into the problem of a Heywood case (negative unexplained variance) for one of my indicators. I hope you can give me some suggestions.

Background information: 

I want to perform a confirmatory factor analysis where I expect three dimensions (RE, RC, and RB) in my data. RE has four indicators, RC and RB both have five. My indicator variables are all binary (levels: 1 or 2). I have about 605 observations.

RE =~ emt1 + emt2 + emt3 + emt4
RC =~ cgn1 +  cgn  2 + cgn3 +  cgn4 + cgn5

RB =~ bhv1 + bhv2 + bhv3 + bhv4 + bhv5

See attachment for an overview of the output after running the sem() function. After running the model, I got a sign from lavaan that one of my items had negative unexplained variance, namely item emt3. I noticed that this item (not surprisingly) had a standardized factor loading bigger than 1. When I looked up information about Heywood cases, I found that I should compute the confidence interval of the point estimate. In that case, I added the following to compute the confidence intervals around the point estimate of the unexplained variance.

RE =~ emt1 + emt2 + a*emt3 + emt4

RC =~ cgn1 + cgn2 + cgn3 + cgn4 + cgn5

RB =~ bhv1 + bhv2 + bhv3 + bhv4 + bhv5

res_var_Emo3 := 1 - (a^2)'

The unstandardized unexplained variance estimate has a CI lower limit and CI upper limit that both don't exceed 0 (see attachment: unstandardized unexplained variance confidence interval), which, as far as I understand, could potentially signal model misspecification.

I also computed the confidence interval around the standardized factor loading and the standardized unexplained variance of the particular item. The standardized factor loading has a CI lower limit that reaches under 1 and the standard unexplained variance exceeds 0 (see attachment: standardized effect and standardized unexplained variance confidence intervals). 

Because the conclusions differ in my case, what is leading here: the unstandardized or standardized estimates and confidence interval? And what does the outcome mean if one of them is leading? I understood it as: if the 95% confidence interval of the unexplained variance point estimate contains values > 0, it is likely caused by sampling error, otherwise by potential misspecification in the model.

I also tried to look for potential misspecifications, although my goodness of fit measures all seem to be fine. The suggested modifications to misspecifications are not helping in the sense that they make the warnings not disappear, or they make theoretically no sense. 

Removing item  emt3  solves the issue, but it is not the most desirable option (I think). Hence, I'm looking for other options possible.

Does anyone have any advice?

Thank you so much!