What does it mean to indicate correlation in lavaan CFA?

Question

I am an undergraduate psychology student and I've been doing basic statistical analysis for professors, post-grad students, etc for their work. I use R for almost everything

I've recently faced myself with exploratory and confirmatory factor analysis for psychological test validation, etc. When talking about EFA, everything went well.

The scale has 10 items, Likert 1 to 5.

I'm going straight to my question:

When I'm creating CFA model in lavaan, there is possibility to create latent factors and observed variables relationship (=~), but you can also create covariance relationship (~~). I tried to explain my professor that we could insert those covariances in the model to improve model fit, etc. But she would not understand (even me, I don't completely comprehend it), because she was saying that the factors on EFA are already created based on item correlations in the instrument, and then it wouldn't make sense to assign those correlations.

What I did is, I ran three models (single factor, two factor and three factor) and also those three including two correlations observed in polychoric correlation matrix. So, in total, 6 CFA models. The ones with the correlations in the model performed better based on fit indices (CFI, RMSEA, SRMR, etc)

Could someone clarify it to me? Is it important to put those covariances in CFA model? Is it good practice? What does it mean in "human/eli5 language"?

If I forgot to give some information, feel free to ask me.

Thanks in advance, love this community!

Answer 1

Here's an example. The PTSD Checklist (PCL) has 17 items. Two of them are:

• Being “super alert” or watchful on guard?
• Feeling jumpy or easily startled?

These two are very similar, and the items are highly correlated. Some people say that this means that these two items form a single factor (hyperactivity). Other people say that these are part of another factor (anxiety and reactivity).

If you put the same question, or a slight rewording of your question, into a factor analysis, you might find a new factor or you might just create a factor from nothing - the factor is an artifact. (And int the case of the PCL, the items are next to each other, which might make it worse). Thurstone called this a 'bloated specific' the IRT literature calls this a 'local dependency'. If you do exploratory factor analysis, you keep the two items, and accept another factor (probably). If you do CFA you can add a correlation - you can say that these are very similar items, not a separate factor.

I was involved in a paper about the PCL which talked about this issue here: https://pubmed.ncbi.nlm.nih.gov/23128035/

Another example: Looking at assessment of Socio-economic status of areas in the US. There are various measures you can use: education, household size, unemployment, median income, poverty. Median income and poverty are more closely related to one another - again, putting a correlation between these two items will improve the model fit. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4794463/ (not trying to blow my own trumpet here, you can easily find many more papers, it's just that these are ones I'm familiar with).