How to define CFA with error terms as sums of exogenous correlations?

CFA
1

I’m working on a complex CFA model in R using lavaan. The tricky part is setting up the error terms for observed variables. Each error should equal the sum of two correlations between unique items.

For example, the error for y1,2 should be the sum of its covariance with y1,3 and y*2,3.

Here’s a snippet of my current lavaan code:

cfa_model <- '
  C =~ C1_12*i1i2 + C1_13*i1i3 + C2_12*i4i5 + C2_13*i4i6
  R =~ R1_12*i1i2 + R1_23*i2i3 + R2_12*i4i5 + R2_23*i5i6
  O =~ O1_13*i1i3 + O1_23*i2i3 + O2_13*i4i6 + O2_23*i5i6

  # Other constraints and covariances...

  i1i2~~i1i3
  i1i2~~i2i3
  i1i3~~i2i3

  i1i2~~1*i1i2
'

# How can I set i1i3's error to equal i1i3~~i1i2 + i1i3~~i2i3?

Any ideas on how to specify these error terms in lavaan? I tried using the ‘equal’ function but it didn’t work as expected.

2

I’ve encountered similar challenges with complex CFA models in lavaan. One approach that might work for your situation is using constraint equations to define the error terms. You can specify these equations in the model syntax using ‘:=’. Here’s an example of how you might modify your code:

cfa_model <- '
  # Your existing model definition...

  # Define error terms as sums of correlations
  err_i1i2 := i1i2~~i1i3 + i1i2~~i2i3
  err_i1i3 := i1i3~~i1i2 + i1i3~~i2i3

  # Set error variances using defined terms
  i1i2 ~~ err_i1i2*i1i2
  i1i3 ~~ err_i1i3*i1i3
'

This method allows you to create custom parameters that are functions of other model parameters. It’s more flexible than the ‘equal’ function for complex constraints. Remember to check your model identification carefully, as these constraints might affect it. If you’re still having issues, consider simplifying your model initially and gradually adding complexity to isolate any problems.

3

have u tried using the ‘:=’ operator in lavaan? it lets u define custom parameters. might work for ur error terms. something like:

err_i1i2 := i1i2i1i3 + i1i2i2i3
i1i2 ~~ err_i1i2*i1i2

just an idea. let us kno if it helps!

4

Hey Jack27! That’s a really intriguing CFA model you’re working on. I’m curious about why you need to set up the error terms as sums of correlations - it sounds like a unique approach!

Have you considered using phantom variables to achieve this? You could create latent variables that represent the sums of correlations, then use those to define your error terms. Something like:

cfa_model <- '
  # Your existing model...

  # Phantom variables
  PE1 =~ 1*i1i3 + 1*i2i3
  PE2 =~ 1*i1i2 + 1*i2i3

  # Set error terms
  i1i2 ~~ PE2*i1i2
  i1i3 ~~ PE1*i1i3
'

This is just a rough idea, though. I’d love to hear more about your research context. What kind of data are you working with? And have you tried any other approaches before this one?

Also, have you considered reaching out to the lavaan mailing list? They might have some package-specific tricks up their sleeves for handling complex error structures like this.

Keep us posted on how it goes! I’m really interested to see how you solve this.