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

CFA
1

I’m working on a complex CFA model in R using lavaan. The tricky part is setting the unique error of observed variables to be the sum of two correlations of unique items.

For example, the error of item y1,2 should equal the sum of its covariances with y1,3 and y*2,3.

Here’s a simplified version of my current lavaan code:

cfa_model <- '
  F1 =~ a*x1 + b*x2 + c*x3
  F2 =~ d*x4 + e*x5 + f*x6
  
  x1 ~~ x2 + x3
  x2 ~~ x3
  
  # How to set x1's error as sum of its covariances?
  # x1 ~~ (x1~~x2 + x1~~x3)*x1
'

I’m stuck on how to explicitly set an item’s error to equal the sum of its covariances in lavaan syntax. Any ideas on how to achieve this?

2

Hey there Oliver63! That’s quite an interesting CFA model you’re working on. :blush:

I’m curious about why you need to set the error terms this way. Is it related to some specific theory or research question you’re exploring? It’s not something I’ve come across before in CFA modeling.

Have you considered using phantom variables to achieve this? You could create latent variables that represent the covariances and then use those to define the error terms. It might look something like this:

cfa_model <- '
  F1 =~ a*x1 + b*x2 + c*x3
  F2 =~ d*x4 + e*x5 + f*x6
  
  # Phantom variables for covariances
  cov12 =~ x1 + x2
  cov13 =~ x1 + x3
  
  # Set x1's error as sum of covariances
  x1 ~~ (cov12 + cov13)*x1
'

This is just a rough idea, though. You’d need to play around with it to get the exact structure you want.

Have you tried asking on the lavaan forums or mailing list? The package developers might have some insights on advanced model specifications like this.

What other approaches have you considered for this problem? I’d be really interested to hear more about your research and what led you to this unique modeling challenge!

3

I’ve encountered similar challenges with complex CFA models in lavaan. One approach that might work is using constraint equations to define the error terms. You could try something like this:

cfa_model <- '
  F1 =~ a*x1 + b*x2 + c*x3
  F2 =~ d*x4 + e*x5 + f*x6
  
  x1 ~~ e1*x1
  x2 ~~ e2*x2
  x3 ~~ e3*x3
  
  x1 ~~ cov12*x2
  x1 ~~ cov13*x3
  
  e1 == cov12 + cov13
'

This explicitly defines the error variance of x1 (e1) as the sum of its covariances with x2 and x3. You might need to adjust the syntax slightly depending on your full model specification. If this doesn’t work as expected, you could also explore using the ‘:=’ operator for more complex constraints. Remember to check model identification carefully with such non-standard specifications.

4

hey oliver63, thats a tricky one! have u tried using the ‘:=’ operator to define custom parameters? something like this might work:

cfa_model ← ’
F1 =~ ax1 + bx2 + cx3
F2 =~ dx4 + ex5 + fx6

x1 ~~ cov12x2 + cov13x3
x1err := cov12 + cov13
x1 ~~ x1err*x1
’

not sure if itll solve ur problem but worth a shot. good luck!