Reciprocal relationships in lavaan

[Duncan Jackson]

Hi everyone.

I'm new to using lavaan and, having used a number of different programs for CFA/SEM, I'm very impressed!  This one is great, fun to use, and it's free!!

On to my question, I'm wondering if anyone can help me out with an issue I'm having with one of my models.  It occurred to me that safety climate might affect incidences of reported bullying behaviour.  However, incidences of bullying behaviour might also influence safety climate.  As such, this could, potentially, reflect a reciprocal relationship between these two variables.

The lavaan code for my model appears below with the reciprocal relationship expressed in the lines reading NAQ_latent ~ PSSC_latent and PSSC_latent ~ NAQ_latent.  

When I try to fit model2, the standardised betas make sense.  However, I get warnings about not being able to compute standard errors with the possibility that the model might not be identified.  I wondered if there's a more appropriate way to model reciprocal relationships in lavaan or if what I'm trying to do is just not possible.

Any recommendations would be gladly received!

Regards,

Duncan

model2 <- '

PSSC_latent =~ pssc_p1 + pssc_p2 + pssc_p3 + pssc_p4 + pssc_p5 + pssc_p6

NAQ_latent =~ naq_p1 + naq_p2 + naq_p3 + naq_p4

POS_latent =~ pos1 + pos2 + pos3 + pos4 + pos5

EOI_latent =~ eoi_p1 + eoi_p2 + eoi_p3

GHQ_latent =~ ghq12_p1 + ghq12_p2 + ghq12_p3 + ghq12_p4

PANAS_latent =~ panas_p1 + panas_p2 + panas_p3 + panas_p4

ITQ_latent =~ asean_quit1 + asean_quit2 + asean_quit3

NAQ_latent ~ PSSC_latent

PSSC_latent ~ NAQ_latent

POS_latent ~ PSSC_latent + NAQ_latent

EOI_latent ~ NAQ_latent + PSSC_latent

GHQ_latent ~ POS_latent + EOI_latent

PANAS_latent ~ POS_latent + EOI_latent

ITQ_latent ~ POS_latent + EOI_latent + GHQ_latent

'

[Edward Rigdon]

Duncan--

     A structural model like this will be not identified without special conditions.  An econometrician using the rank and order conditions would want each of the two reciprocal dependents to have at least one other predictor that is not common to both reciprocal dependents, but it is possible to estimate this model as long as at least one of the two has an unshared predictor.  So a model like:

A ~ B

B ~ A

is not identified.

An econometrician would expect at least

A ~ B + C

B ~ A + D

C ~~ D   # probably

At a minimum, you would need:

A ~ B + C

B ~ A

and you might need to help that toward convergence by providing starting values.

Rigdon, Edward E. (1995), "A Necessary and Sufficient Identification Rule for Structural Models Estimated in Practice."  Multivariate Behavioral Research 30 (4), 359-84.

--Ed Rigdon

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[Duncan Jackson]

Prof. Rigdon,

Thanks very much indeed!  This is very helpful. 

After a little experimentation, I've noticed that, at least in my case, A ~ B, B ~ A, or A ~~ B all return identical results.  Thus, I wonder if A ~~ B might be an appropriate compromise.

I'd be interested to hear your thoughts.

Thanks again!

Duncan

[Edward Rigdon]

Duncan--

     All three options really are saying the same thing in this case.  With no other predictors for A or B, all you have is a covariance, however you model it.  Making A or B predictor sounds like an arbitrary choice.  So the most correct choice might be to simply allow them to covary, A~~B.  At least, your model properly represents your uncertainty.

--Ed R.

[Duncan Jackson]

Prof. Rigdon, that makes total sense.  Many thanks for your help!

Regards, Duncan