I am trying to run both a phenotypic 3 latent factor cholesky and then an ACE Cholesky using the same factor structure and data. My second order F2 and F3 factors are correlated at about .92 and my phenotypic cholesky model runs fine if I allow the error terms of an indicator of F1 and an indicator of F3 to correlate within twin 1 and within twin 2 (to control for shared method variance). However, if I remove the correlations, then the correlation between F3 for twin 1 and F3 for twin 2 exceeds 1.0 and I got a messaging saying that the covariance matrix for these two factors is not positive definite. Thus, my first question is: Am I violating any assumptions of the cholesky models by allowing the error terms across F1 and F3 to correlate? It seems like this method works out well in my case because it somehow removes the multicollinearity issue and the error message goes away. However, I want to make sure that my method is conceptually sound and that it doesn't somehow screw up my ACE estimates.
Another related question is how come all the Fs factors in Cholesky Models usually not allowed to correlate (i.e. no double headed arrows above these factors)? In reality, these factors tend to correlate and in my case, very highly and yet, what I have seen so far (mostly in journal articles and Hermine Maes's power point slides on Multivariate Genetic Analysis) is that these Fs are assumed to be independent.
My third question is does it make a difference that I use both twins' data to conduct phenotypic choleskys vs. just using twin 1 (randomly selected) data? I have tried it both ways and obtained very similar estimates.
Thank you in advance for your insights on this,