I am trying to run a simulation of power/bias for a particular SEM, in which a number of correlated uniquenesses will need to be specified. I know how to determine the appropriate values of the measurement model parameters when these correlated uniquenesses are omitted--the squared standardized loading for an indicator and it's uniqueness should sum to one (e.g., $\lambda = .75, \theta$ = .4375). So if I know the value of the $\lambda$s that I want, I can easily solve for the $\theta$s. What's not clear to me is how to adjust my population estimates when these correlated uniquenesses are involved. In particular, I would like to know, given a particular $\lambda$ for an indicator, how to:

  1. Determine an appropriate value for the correlation between uniquenesses that corresponds to a particular % of an indicator's observed variance
  2. Determine an appropriate value for $\theta$ given the specified $\lambda$ and correlated uniqueness.

I've gone ahead and included a hypothetical population model (* marks specified parameter values) below. I'd like to know in this example, given $\lambda = .75$ for all loadings, how to determine the values for $\theta_{14}$, $\theta_{25}$, and $\theta_{36}$ that corresponds to say, 20% of variability in $X_1$-$X_6$, leaving 23.75% in each of $\theta_{11}$-$\theta_{66}$.

One thing that I've already tried is specifying this residual association in the form of latent method factors in the population model (i.e., one onto which $X_1$ and $X_4$ would load, another for $X_2$ and $X_5$, and a third for $X_3$ and $X_6$) with particular loading values corresponding to the % of variability that I want. However, this makes my simulation software cranky later, as it considers the population model w/ method factors as appreciably different from my model to analyze.

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