# How to Determine Appropriate Values for Correlated Uniqueness in SEM Simulation?

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.