Residual Variance as a Function of a Latent Variable

First, in the item factor analysis utilities (AKA item response theory, `mxExpectationBA81()`) you can have access to the latent variable estimates via a dynamic data interface. Some further details are in the appendices of Pritikin et al. (2014, DOI: [10.1177/0013164414554615](https://doi.org/10.1177/0013164414554615)).

Second, you could write an `mxAlgebra()` that computes the factor scores, and then make the residual variance a function of those factor scores. This would not be terribly hard for all-continuous data. However, it would be difficult for ordered categorical data, and you could run into a recursion problem. The factor scores are a function of the residual variances, which you want to be a function of the factor scores. You'd probably have to write a compute plan (e.g., with `mxCompute()`) to make sure things happen in the right order. At some point, you're also just writing your own EM algorithm (see `mxComputeEM()`).

I am a little curious about why you'd want to do this. The only use-case I can think of is a way to handle binary observed variables: the variance of a binary variable is p*(1-p) where p is the probability of a correct response, and in this case would be the factor score. The built-in utilities for item response theory, ordered categorical data thresholds, and weighted least squares would all be better though.

Hope that gets you started!