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I have a SEM where various latent variables are correlated with each other. I want to use a subset of these correlations to run a regression. This can be done easily enough to find point estimates, but standard regressions from correlation matrices assume that the only source of error in the correlations is from sampling error (i.e., not also error in estimating the latent variable), so the standard errors of the regression coefficients are biased.

Is there a way to adjust standard errors for such a regression to account for the additional source of error due to using latent variable correlations?

Note: In this case, I cannot simply run the regression within the SEM as it is somewhat susceptible to 'Interpretational Confounding' (Burt, 1976), whereby the measurement parameters change to optimize the fit when run as a regression, where the problem is not present when run as correlations. I can clarify this in the comments if anyone's interested.

Reference

Burt, R. S. (1976). Interpretational Confounding of Unobserved Variables in Structural Equation Models. Sociological Methods & Research, 5(1), 3–52. https://doi.org/10.1177/004912417600500101

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    $\begingroup$ If it's a saturated regression model (saturated in the structural part of the model) it shouldn't change the measurement parameters. You could, I suppose, constrain the measurement model to their estimated values in the saturated model, then they won't change (but changing seems, to me, like a feature, not a bug). $\endgroup$ – Jeremy Miles Dec 8 '19 at 1:30
  • $\begingroup$ Some further details: all factors on one half of the model are correlated with all factors on the other. I'm only changing some of these to regressions, which does seem to allow measurement parameters to change. Given that I'm using a method to specifically avoid measurement parameters changing without fixing them, this is not ideal (Nagy, 2017, Extension Procedures for Confirmatory Factor Analysis, doi: dx.doi.org/10.1080/00220973.2016.1260524). $\endgroup$ – Tim Bainbridge Dec 9 '19 at 4:09
  • $\begingroup$ I could perhaps try changing all correlations to regressions, but this would mean that the constraints required by Nagy's method may also need to change, and I wouldn't trust myself to figure that out (although I guess I could test such a model). $\endgroup$ – Tim Bainbridge Dec 9 '19 at 4:12
  • $\begingroup$ I don't have access to the Nagy paper, so I can't comment. $\endgroup$ – Jeremy Miles Dec 11 '19 at 15:10
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    $\begingroup$ No problem. Thanks for your comments. I think I'll leave this unresolved for now and just use a simpler model where this question is not an issue. $\endgroup$ – Tim Bainbridge Dec 12 '19 at 6:43

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