There are a number of issues with regards to standard errors. Assuming the
model is correct, probably the most important is that under maximum likelihood
estimation these are the obtained from the Information matrix AND are
asymptotic! This brings TWO issues:
1) Relationship between the AI matrix and the Information matrix as they are
not the same.
2) What is meant by asymptotic? I.E. when is N large enough?
Variance components are usually skewed and one can only guess about any
ratios. Also, standard errors are usually a function of the true value of the
estimate and the standard errors for ratios are usually based on Taylor series
approximations - usually first order (correct me if I am wrong).
With your estimates Kim, are the correlations actually different (0.91 is 1
standard deviation of 0.83 +-0.08)? Without knowing the data structure, traits
and the estimated variance components, it is hard to draw any conclusions
about the results. You may have a part-whole relationship between variables.
Further, you clearly have more statistical information about one of the traits
than the others and it probably has a higher heritability as well.
Bayesian would overcome some (all) of the problems as the posterior would tell
you considerable amount of information. But introduce other issues e.g
proving that you have converged if using MCMC,...
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