I have found that random effects terms can affect other coefficients within a model from here.
I see how in this example the coefficients change with the addition of a random effect; I'm still not sure why this happened though.
I've seen how the Frisch–Waugh–Lovell theorem can show how coefficients may change in response to additional covariates (I've not gone through the proof but I can sort've connect the dots); I don't really understand how this could translate to random effects though, as I've been told random effects aren't observed and are integrated out in the estimation.
I tried reading around further to see if seeing what the likelihood function looks like to see if that would help and found this - but it doesn't really clear things up. I don't know what d(theta) is, or why (b^t)D(theta)(b) is in the formula for the random effect.
I suppose what I'm looking for is a an equation that shows the coefficients are a function of the random effects; I assume this doesn't exist and there is some misunderstanding on my side for even asking this. Further, does something like this exist in the linear regression setting.
Any help, or recommendations to textbooks/online resources, would really be appreciated
