[R-sig-ME] A conceptual question regarding fixed effects

[Simon Harmel]

Dear Colleagues,

Can we say in mixed-effects models, a fixed-effect coef. is some kind of
(weighted) average of its individual regression counterparts fit to each
level of a grouping variable and that is why fixed-effect coefs in
mixed-effects models can prevent things like a Simpson's Paradox case (
https://stats.stackexchange.com/a/478580/140365) from happening?

If yes, then, would this also mean that if we fit models with the exact
same fixed-effects specification but differing random-effect
specifications, then the fixed coefs can be expected to be different in
value but also meaning (i.e., what kind of [weighted] average they
represent)?

For example, would the meaning of a fixed-effect coef. for variable X
change if it has a corresponding random-effect in the model vs. when it
doesn't, or if we allow X to vary across levels of 1 grouping variable (X |
ID1) vs. those of 2 nested grouping variables (X | ID1/ID2)?

Many thanks for helping me understand this better,
Simon

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