Binomial GLMM vs GLM question (Ken Beath)

For example, if you make the effects random then you're effectively
marginalizing them, whereas if you make them fixed you're forced to
condition on them

Hello all!

I'm cherry picking this one line from a recent discussion because it's
the most recent example of people discussing fixed and random effects in
terms of conditioning and marginalization.  The use of this terminology
on this list seems to have increased in the past year or so (or, more
likely, I've just started noticing it), and it's time for me to confess
that I'm not sure I understand it.

If we have a model

#v+
y ~ 1 + x + (1 | z)
#v-

the suggested correspondences between fixed effects and
conditionalizing, and between random effects and marginalization suggest
to me that at some point we are interested in

#v+
\sum_{z} P(y | x, z)
#v-

This is my guess at the correspondence suggested by the quote above, but
it's based solely on the fact that I think I know what conditional
probabilities and marginalization are.  It could be 100% off base.

I guess I have four questions:

1.  Is this the correct understanding of how fixed and random effects
    translate into conditionalizing and marginalizing?
2.  In mixed logit models, we are modeling probabilities (or, log odds
    of probabilities) , so this specification maybe makes some sense to
    me.  But how does it fit into a linear mixed model?
3.  What role does this probability play in fitting the model?
4.  Do the coefficients for fixed effects from the fitted model have an
    interpretation in terms of the above probability model?

Sorry if this query is so misinformed as to be nonsensical.  If you have
a feeling for what I'm trying to ask, feel free to answer that question
instead ;)  Also, if this is a RTF*-type question, please let me know
what the appropriate value of * is in this case (V for vignette,
maybe?).

Thanks for any help,
/au

Austin Frank
http://aufrank.net
GPG Public Key (D7398C2F): http://aufrank.net/personal.asc