[R-sig-ME] Random vs Fixed effect correlations

[Mike Lawrence]

Hi Folks,

I'm in the process of learning mixed modelling, and I'm having trouble
getting my head around the meaning of the correlations produced for
fixed effects when you do something like print( my_lmer_model , corr=T
). To possibly help you help me, I'll try to explain what I think I
*do* understand.

I understand that when you fit a model like "A ~ B + (1|C)", you're
letting each level of C have it's own intercept, but specify that
levels of C should be fit to a common slope for the effect of B.
Furthermore, I understand that when you fit a model like "A ~ B +
(1+B|C)", you're letting each level of C have it's own intercept *and*
it's own slope for the effect of B. I understand that this differs
from simply fitting individual models to each level of C in that in
mixed effects modelling, the variance observed in the data at each
level of C affects your confidence in that level's estimate of the
slope of B, and thus estimates with low confidence are shrunk towards
the mean estimate of B across levels of C. Finally, I understand that
when you have multiple random effects like "A ~ B + D + (1+B+D|C)",
you can compute a correlation between the per-Ss estimates of B and D
that take into account the shrinkage applied to the estimates.

What I don't understand is the fixed effects correlations obtained
when you fit a model like "A ~ B + D + (1|C)". In that model, I
thought that Ss are fit to common estimates of the slope for B and D,
so I don't understand over what these estimates are being correlated.

Could it be that the fixed-effects correlation between B & D is the
correlation across trials, ignoring levels of C? If so, would it then
be accurate to describe this correlation as a measure of the degree to
which measurement error in B & D are correlated? Furthermore, if there
are true correlations at the random effects level between B & D (for
example: such that levels of C with high B's also have high D), would
a model like "A ~ B + D + (1+B+D|C)" take this random effects
correlation into account when reporting the fixed effects correlation
between B & D? If not, I can imagine that in presence of correlations
at the random effects level would confound investigation of
correlations at the fixed effects level.

Would anyone care to attempt to ameliorate my confusions?

Mike

-- 
Mike Lawrence
Graduate Student
Department of Psychology
Dalhousie University

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~ Certainty is folly... I think. ~