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

Shujuan Feng fengsj at mail.utexas.edu
Tue Jun 8 19:25:20 CEST 2010

I was confused about the correlation between fixed coefficients. I am still 
not very sure about it.....

One thing is that the correlations between fixed coefficients also exist in 
the model without random effects. It is more like: how one coefficient 
estimate influence the other coefficient estimate. (We get point fixed 
estimate, but a fixed coefficient have a distribution and can vary (CIs).) 
Strong correlations between fixed coefficients may show there is 
collinearity between the predictors.

I may be totally off. I am in the process of learning, too.

----- Original Message ----- 
From: "Mike Lawrence" <Mike.Lawrence at dal.ca>
To: <r-sig-mixed-models at r-project.org>
Sent: Tuesday, June 08, 2010 11:30 AM
Subject: [R-sig-ME] Random vs Fixed effect correlations

> 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
> Looking to arrange a meeting? Check my public calendar:
> http://tr.im/mikes_public_calendar
> ~ Certainty is folly... I think. ~
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