fixed effects correlated with the intercept

And hello again!

I'm getting a result that is very confusing to me and I'm hoping for
some advice or clarification.  I have two covariates that I consider
to be controls in my model.  When I include either in the model, the
fixed effect shows a strong correlation ( > .85 ) with the intercept.
The result of including these factors is that the estimated intercept
is much lower than I would expect.  Is there any conclusion to be
drawn from these correlations?  Normally when I see correlations among
fixed effects I worry about collinearity.  I'm absolutely confused
about what it would mean for a covariate to be collinear with the
estimated population mean.  Any help is appreciated in clearing this
up.

It's possible that the appropriate conclusion is that I'm overfitting.
I'm not sure this is the case.  The degrees of freedom in the model is
still relatively low compared to the number of data points (12 df on
~2500 observations).  Is overfitting still the most likely culprit?

One attempt at dealing with the above problem was to remove the
intercept from the model.  This causes lmer to estimate a coefficient
for each of the levels in the first factor in the model.  I think that
this treatment did not resolve whatever problem there is with these
two covariates-- now instead of being correlated with the intercept,
they are correlated with both levels of the split factor.

While this approach didn't resolve my original issue, it did bring up
a few others.  First of all, the coef() method fails on a model with
no intercept for the fixed effects, giving the error "unable to align
random and fixed effects".  Is this a known issue?  Is there a
workaround?

Second, while the estimates for both levels of the split factor are
shown to be significantly different from zero using mcmcsamp, I'm
still interested in whether there is a difference between the two
levels.  What's the appropriate test to check the null hypothesis that
the difference between the two parameter estimates is zero?

Thanks again,
/au

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

Is'nt this what might be expected.  Center the covariate about
its mean and, depending on the detailed variance-covariance
structure, the correlation may well reduce to zero.
...
[snip useful demonstration]
...
the correlation can be made arbitrarily close to -1 or 1,
respectively.

What do you mean when you say "I have two covariates that I consider
to be controls in my model." Do you mean that these code for
observations that you are treating as controls?  Or what?

On Fri, Mar 23 2007, John Maindonald wrote:

Is'nt this what might be expected.  Center the covariate about
its mean and, depending on the detailed variance-covariance
structure, the correlation may well reduce to zero.
...
[snip useful demonstration]
...
the correlation can be made arbitrarily close to -1 or 1,
respectively.

Thanks, this explains a lot.  I also appreciate the general point
about thinking in terms of lm when trying to reason about the fixed
effects in a model.

What do you mean when you say "I have two covariates that I consider
to be controls in my model." Do you mean that these code for
observations that you are treating as controls?  Or what?

I guess more precise terms might be "post-hoc controls" or "possible
confounds".  Our stimulus selection process did not take into account
certain properties of the stimuli that may have influenced the
observed behavior.  By adding these properties into the model as
post-hoc controls we can test whether the factors of interest have a
significant effect on the observed behavior even when these other
properties of the stimuli are accounted for.

Thanks again,
/au

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