[R-sig-ME] Point estimate outside of its confidence interval with lmer

[Emmanuel Curis]

Hello,

I have encountered an unexpected result for some datasets when using
confint after fitting a model with lmer: the confidence intervals for
the standard deviations in the model did not included the point
estimate, given by summary for instance.

I think the problem is a mix of small sample size and REML vs ML, but
I would be happy to have confirmation that my interpretation is
correct, since I'm not very familiar with profiling and REML vs
ML... and wonder if something more problematic is occuring.

My interpretation is that lmer fits using REML by default, hence the
variances are estimated "unbiased" (the equivalent of dividing by n -
k for the residual variance of a linear model, but I'm not sure
exactly what is n and k for random effects variance estimation).  But
when confint is called, using profile, the ML profile is used, hence
variance confidence intervals are estimated "biased" (the equivalent
of dividing by n).  However, when n is small, dividing by n - k and n
may give very different results, hence in some cases the profiled
confidence interval for ML estimate does not include the REML
estimate, for the standard deviations.  I guess in practice the
difference between REML and ML is more complex than just using n or
n-k, but would it be idea?

Support for this is that
 1) when using bootstrap, there seems to be no such discrepancy
 2) when fitting using REML = FALSE, the profiled IC is the same and
    does include the (different) point estimate

Does it sound correct?

Additionnal questions, if this interpretation is correct:
 - would it make sense to make confidence intervals based on REML
   profiles, and not ML profiles? if so, how?

 - wouldn't a warning be a good idea when point estimates are outside
   CI, with the explaination if it is indeed REML vs ML?

 - if this is indeed a "small sample size" problem, I guess in such
   cases any asymptotic result is difficult to trust, right?  Does it
   mean profiled interval cannot be trusted also, neither nested
   models tests, and that only bootstrap may be used?
   
 - in such cases, is there any argument to prefer REML over ML or
   vice-versa?

Thanks in advance for your help,

-- 
                                Emmanuel CURIS
                                emmanuel.curis using parisdescartes.fr

Page WWW: http://emmanuel.curis.online.fr/index.html