[R-sig-ME] Peak in posterior density produced by pvals.fnc

[Romain.Piault]

Dear R users,

I have a question with respect to the pvals.fnc from the package
languageR. 

I am working on a small and unbalanced dataset (18 nests of eurasian
kestrels with a total of 64 nestlings; one fixed factor with 2 levels; 6
nests (25 nestlings) for the first level, 12 nests (39 nestlings) for
the second level).

I have run a simple mixed-effects model with "nest" as a random factor,
and then obtained the p-values and 95% confidence intervals for the
fixed and random effects using the pvals.fnc.
The pvals function gives a graph of the posterior density for the
intercept, fixed factor, random factor and residuals (I can send the
graph to those who would be interested; it was too big to pass it to the
mailing list). The density for the random factor shows a peak
near zero. I have looked for an explanation on the mixed-models list and
found this e-mail from Douglas Bates:

"Forwarded message ----------
From: Douglas Bates <bates at stat.wisc.edu>
Date: Tue, Apr 1, 2008 at 1:40 PM
Subject: Re: [R-sig-ME] lme4::mcmcsamp + coda::HPDinterval
To: Sundar Dorai-Raj <sundar.dorai-raj at pdf.com>
Cc: r-sig-mixed-models at r-project.org


May I suggest that you repeat the experiment in the development
 version of the lme4 package?  In that version the HPDinterval function
 has been moved to lme4 and it is no longer necessary to attach the
 coda package.

 I think it is easier to see what is happening when you use that
 version of the package because you can use the xyplot method to
 examine the evolution of the sampler.  I enclose a modified version of
 your code.  Running this version produces the enclosed plot.  You will
 see that the (relative) standard deviation of A (labelled 'ST2') gets
 stuck near zero.  This is a known problem with MCMC sampling of
 variance components.  The prior distribution corresponds to a "locally
 constant" uninformative prior on log(sigma_A).  As long as the
 likelihood at sigma = zero is sufficiently small to prevent the MCMC
 sampler getting near there the sampler proceeds happily.  However, if
 the likelihood is not sufficiently small then the MCMC sampler may
 wander into the "sigma near zero" region where the posterior density
 of log(sigma) is essentially flat and it gets stuck there.  The recent
 paper by Gelman et al. (JCGS, 2008) provides a suggestion of avoiding
 this problem by overparameterizing the model for the MCMC sampler but
 I haven't yet implemented." 

Can anyone confirm me that this is indeed the explanation for the
strange pattern I find with my data?
Is there a way to correct this problem?

Thanking you in advance for your answer.

Best regards,

Romain