# [R] storing the estimates from lmer

Douglas Bates bates at stat.wisc.edu
Mon Jul 17 21:42:11 CEST 2006

On 7/17/06, Göran Broström <goran.brostrom at gmail.com> wrote:
> On 7/15/06, Douglas Bates <bates at stat.wisc.edu> wrote:
> [....]
> > <rant>
> > Some software, notably SAS PROC MIXED, does produce standard errors
> > for the estimates of variances and covariances of random effects.  In
> > my opinion this is more harmful than helpful.  The only use I can
> > imagine for such standard errors is to form confidence intervals or to
> > evaluate a z-statistic or something like that to be used in a
> > hypothesis test.  However, those uses require that the distribution of
> > the parameter estimate be symmetric, or at least approximately
> > symmetric, and we know that the distribution of the estimate of a
> > variance component is more like a scaled chi-squared distribution
> > which is anything but symmetric.
>
> You should add ..."when the true value of the variance is (close to)
> zero", I guess. Or does not standard asymptotic ML theory apply to
> these models? BTW, what is a
> "scaled chi-squared distribution"?

Consider a simple case of an iid sample from a normal distribution
with mean $\mu$ and variance $\sigma^2$.  In that case the sample
variance $s^2$ has a $\sigma^2\chi^2$ distribution with n-1 degrees of
freedom.  (Either that or I have been seriously misinforming my intro
statistics classes for several years now.)  That's all I meant by a
"scaled chi-squared distribution".

All I am claiming here is that estimates of other variance components
in more complicated models have a similar behavior, not exactly this
behavior.  The point is that they would not be expected to have nice,
symmetric distributions that can be characterized by the estimate and
a standard error of the estimate.  If you create a Markov chain Monte
Carlo sample from a fitted lmer object you generally find that the
logarithm of a variance component has a posterior distribution that is
close to symmetric.  Depending on how precisely the variance component
is estimated, the distribution of the variance component itself can be
far from symmetric.

If it still seems that I am stating things too loosely then perhaps we
could correspond off-list and I could try to explain more clearly what
I am claiming.