[R-sig-ME] Missing values in lmer vs. HLM

[Douglas Bates]

My apologies for making such a statement then not following up.  As has
been mentioned, this is a holiday weekend in the U.S.

The section that Landon quoted does get at the point of my comment.

The usual justification for REML is that REML estimators of variance
components are less biased than are the maximum likelihood estimators
(mle).  On the surface this seems to be a convincing argument, for who
would want to use a "biased" estimator?

But why should we be concerned with the estimator of the variance?  Why not
the estimator of the standard deviation, or the logarithm of the standard
deviation?  The distribution of variance estimators are highly skewed in
most cases.  Consider the simplest case of estimating the variance from an
i.i.d. sample from a Gaussian distribution.  The distribution of the
estimator is a Chi-squared distribution, which is highly skewed.  The
distribution of the estimator of σ is less skewed.  The distribution of the
estimator of log(σ) is more-or-less symmetric.

The important point here is that "bias" relates to the expected value of
the estimator.  The argument for REML is based on the expected value of a
quantity with a highly skewed distribution, but we know that this is a poor
measure of location for such a distribution.  That's why it is more
informative to consider median salaries instead of average salaries.  The
fact that the average wealth of members of LeBron James's high school
basketball team is very high doesn't make them all rich.

Mle's have an invariance property in that the mle of σ is the square root
of the mle of σ²; the mle of log(σ²) is the logarithm of the mle of σ²,
etc.  Unbiased estimators aren't invariant under transformation.  The
square root of an unbiased estimator of σ² is not an unbiased estimator of
σ.

If an unbiased estimator were so important then we should probably consider
the estimate of log(σ²), not σ² itself.  The reason for our being fixated
on σ² is more computational than practical.  When using hand calculations
it is easiest to estimate σ² then derive an estimate of σ from that.  These
considerations are less convincing when using computers.

In summary, the case for REML is less convincing than it seems at first
glance.  It is a consequence of a certain type of mathematical exposition,
where your assumptions are never questioned.  You only care about going
from "if" to "then".  In mathematical statistics you say, "assuming that
the model is correct, these are the consequences" and that is all there is
to it.  The way that the game is actually played is that, when you get to
the end of the proof and discover that you need some conditions to make it
work, you go back to the beginning and add those conditions.  It helps if
you call this case the "regular" case or the "normal" case or some other
word with favorable connotations.

So if you want to characterize the "best" estimator you do it by peeling
off properties related to the first moment, the second moment, etc. For the
first moment you say that the expected value of the estimator must be equal
to the parameter being estimated and you call that the "unbiased" case.
Technically this is just a mathematical property but the connotation of the
word gives it much more heft than the mathematical property.  In
mathematical statistics it is irrelevant to question why it is this
particular estimator or this particular scale that is of interest - the
only objective is to prove the theorem and publish the result.

(The folklore in our department is that George Box's famous statement about
"all models are wrong" originated in a thesis defense where the candidate
began by stating that "Assuming that the model is correct" and George
interrupted to say "But all models are wrong".  It wasn't a good day for
the candidate.  I'm sorry to say that I don't know if this story is
accurate as I never took the opportunity to ask him.)

On Sat, Jul 4, 2015 at 11:36 PM landon hurley <ljrhurley at gmail.com> wrote:

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> On 07/05/2015 12:14 AM, Phillip Alday wrote:
> > On Sat, 2015-07-04 at 21:21 +0200, Karl Ove Hufthammer wrote:
> >> Den 04. juli 2015 18:18, Douglas Bates skreiv:
> >>> Having said all this I will admit that the original sin, the
> >>> "REML" criterion, was committed by statisticians.  In retrospect
> >>> I wish that we had not incorporated that criterion into the nlme
> >>> and lme4 packages but, at the time we wrote them, our work would
> >>> have been dismissed as wrong if our answers did not agree with
> >>> SAS PROC MIXED, etc.  So we opted for bug-for-bug compatibility
> >>> with existing software.
> >>
> >> Hm. I find this statement surprising. I was under the impression
> >> REML is *preferred* to ML in many situations (e.g. in simple
> >> random intercept models with few observations for each random
> >> intercept), and that *ML estimation* may result in severe bias. Do
> >> you consider maximising the REML criterion as a bug?
> >>
> >
> > This was my question as well. My understanding was that REML, like
> > Bessel's correction for the sample variance, was motivated by bias in
> > the maximum-likelihood estimator for small numbers of observations.
> > The corrected estimator is in both cases no longer the MLE, so that
> > the ML part is bit of a misnomer, but if you take "residualized"
> > expansion of RE instead of "restricted", then REML seems more like a
> > function of ML and not a "type" of ML.
> >
> > IIRC, the default in MixedModels.jl is now ML -- have you changed
> > your opinion about the utility of REML? Is there some type of weird
> > paradoxical situation with REML like with Bessel's correction -- the
> >  variance estimates are no longer biased, but the s.d. estimates
> > are?
> >
> >
> > Or is the original sin the use of the name REML when REML is no
> > longer *the* maximum likelihood?
> >
>
> I had assumed that he would have responded by now, but it is a holiday
> in the US. The position Bates is taking is explained (I think) in his
> 2010 report
> lme4: Mixed effects modelling with R in Section 5.5 `The REML
> Criterion', roughly page 123-124 in the pdf [0]. It's a short read, but
> the most relevant bit I think is:
>
> > The argument for preferring σ_R to σ_L as an estimate of σ**2 is
> > that the numerator in both estimates is the sum of squared
> > residuals at β and, although the residual vector, yobs − Xβ , is an
> > n-dimensional vector, the residual at θ satisfies p linearly
> > independent constraints, X**{T} (yobs − Xβ ) = 0. That is, the residual
> > at θ is the projection of the observed response vector, yobs , into
> > an (n − p)-dimensional linear subspace of the n-dimensional response
> > space. The estimate σR takes into account the fact that σ**2 is
> > estimated from residuals that have only n − p degrees of freedom.
> >
> > Another argument often put forward for REML estimation is that σ_R is
> > an unbiased estimate of σ**2 , in the sense that the expected value of
> > the estimator is equal to the value of the parameter. However,
> > determining the expected value of an estimator involves integrating
> > with respect to the density of the estimator and we have seen that
> > densities of estimators of variances will be skewed, often highly
> > skewed. It is not clear why we should be interested in the expected
> > value of a highly skewed estimator. If we were to transform to a
> > more symmetric scale, such as the estimator of the standard deviation
> > or the estimator of the logarithm of the standard deviation, the
> > REML estimator would no longer be unbiased. Furthermore, this
> > property of unbiasedness of variance estimators does not generalize
> > from the linear regression model to linear mixed models. This is all
> > to say that the distinction between REML and ML estimates of
> > variances and variance components is probably less important that
> > many people believe.
>
>
> best,
>
> landon
> [0]:
>
> www.researchgate.net/publictopics.PublicPostFileLoader.html?id=53326f19d5a3f206348b45af&key=6a85e53326f199010f
>
> > Best, Phillip Alday _______________________________________________
> > R-sig-mixed-models at r-project.org mailing list
> > https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models
> >
>
>
> - --
> Violence is the last refuge of the incompetent.
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