Missing values in lmer vs. HLM
There has been a reasonable amount of work on comparing different methods for random-effects meta-analysis. Using REML does produce better estimates of the effect size, in terms of coverage than maximum likelihood. Using the profile likelihood produces better results again, so maybe that is what should be used. On the other hand it is likely that the distribution of the random effects isn't normal, so it probably isn't important. If it is important there are now more general ways of fixing the bias and coverage, parametric bootstrap should work nicely, so it doesn't seem useful to use a technique that only has limited application. Maximum likelihood is applied to mixed effects logistic where it has much the same problems, and everyone just seems to ignore them.
On 6 July 2015 at 10:31, John Maindonald <john.maindonald at anu.edu.au> wrote:
Quoting Douglas:
. . . 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.
So mathematical statistics is a ?game?. That is surely a rather damning comment! It does however raise important points. My perception is that the situation has improved greatly from the ?that is all there is to it? typical stance in the published literature of the 1960s and 1970s. There?s greater pressure to back up theoretical development with computations with (often, somewhat) real data. The situation is though uneven. In some parts of the literature though (the literature on smoothing seems to me particularly rife with this problem) serious issues with the unreality of iid or at least id (independence) assumptions, for time series and/or spatial data, are just ignored! That is just one example. [For interpolation, maybe the iid assumption often makes reasonable sense for spatial data.] More important than whether the estimator has likelihood in its name, or whether it is misleading to call it some kind of likelihood estimator, is whether it serves the intended purpose. Use the median for sure where it makes sense, which incidentally is neither unbiased nor ML. I do not think that one would get away with quoting maximum likelihood estimators (eg, for wages and wage differentials in various sectors) in a set of national account figures. Here, one might be tempted to make politically fraught comments about the malign effects of highly skew distributions! Nuf sed. In many contexts, it is thought important to have numbers that add up. That, after all is what analysis of variance breakdowns of the mean are all about. Medians do not work well in that context; they do not give a table that adds up. But of course, just because one is presenting a table where effects add up to give a grand mean, distributions that are close to symmetric are important, for conceptual as well as for theoretical (normality of sampling distribution) reasons. This all becomes a whole lot more fraught for GLM models. The main benefit of REML may be that it matches what comes out computationally to the theory. Does this do serious damage to the numbers that come out, relative to the way that they will be used? Doug, do your strictures apply also to the t-statistic, which is a REML type statistic? (One uses an unbiased estimate of the variance in the denominator.) Or is the issue that it is just a means to an end? On moment estimators, comments made by Tukey seem to me relevant: "Do not assume ?that we always know what in fact we never know ? the exact probability structure . . . No data set is large enough to tell us how it should be analysed.? [Tukey: More honest foundations for data analysis. Journal of Statistical Planning and Inference, vol. 57, no. 1, pp. 21-28, 1997] Nor, I want to add, do we commonly have all the needed background information. Moment estimators can be a way to get an estimator that applies to a wide class of distributions. I and many others think this more than sufficient justification for the dispersion estimate that is widely used in quasi-likelihood computations, notable to GLM models with quasi-Poisson or quasi-binomial distributions. A key question is of course whether the dispersion might vary with the mean. And yes, this does make a whole lot more sense if one is working on a scale where the sampling distribution(s) is(are) symmetric. So perhaps what is wrong with standard quasi models is that they inflate the variance on a scale where distributions are nothing like symmetric! What is totally wrong is any failure to adjust for an inflated variance in cases (in most areas, e.g., ecology, the great majority) where the variance clearly is inflated relative to the Poisson or binomial. Note that this applies also to glmer models, notably where Poisson or binomial errors are specified. One can specify an observation level random effect. I suspect that results are often compromised because of failure to attend to this issue. In summary, there are some very important issues here, but I do not see that substituting one mathematical simplification for another is an answer. In the end, we want our models to be useful, useful I would hope for more than purposes of getting promotion! John Maindonald email: john.maindonald at anu.edu.au
On 6/07/2015, at 01:21, Douglas Bates <bates at stat.wisc.edu> wrote: 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 ? is less skewed. The distribution of
the
estimator of log(?) 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 ? is the square root of the mle of ??; the mle of log(??) is the logarithm of the mle of ??, etc. Unbiased estimators aren't invariant under transformation. The square root of an unbiased estimator of ?? is not an unbiased estimator
of
?. If an unbiased estimator were so important then we should probably
consider
the estimate of log(??), not ?? itself. The reason for our being fixated on ?? is more computational than practical. When using hand calculations it is easiest to estimate ?? then derive an estimate of ? 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:
-----BEGIN PGP SIGNED MESSAGE----- Hash: SHA512 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 ?_R to ?_L as an estimate of ?**2 is
that the numerator in both estimates is the sum of squared
residuals at ? and, although the residual vector, yobs ? X? , is an
n-dimensional vector, the residual at ? satisfies p linearly
independent constraints, X**{T} (yobs ? X? ) = 0. That is, the residual
at ? is the projection of the observed response vector, yobs , into
an (n ? p)-dimensional linear subspace of the n-dimensional response
space. The estimate ?R takes into account the fact that ?**2 is
estimated from residuals that have only n ? p degrees of freedom.
Another argument often put forward for REML estimation is that ?_R is
an unbiased estimate of ?**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
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