[R-sig-ME] Question about what is "shrinkage"...

[Emmanuel Curis]

Hello,

Thanks Daniel for the link on the clear article (despite I indeed do
not know anything about baseball) and Douglas for the detailed
answer. Quite interestingly, the article is more on the side of the
estimator and Douglas' answer on the side of the reduced variance, at
least as I understand it, but I think I begin to understand the link
between the two.

But there are still a few questions I have, some of them
philosophical...

When reading the paper, the two examples correspond to setups that
could be handled by random-effect models (the baseball player or the
town). In fact, in the end of the paper, individual mean values coming
from a random variable is mentionned.

Does it mean that individual means obtained by random effect models as
used in lmer, for instance, are themselves a kind of shrinkage
estimator --- that is, already corrected by a shrinkage factor, but
not given by a formula similar to the one cited in the paper? I know
that random effects themselves are not (conditionnal) means, but
modes, but when added to the fixed effects parts, corresponding to the
mean (at least in linear models), aren't they comparable to (shrinked)
means?

Would it be, in this case, an argument for prefering random effects
over fixed effects when the number of modalities is « high » (>= 3 if
I read correctly the paper, but may be another limit for such models
and for cases of unkwnown, estimated variance?), beside convergence
problems, and instead prefer fixed effects below even if
philosophically a random effect would be needed (experiments on two
patients only) --- and that there is a link between the efficiency of
the shrinkage effect and the ability to estimate correctly the
variance?

This would also explain how it is possible to associate a shrinkage to
each random effect...

As far as I could see, however, the shrinkage estimator can also
improve regression coefficients, when they are more than 3. Does it
still holds when dealing with multidimensionnal vectors of which each
composent represent very different things ? And for regression
coefficients, if shrinked version gives better values, wouldn't it be
logical to build tests on these coefficients on the shrinked values?
Is it possible? (but these questions are on the frontier to be
off-topic I guess).

My other concern is about the usage of shrinkage as a diagnostic. If I
understood correctly Douglas answer, size of shrinkage measures how
informative is the data of a single patient to estimate its own
value. Hence, if shrinkage is important, does it mean that the model
is not suitable for looking into individual predictions, but only
average ones --- hence, useless in PK-pop for adaptating doses for
instance? Is there any guides to define what is an acceptable
shrinkage? And does it have other values for model's diagnostic and
interpretation?

Last point: I understand well in the paper how to calculate the
shrinkage factor (there seems to be several different but close
formulas according to the reference, but I guess these are only
variants?), using obtained values for each individual. But for several
linear models, as mentionned by Douglas, it is not possible to obtain
individual parameters. In such case, how is shrinkage
computed/estimated ?

Thanks again in advance for any answer,

-- 
                                Emmanuel CURIS
                                emmanuel.curis at parisdescartes.fr

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