It turns out that population pharmacokinetics is a good subject area in
which to discuss the concept of shrinkage because the information on each
individual subject may be sparse.  To avoid inconveniencing a patient or
test subject and also to avoid the expense of multiple assays we would like
to keep the number of samples per patient low.  If we ignore the fact that
the patients are drawn from a population and treat each patient as
essentially a brand new study then the parameter estimates we can expect to
obtain from such sparse data will be poorly defined.  In fact, we may not
even get parameter estimates if the number of data values is less than the
number of parameters in the model.

But if we model the data as if there were "typical" values of the
pharmacokinetic values in the population and each patient's specific values
of these parameters represent a deviation from the typical values then we
can "borrow strength" from the population, to use John Tukey's phrase.  In
a mixed-effects model the deviations are modeled as a sample from some
population, which creates a penalty on very large deviations.  This has the
effect of pulling the estimates for each individual toward the population
mean.  The figure in the book that you mention shows that when an
individual's data determine the parameter estimates well, the amount of
this shrinkage toward the population values will be small.  However, if the
individual's data produce very imprecise estimates by themselves then the
"shrinkage" toward the population values will be greater. Essentially we
are assuming that their parameter values will be similar to those of others
in the population unless their data clearly shows that they are different.

In practice this balancing of the population values and the indications
from the individual's measurements can be used to get more accurate
predictions.  When I first started working in nonlinear mixed-effects I sat
in on a pharmacokinetics class and was surprised and gratified one day to
see a guest presentation about a way of calibrating dose of a drug with a
narrow therapeutic window using early results from the patient and the
population pharmacokinetic estimates.  The drug is Theophylline which is
used to treat asthma.  If a person arrived at the ER with a severe asthma
attack they would load them up with a bolus dose and then switch to an IV
drip where they could control the infusion rate.  Early measurements of
serum concentration would not be sufficient to fit a pharmacokinetic model
of the serum concentration, which needs to kept within a narrow range to be
of benefit to the patient but not harmful to them. However if you combined
early measurements on the patient's serum concentration with the population
model you could get a much better estimate of the steady-state
concentration.

Another instance of shrinkage that I see is in the so-called "Item Response
Theory" models to derive estimates of a student's ability as measured by
their correct/incorrect responses on a test.  The way I would advocate
doing this is to fit a generalized linear mixed model with random effects
for the participant and for the item (i.e. question).  It is not a common
practice to do this.  Most of the time the model has fixed-effects for the
participant and random effects for the item.  In such a case the ability
estimate for a student who gets all the questions correct is infinite.
 This may not seem like an important case but when hundreds of thousands of
students are taking achievement tests in an American state there will be
some who do get all the questions correct.  Modeling the ability as a
random effect would shrink these estimates back from infinity to a large,
but finite, value.



On Wed, Sep 18, 2013 at 3:20 PM, Emmanuel Curis <
emmanuel.curis@parisdescartes.fr> wrote:

> Hello,
>
> I've read several time the term "shrinkage", either on this list or,
> even more often, when dealing with population pharmacokinetics, and I
> am not quite sure what it means and what is its usage... Could it be
> possible to have either some references or some explanations? I give
> below a longer version, with how far I could get and where I am
> stopped... Thanks in advance for any help!
>
> I've search a little bit on the net; shrinkage seems related to the
> fact that after regression, it is possible to obtain more precise, but
> slightly biased, estimators of the coefficients, by making them a
> little bit smaller than the actual value (hence Ť shrinkage ť).
> However, in the discussions especially about PK-pop models, the usage
> of "shrinkage" does not seem to me coherent with this meaning...
> Instead, it seems to be a property of mixed-models, linked to
> variances estimations, and used to check the model quality or validaty
> in some way, with sentences like "this model increased the shrinkage"
> and mentions of something like "random effects parameters shrinkage"
> and "residuals shrinkage" (eta-shrinkage and epsilon-shrinkage)...
>
> My other idea was related to the fact that when modeling a set of
> repeated measures on several patients, with a straight line, the set
> of slopes shows less variability when using a mixed model on the whole
> set, than using separate lines for each patient --- as exemplified for
> instance in Douglas Bate's book. Hence, variance of slopes is shrinked
> in the mixed model approach compared to the variance obtained from the
> sample of all individual slopes. This idea seems closer to the use and
> terminology above, but I can't see if shrinkage is a good or bad
> thing...
>
> I mean, since one imposes a given distribution, hence a constraint, on
> slopes, the fact that variance is smaller is not a surprise and it
> could be a drawback of the estimation, leading to underestimation.
> Conversely, variance of individual slopes also includes the residual
> variability, hence is expected to be higher. Is it true then that the
> mixed-model estimation is better? But in that case, how shrinkage can
> be used to quantify the correctness of a model?
>
> Thanks in advance,
> Best regards,
>
> --
>                                 Emmanuel CURIS
>                                 emmanuel.curis@parisdescartes.fr
>
> Page WWW: http://emmanuel.curis.online.fr/index.html
>
> _______________________________________________
> R-sig-mixed-models@r-project.org mailing list
> https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models
>
>

	[[alternative HTML version deleted]]

