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

[Daniel Wright]

I like the baseball example in here (and the paper too), but if you don't know baseball, similar examples could be thought up in other sports.

http://www-stat.stanford.edu/~ckirby/brad/other/Article1977.pdf

Daniel B. Wright, Ph.D.
Senior Research Associate - Learning Insights Team 
500 ACT Drive, Iowa City, IA 52243-0168
512.320.1827





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-----Original Message-----
From: r-sig-mixed-models-bounces at r-project.org [mailto:r-sig-mixed-models-bounces at r-project.org] On Behalf Of Emmanuel Curis
Sent: Wednesday, September 18, 2013 3:20 PM
To: r-sig-mixed-models at r-project.org
Subject: [R-sig-ME] Question about what is "shrinkage"...

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 « shrinkage »).
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 at parisdescartes.fr

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

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