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What is Shrinkage?

In 2007, Mats Karlsson and Radojka Savic published a perspective in Clinical Pharmacology & Therapeutics titled “Diagnosing Model Diagnostics.” In this article they examined the use of diagnostic plots to evaluate the adequacy of model fits for nonlinear mixed-effects analysis. Although there is a wealth of information in this article, the population PK analysis community latched on to the term “shrinkage” that was used to describe the phenomenon that occurs when a model is over-parameterized for the amount of information contained in the data. They described ε-shrinkage and η-shrinkage which I will try to summarize here.


ε is the term that refers to the residual error in the model. ε-shrinkage is calculated as 1-SD(IWRES) where IWRES is an individual weighted residual. When data is very informative, ε-shrinkage is zero and it moves toward 1 when data is less informative. Thus ε-shrinkage can range from 0% to 100%. But what is ε-shrinkage? When ε-shrinkage is large, the individual predictions are of little value for assessing model adequacy because the individual predictions “shrink” back toward the observation, meaning that IPRED ≈ DV (observation).


η is the term that refers to the between individual variation in the model, in other words, how patients differ from one another. η-shrinkage is calculated as 1-SD(η)/ω where η are the between individual variation terms and ω is the population model estimate of the standard deviation in η. When data is very informative, η-shrinkage is zero and it moves toward 1 when data is less informative, meaning that η-shrinkage can range from 0% to 100%. When η-shrinkage is high, the individual parameter estimates “shrink” back toward the population parameter estimate, meaning that CLi ≈ CLpopulation. When η-shrinkage is large, diagnostic plots of individual parameter estimates and covariates could be misleading.

What to do about shrinkage?

So, how much shrinkage is too much? Karlsson and Savic suggest that bias can result with only 20-30% shrinkage. What should you do if you see shrinkage? In general, shrinkage indicates that the model is over-parameterized for the data that is available. The first recommendation is to simplify the model if possible. If that doesn’t resolve the issue, the second recommendation is to remember that the diagnostic plots may be misleading.

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