1. What Is Shrinkage?
Shrinkage in population pharmacokinetic (PK) modeling describes the tendency of individual parameter estimates to move toward the typical population value when the individual data contain limited information about that parameter.
This is most commonly discussed for empirical Bayes estimates (EBEs), also called individual estimates or post hoc estimates. In a population model, an individual's parameter is informed by two sources:
- the individual's observed concentration data, and
- the population distribution of the parameter.
When the individual's data are highly informative, the individual estimate can be substantially different from the population typical value. When the individual's data provide little information, the population distribution contributes more strongly and the estimate tends to remain closer to the population value.
2. Why Does Shrinkage Occur in Population PK?
A basic population PK model separates typical population parameters from between-subject variability.
For example, suppose clearance for individual \(i\) is modeled using an exponential random-effects model:
Here:
- \(CL_{\mathrm{pop}}\) is the typical population clearance.
- \(\eta_{CL,i}\) describes how individual \(i\)'s clearance differs from the typical value.
- The distribution of \(\eta_{CL}\) represents between-subject variability in clearance.
Typically, the random effect is assumed to have mean zero:
The observed concentration data provide information about \(\eta_{CL,i}\). If the data are sparse, noisy, or poorly positioned in time to estimate clearance, the model has limited evidence that the individual's \(\eta\) differs substantially from zero.
Consequently, the individual's EBE can be pulled toward zero.
Conceptually, less informative individual data can cause EBEs to move toward the population typical value. The amount of shrinkage depends on the information available in the individual data relative to the population variability and residual error.
3. What Is Eta Shrinkage?
Eta shrinkage describes the degree to which individual estimates of between-subject random effects are compressed toward their population mean, conventionally zero.
If an individual's true deviation from the population mean is represented by \(\eta_i\), the model produces an estimate such as \(\hat{\eta}_i\). With substantial information, \(\hat{\eta}_i\) can reflect the individual's underlying deviation. With weak information, \(\hat{\eta}_i\) tends to be closer to zero.
A commonly used descriptive measure of eta shrinkage is:
where \(\omega\) is the estimated population standard deviation of the corresponding eta distribution.
Equivalent expressions are sometimes written using the variance rather than the standard deviation, so the exact convention should always be checked when comparing software outputs or publications.
| Situation | Individual information | Typical effect on EBE |
|---|---|---|
| Rich sampling with informative PK phases | High | EBEs can vary substantially across individuals |
| Moderate sampling | Intermediate | Some movement toward the population mean |
| Very sparse sampling | Low | EBEs tend to cluster more closely around zero |
High eta shrinkage therefore means that the observed spread of individual EBEs can be much smaller than the estimated population variability.
4. What Is Epsilon Shrinkage?
Epsilon shrinkage concerns the individual residuals rather than the between-subject random effects.
A simple observation model might be written as:
where \(C_{ij}\) is an observed concentration, \(f(\cdot)\) is the structural PK model, \(\eta_i\) represents between-subject variability, and \(\epsilon_{ij}\) represents residual unexplained variability.
Individual residuals are often estimated conditional on the model and can also be pulled toward zero when the data contain limited information.
A commonly used descriptive expression is:
where \(\sigma\) represents the estimated residual-error standard deviation under the relevant error model.
5. Shrinkage From a Bayesian Perspective
One of the clearest ways to understand shrinkage is through the Bayesian interpretation of an individual parameter estimate.
Suppose the individual parameter is represented by \(\eta_i\), with a population distribution:
The individual's concentration observations provide a likelihood for \(\eta_i\). Combining that likelihood with the population distribution produces a conditional distribution for the individual's random effect:
The EBE can be viewed as a mode or conditional estimate of this individual-specific distribution, depending on the estimation framework.
When the individual data are highly informative, the likelihood contributes substantial information and the conditional estimate can move away from zero.
When the individual data are weakly informative, the population distribution exerts greater influence and the estimate remains closer to zero.
6. What Causes More or Less Shrinkage?
Shrinkage is influenced by the amount of information available for estimating an individual parameter and by the relative magnitude of the model's variability components.
6.1 Sparse sampling
If only a small number of concentrations are available for an individual, there may be limited information about parameters such as clearance, volume, or absorption rate.
6.2 Sampling times
The number of samples is not the only issue. Their timing matters. For example, observations concentrated around a single PK phase may provide little information about another parameter.
6.3 Residual unexplained variability
Large residual error makes it harder to distinguish an individual's underlying PK behavior from measurement and model noise. This can increase shrinkage.
6.4 Between-subject variability
The estimated magnitude of the population variability also matters. The balance between \(\omega^2\), residual error, and individual information affects how strongly estimates are pulled toward the population distribution.
6.5 Parameter identifiability
If two parameters produce similar effects on the observed concentrations over the available sampling schedule, individual estimates may be weakly identified even when the structural model fits the population data adequately.
6.6 Dose and concentration range
The dose and resulting concentrations can influence how much information the observations provide about particular parameters, especially in nonlinear models.
| Factor | Why it can affect shrinkage |
|---|---|
| Few observations | Less individual information is available |
| Poorly timed observations | Important PK processes may not be adequately observed |
| Large residual error | Individual PK differences become harder to distinguish from noise |
| Weak parameter identifiability | Individual parameters cannot be estimated precisely from the available data |
| Extensive parameter correlation | Information about one parameter may be difficult to separate from another |
7. How Shrinkage Changes the Distribution of EBEs
Consider a population model in which the true between-subject random effects for clearance have a standard deviation of \(\omega=0.5\).
If the individual data are highly informative, the estimated EBEs might have a distribution that resembles the population random-effect distribution.
With substantial shrinkage, the EBE distribution becomes narrower.
Conceptual illustration: shrinkage reduces the apparent spread of individual EBEs relative to the underlying population random-effect distribution.
This narrowing is important because an EBE distribution should not automatically be interpreted as a direct empirical estimate of the true between-subject variability.
8. Why Does Shrinkage Matter for PK Diagnostics?
Individual EBEs are often used to create diagnostic plots. For example, a modeler may plot an EBE for clearance against body weight, age, renal function, or another covariate.
This can be useful when shrinkage is modest. However, substantial shrinkage changes how much information the EBE contains about the individual's underlying parameter.
When EBEs are strongly compressed toward zero, relationships between EBEs and covariates can become attenuated or difficult to detect.
For example, suppose the true relationship is:
If the estimated \(\hat{\eta}_{CL,i}\) values are strongly shrunk toward zero, an EBE-versus-weight plot may show a weaker apparent relationship than would be obtained from more informative individual data.
9. Shrinkage and Covariate Relationships
Covariate analysis is one of the most important settings in which shrinkage needs to be considered.
Suppose a population PK model contains a covariate relationship for clearance:
A modeler might initially examine the relationship between estimated individual clearance and body weight. But if the individual estimates have substantial shrinkage, the observed spread of the estimates may be substantially smaller than the true underlying spread.
This creates an important distinction:
- EBE-based exploration uses individual estimates and can be affected by shrinkage.
- Model-based covariate evaluation estimates the covariate relationship directly within the population model and does not rely solely on the visual spread of EBEs.
Consequently, high shrinkage should not be interpreted as evidence that covariates have no relationship with PK parameters.
10. Why Shrinkage Matters for PK/PD Modeling
Shrinkage becomes particularly important when individual PK estimates are subsequently used as inputs to a pharmacodynamic analysis.
Suppose a modeler first estimates individual clearance values and then uses those EBEs as predictors in a separate exposure-response analysis.
If the clearance EBEs are highly shrunk, they may contain limited information about true individual differences in clearance. This can affect downstream analyses.
For example, an analysis might attempt to relate an estimated individual exposure metric to an observed response:
If \(AUC_i\) is calculated using highly shrunk individual parameter estimates, the resulting exposure measure may not preserve all of the relevant between-subject variation.
This is one reason integrated population PK/PD modeling can be preferable to a simple two-stage procedure when the scientific objective requires accurate propagation of parameter uncertainty and individual variability.
11. Common Problems Caused by High Shrinkage
High shrinkage is not automatically a reason to reject a model. It becomes important when the modeler's intended use of individual estimates requires information that the data do not adequately provide.
11.1 Compressed EBE distributions
EBEs can appear much less variable than the estimated population distribution.
11.2 Weak EBE-covariate relationships
Relationships can be visually attenuated when individual estimates contain little information about individual deviations.
11.3 Diagnostic interpretation
Plots based on individual random effects may be less informative when shrinkage is substantial.
11.4 Downstream exposure-response analyses
Highly shrunk individual parameters may provide poor substitutes for directly estimated individual exposure or PK quantities in a two-stage analysis.
11.5 Misleading residual diagnostics
When individual predictions depend heavily on population predictions because of shrinkage, diagnostic behavior involving conditional predictions and residuals can require careful interpretation.
12. Shrinkage and Residual-Based Diagnostics
Population PK diagnostics often include residual-based measures such as individual weighted residuals (IWRES), conditional weighted residuals (CWRES), and related quantities.
The purpose of these diagnostics is to assess whether observed concentrations are consistent with model predictions after accounting for the specified variability structure.
High eta shrinkage can affect diagnostics that depend strongly on individual parameter estimates because the individual predictions may become increasingly similar to population predictions.
For this reason, diagnostic interpretation should consider:
- the amount of eta shrinkage;
- the amount of epsilon shrinkage;
- the sampling design;
- the residual error model;
- the number and timing of observations per individual; and
- whether the diagnostic requires reliable individual parameter estimates.
Shrinkage is therefore best considered alongside visual predictive checks, prediction-corrected visual predictive checks, residual diagnostics, parameter estimates, and other model-evaluation tools rather than in isolation.
13. Worked Example: Calculating Eta Shrinkage
Suppose a population PK model estimates the between-subject variability in clearance as:
After obtaining individual EBEs, suppose their empirical standard deviation is:
Step 1: Apply the shrinkage formula
Step 2: Substitute the values
Step 3: Calculate
Thus, the estimated eta shrinkage is:
Conceptually, the observed spread of the EBEs is about 60% of the estimated population random-effect standard deviation under this descriptive calculation.
14. How Shrinkage Affects an Individual Parameter Estimate
Suppose the typical population clearance is:
Assume an individual's underlying deviation would correspond to an individual clearance of approximately 8 L/h. In the exponential model:
the corresponding random effect is:
Now suppose the individual's data are sparse and the EBE is estimated as only:
The EBE-based individual clearance becomes:
The estimated individual clearance is therefore closer to the population value of 5 L/h than the hypothetical underlying value of 8 L/h.
This illustrates the central idea of shrinkage: the EBE reflects the combination of individual information and population information, rather than being a direct measurement of the individual's true parameter.
15. Shrinkage Is Parameter-Specific
It is important not to describe a population PK model simply as having "high shrinkage" without specifying which random effect is shrinking.
For example, a model could have:
| Random effect | Example shrinkage | Possible interpretation |
|---|---|---|
| \(\eta_{CL}\) | 15% | Individual clearance estimates retain relatively substantial information |
| \(\eta_V\) | 35% | Individual volume estimates are more compressed |
| \(\eta_{ka}\) | 70% | Individual absorption-rate estimates are strongly informed by the population distribution |
Different parameters can have very different levels of individual identifiability.
For example, a sparse sampling design may contain enough information about overall exposure to estimate clearance reasonably well while providing little information about an absorption rate constant.
16. Shrinkage and Sampling Design
Shrinkage is strongly connected to the information content of the sampling design.
Consider a one-compartment oral PK model with first-order absorption:
To estimate \(k_a\), the study needs observations that provide information about the absorption portion of the concentration-time profile.
If all samples are collected long after the absorption phase, the data may provide little information about \(k_a\). The model can still estimate a population-level absorption rate, but individual \(k_a\) EBEs may exhibit substantial shrinkage.
Similarly, if there are few samples during the terminal elimination phase, individual clearance or terminal-rate estimates may be weakly informed.
This leads to an important design principle:
17. How Can Shrinkage Be Reduced?
If high shrinkage limits the intended use of individual estimates, several approaches may improve the information available to the model.
17.1 Improve sampling
Collect samples at times that are informative for the parameters of interest. This can be guided by optimal design or simulation-based design evaluation.
17.2 Increase the number of informative observations
Additional samples can provide more information, although simply increasing the number of samples is not sufficient if the new observations are redundant.
17.3 Improve the residual error model
An inappropriate or overly large residual-error model can reduce the apparent information available from individual observations.
17.4 Reconsider the structural model
An unnecessarily complex model can create parameter-identification problems. Conversely, an oversimplified model can leave systematic patterns in the residuals that obscure the intended interpretation.
17.5 Incorporate important covariates
Explaining systematic between-subject differences through measured covariates can reduce unexplained heterogeneity, although adding covariates should be based on scientific and statistical considerations rather than used solely to manipulate shrinkage.
17.6 Use an integrated model when appropriate
For exposure-response analyses, directly modeling PK and PD together can avoid treating highly shrunk individual PK estimates as though they were observed quantities.
18. Is High Shrinkage Always Bad?
No. High shrinkage is not automatically evidence that a population PK model is invalid.
A model can adequately describe population concentration-time data while producing highly shrunk individual estimates for one or more parameters.
This may be entirely expected when the study design contains limited information at the individual level.
The important question is therefore:
If the objective is population-level prediction, high shrinkage for an individual parameter may be less problematic than if the objective is to use that individual's EBE as a reliable measure of their actual PK parameter.
| Use of model | Potential relevance of shrinkage |
|---|---|
| Population-level concentration prediction | May be relatively limited if the population model adequately describes the data |
| EBE versus covariate plots | Can be important because shrinkage compresses individual variability |
| Individual dosing based on EBEs | Important when individual parameter estimates are expected to be reliable |
| Two-stage PK/PD analysis | Potentially important because shrunk estimates may lose individual information |
| Simulation of population behavior | Depends primarily on the population model and its variability structure, not simply on the empirical spread of EBEs |
19. Common Misinterpretations of Shrinkage
Mistake 1: "High shrinkage means the model is wrong."
Not necessarily. High shrinkage can be a consequence of sparse or poorly informative data.
Mistake 2: "Shrinkage means individuals are actually becoming more similar."
No. Shrinkage concerns the estimated individual parameters. It does not imply that biological between-subject variability has disappeared.
Mistake 3: "A narrow EBE distribution means low population variability."
Not necessarily. High shrinkage can make the EBE distribution substantially narrower than the estimated underlying random-effect distribution.
Mistake 4: "No EBE-covariate relationship means the covariate does not matter."
High shrinkage can attenuate EBE-covariate relationships.
Mistake 5: "Every parameter has the same shrinkage."
Different parameters can have very different levels of individual information and therefore different shrinkage.
Mistake 6: "Shrinkage is a property of the drug."
Shrinkage is a property of the model-data combination. Sampling design, residual variability, model structure, parameter variability, and the population all contribute.
20. How Should Shrinkage Be Reported?
When shrinkage is important to the interpretation of a population PK analysis, it is useful to report it explicitly and identify the associated parameter.
For example, a report might state that eta shrinkage was approximately 20% for clearance, 35% for volume, and 65% for absorption rate.
Good reporting should make clear:
- which random effect was evaluated;
- the definition or formula used for shrinkage;
- whether the reported quantity is eta or epsilon shrinkage;
- which estimation method and software generated the individual estimates; and
- how shrinkage affected the interpretation of individual diagnostics or downstream analyses.
Because different shrinkage definitions and estimation procedures can be used, numerical values should be interpreted in the context of the calculation method.
21. Worked Example: Interpreting Different Shrinkage Values
Suppose a population PK model produces the following results:
| Parameter | Population variability | Eta shrinkage |
|---|---|---|
| Clearance | \(\omega_{CL}=0.40\) | 12% |
| Volume | \(\omega_V=0.35\) | 28% |
| Absorption rate | \(\omega_{ka}=0.70\) | 72% |
The results suggest that individual clearance estimates contain substantially more information than individual absorption-rate estimates in this hypothetical dataset.
The high shrinkage for \(k_a\) might be expected if the study has few observations during the absorption phase.
It would therefore be inappropriate to conclude that absorption-rate variability is necessarily small simply because the \(k_a\) EBEs show a narrow distribution.
22. Shrinkage as a Balance of Information
A useful conceptual framework is to think of an individual estimate as a balance between two sources of information:
An EBE combines information from the individual's observations with information supplied by the population model. Shrinkage becomes more pronounced when individual information is weak relative to the population information.
This perspective is useful because it explains why shrinkage can change when the same drug is studied using a different sampling design, population, residual-error model, or dosing regimen.
23. A Practical Workflow for Evaluating Shrinkage
- Identify the individual estimates. Determine which EBEs or post hoc estimates are being evaluated.
- Calculate or obtain shrinkage. Identify whether eta, epsilon, or another shrinkage metric is being reported.
- Associate shrinkage with the parameter. Examine clearance, volume, absorption parameters, and other random effects separately.
- Inspect the sampling design. Determine whether the data contain adequate information for the parameter.
- Review residual variability. Large unexplained residual variability can contribute to weak individual information.
- Examine EBE distributions. Compare the empirical spread of EBEs with the estimated population random-effect distribution.
- Interpret EBE-covariate plots cautiously. High shrinkage can weaken apparent relationships.
- Consider the intended use. Ask whether individual estimates are being used for diagnostics, covariate exploration, dosing, or downstream PK/PD analysis.
- Use integrated modeling when appropriate. Avoid treating highly shrunk EBEs as directly observed individual parameters in analyses where individual information is critical.
24. Key Takeaways
- Shrinkage describes the tendency of individual population-PK estimates to move toward the population typical value when individual data contain limited information.
- Eta shrinkage concerns individual estimates of between-subject random effects, which conventionally have a population mean of zero.
- Epsilon shrinkage concerns individual residual estimates and reflects limited information about individual residual deviations.
- Shrinkage arises naturally from the hierarchical structure of population PK models, where individual observations are combined with population-level information.
- Sparse sampling, poorly timed observations, high residual variability, and weak parameter identifiability can increase shrinkage.
- Different parameters in the same population PK model can have very different levels of shrinkage.
- A narrow EBE distribution does not necessarily imply that the underlying population variability is small.
- High shrinkage can attenuate apparent relationships between EBEs and covariates.
- High shrinkage is not automatically evidence that a population PK model is inadequate.
- The importance of shrinkage depends strongly on how individual estimates are being used.
- Highly shrunk individual PK estimates should be interpreted cautiously in two-stage PK/PD or exposure-response analyses.
- Improving sampling design and individual information is often more effective than simply changing estimation procedures when the objective is to reduce shrinkage.
- Shrinkage should be interpreted together with model diagnostics, parameter estimates, sampling design, residual variability, and the scientific purpose of the analysis.
Where to Go Next
A natural next step is to study Empirical Bayes Estimates in Population PK in more detail, including MAP estimation, post hoc parameter estimation, conditional distributions, and the distinction between population parameters and individual parameters.
From there, the concepts connect naturally to Covariate Modeling in Population PK, Visual Predictive Checks, Model Diagnostics, Optimal Sampling Design, and PK/PD Modeling.
References
- Sheiner LB, Ludden TM. Population pharmacokinetics/dynamics. Annual Review of Pharmacology and Toxicology. 1992;32:185–209.
- Sheiner LB, Ludden TM, Beal SL. Hierarchical statistical models for the analysis of population pharmacokinetic data. Journal of Pharmacokinetics and Biopharmaceutics. 1979;7:305–318.
- Ette EI, Williams PJ. Population pharmacokinetics I: background, concepts, and models. Annals of Pharmacotherapy. 2004;38:1702–1706.
- Ette EI, Williams PJ. Population pharmacokinetics II: estimation methods. Annals of Pharmacotherapy. 2004;38:1907–1915.
- Food and Drug Administration. Population Pharmacokinetics: Guidance for Industry. U.S. Department of Health and Human Services.
- European Medicines Agency. Guideline on Reporting the Results of Population Pharmacokinetic Analyses.
Definitions and shrinkage calculations can vary across software and publications. When reporting a numerical shrinkage estimate, specify the calculation used and the parameter to which it applies.