What Shrinkage Describes
An individual Bayesian estimate combines information from a patient’s observations with information from a population model. When the observations provide limited information about an individual parameter, the posterior estimate is pulled toward the population value. This movement is called shrinkage. It describes how an estimate is formed under the model; it does not, by itself, show that the patient’s true parameter is close to the population value.
The size of the pull depends on both the information in the individual data and the spread allowed by the population prior. A narrowly distributed prior can exert substantial influence, as can a weakly informative likelihood. To quantify the pull, we must specify the parameter scale and the model. The calculations below use a normal approximation on the log scale, a useful way to represent a positive PK parameter such as clearance.
A Model That Makes the Pull Explicit
Let \(\theta_i\) be an individual’s log clearance. Suppose, after accounting for the PK model and dosing history, the population prior for this individual is normal, and the individual likelihood can be summarized by an approximately normal estimate \(z_i\):
Here \(\mu\) is the population mean on the log scale, \(\omega^2\) is the between-individual prior variance on that scale, \(z_i\) is the data-only estimate, and \(s_i^2\) is its sampling variance. For this conjugate calculation, treat \(\mu\), \(\omega^2\), and \(s_i^2\) as known. The likelihood summary must be approximately unbiased for \(\theta_i\), with its uncertainty adequately represented by \(s_i^2\); the prior and likelihood are combined as specified, with no additional dependence between them.
Completing the square in the product of the prior density and the likelihood gives a normal posterior. Its mean is a weighted average of the prior mean and the data-only estimate:
Define \(B_i=s_i^2/(\omega^2+s_i^2)\) as the prior-weight fraction and \(A_i=\omega^2/(\omega^2+s_i^2)\) as the data-weight fraction. Since \(A_i+B_i=1\), the posterior mean can also be written \(m_i=\mu+A_i(z_i-\mu)\). Thus \(A_i\) is the fraction of the data-only displacement from the prior mean retained in the posterior, and \(B_i\) is the fraction removed by shrinkage. The posterior variance is:
This result also explains the direction of the pull. If \(z_i>\mu\), then \(m_i\) lies between \(\mu\) and \(z_i\); if \(z_i<\mu\), it lies between \(z_i\) and \(\mu\). When the data are precise relative to the prior spread, \(s_i^2\) is small and \(A_i\) approaches 1. When the likelihood is weak relative to the prior, \(s_i^2\) is large and \(A_i\) approaches 0.
Worked Examples on the Log-Clearance Scale
Worked Example: Moderate Information About Clearance
Suppose a hypothetical patient has a prior for log clearance with mean \(\mu=\log(8)\), corresponding to a prior median clearance of 8 L/h, and standard deviation \(\omega=0.30\) log units. The individual data yield \(z_i=\log(12)\) with standard error \(s_i=0.20\) log units. Assume the normal model above is adequate.
The data weight and prior weight are:
The posterior mean on the log scale is:
The data-only displacement from the prior mean is retained by about 69.23%; about 30.77% is removed. Exponentiating the posterior log mean gives a posterior median clearance of \(\exp(2.3602)\approx10.59\) L/h. The posterior standard deviation is \(\sqrt{0.09(0.04)/0.13}\approx0.1664\) log units. A normal 95% posterior interval on the log scale is \(2.3602\pm1.96(0.1664)\), or approximately \((2.034,2.686)\). Exponentiating gives approximately \((7.64,14.68)\) L/h. Under this model, the interval contains 95% of the posterior probability for clearance; it is not a statement that there is a 95% frequentist probability that a fixed parameter lies in the computed interval.
Worked Example: A Sparse Estimate Pulled Toward the Population
Now consider a different hypothetical patient. The prior log-clearance mean is \(\mu=\log(10)\), with \(\omega=0.25\) log units. The data-only estimate is \(z_i=\log(16)\), but its standard error is \(s_i=0.50\) log units. Assume the same normal prior-likelihood model.
The data weight is \(A_i=0.25^2/(0.25^2+0.50^2)=0.0625/0.3125=0.2000\), so the prior weight is \(B_i=0.8000\). The posterior mean is:
The posterior median clearance is \(\exp(2.3966)\approx10.99\) L/h, much closer to the prior median of 10 L/h than to the data-only estimate of 16 L/h. That is a consequence of the weak likelihood relative to the prior spread. The posterior standard deviation is \(\sqrt{0.0625(0.25)/0.3125}=\sqrt{0.05}\approx0.2236\) log units. A 95% posterior interval is \(2.3966\pm1.96(0.2236)\), or approximately \((1.958,2.835)\) on the log scale, which corresponds to about \((7.09,17.03)\) L/h. The interval remains broad: the point estimate’s closeness to 10 L/h should not be mistaken for precise evidence that the patient’s clearance is near 10 L/h.
Worked Example: More Precise Data, Less Shrinkage
Use the same prior median of 10 L/h and \(\omega=0.25\) log units, and the same data-only estimate \(z_i=\log(16)\), but suppose its standard error is \(s_i=0.08\) log units. This represents a more informative likelihood under the stated model.
The data weight is \(A_i=0.0625/(0.0625+0.0064)=0.0625/0.0689\approx0.9071\); the prior weight is \(B_i=0.0064/0.0689\approx0.0929\). Therefore:
The posterior median clearance is \(\exp(2.7289)\approx15.31\) L/h. About 90.71% of the data-only displacement from the prior mean is retained, compared with 20.00% in the sparse-data example. The posterior standard deviation is \(\sqrt{0.0625(0.0064)/0.0689}\approx0.0762\) log units. Both the smaller shrinkage fraction and the narrower posterior reflect the greater information in the likelihood, conditional on the model assumptions.
Why a Shrunken Estimate Is Not Evidence of a Typical Patient
The posterior estimate is a compromise between prior and likelihood information. In the sparse-data example, its proximity to the prior median primarily reflects the low data weight. It does not establish that the patient’s underlying clearance is typical, nor does the posterior point estimate reveal how much uncertainty remains. To assess what the data say about the individual, consider the posterior distribution and its uncertainty, not only the location of its mean or mode.
A related issue arises when examining empirical Bayes estimates across a group. Their spread can be smaller than the population random-effects spread because each estimate is pulled toward the population mean, especially for individuals with sparse or uninformative data. A common descriptive measure for a random effect is \(1-\operatorname{SD}(\widehat{\eta}_i)/\omega\), where \(\widehat{\eta}_i\) are individual estimates of a mean-zero random effect and \(\omega\) is its population standard deviation. Another uses variances, \(1-\operatorname{Var}(\widehat{\eta}_i)/\omega^2\). These group-level summaries are not the same as the individual weight \(B_i\) derived above, and their numerical values depend on the data, estimator, and model. A narrow spread of individual estimates may therefore reflect estimation shrinkage rather than genuinely similar patients.
The log scale also matters. Here, \(\exp(\mu)\) is the prior median clearance when log clearance is normal; the prior arithmetic mean clearance is \(\exp(\mu+\omega^2/2)\). The weighted-average result applies to the log parameter, not directly to clearance in L/h. Likewise, the exponentiated posterior log mean is the posterior median clearance, not its arithmetic mean. Under a normal posterior for log clearance with mean \(m_i\) and variance \(v_i\), the arithmetic mean clearance is \(\exp(m_i+v_i/2)\). State which summary is being reported and its units.
Comparisons, Diagnostics, and Limits
| Summary | What it represents | What it does not establish |
|---|---|---|
| Prior mean or median | Population information before using this individual’s observations | The patient’s individual parameter value |
| Data-only estimate \(z_i\) | The likelihood-based summary without the individual prior | A precise estimate when the data are sparse |
| Posterior estimate \(m_i\) | A combination of prior and likelihood information under the model | That the patient is typical when the estimate is near the population value |
In the normal model, the posterior mean and posterior mode coincide. In a nonlinear PK model, posterior estimates may be computed by methods such as those introduced in the earlier material on MAP estimation, posterior means, Laplace approximation, and posterior sampling. A single normal weight may then be only a local approximation; do not assume that the same formula exactly describes a nonlinear posterior.
Useful checks include comparing the data-only and posterior estimates on the same scale, checking whether the reported posterior uncertainty is consistent with the stated information, and verifying that the estimate lies between prior mean and data-only estimate when the scalar normal model applies. If it does not, check the scale, the calculation, and whether the assumptions of that model are actually being used. In more complex PK models, a posterior estimate need not lie between two scalar summaries because the likelihood and parameterization may be nonlinear.
The formula relies on a correctly specified prior, an adequate likelihood summary, and known variances. In practice these quantities are estimated, and nonlinear models may produce asymmetric posteriors or parameter-dependent information. The weight is therefore an interpretation under a specified model, not a direct measurement of how “typical” a patient is. Shrinkage alone also does not determine whether the prior and data are compatible; that question requires separate assessment.
Common Pitfall
- Wrong reasoning: “The posterior clearance is close to the population median, so the patient probably has typical clearance.”
- Correction: A posterior estimate can be close to the population value because the individual data are weak relative to the prior. Examine the data weight and posterior uncertainty before drawing an individual-level conclusion.
- Wrong reasoning: “The individual shrinkage fraction is the same as one minus the spread of all individual estimates divided by the population spread.”
- Correction: The first is defined here for an individual under a scalar normal model; the second is a group-level descriptive measure. They answer related but different questions.
Check Your Understanding
Use the normal prior-likelihood model in this tutorial unless a question says otherwise. The solutions follow the questions.
- If \(\omega=0.20\) and \(s_i=0.10\), calculate the data weight and shrinkage fraction.
- A prior log-parameter mean is \(\mu=1.5\), and a data-only estimate is \(z_i=2.0\). If the data weight is 0.40, calculate the posterior mean and the fraction of the data-only displacement removed.
- In one sentence, explain why a posterior estimate near the prior mean does not by itself show that an individual parameter is typical.
- For a lognormal parameter with posterior log mean \(m_i\) and variance \(v_i\), which exponentiated value is the posterior median, and which is the posterior arithmetic mean?
Solutions. (1) \(A_i=0.20^2/(0.20^2+0.10^2)=0.04/0.05=0.8000\); hence \(B_i=0.2000\). (2) \(m_i=\mu+A_i(z_i-\mu)=1.5+0.40(2.0-1.5)=1.70\). The removed fraction is \(1-A_i=0.60\). (3) It may be near the prior mean because the likelihood is weak relative to the prior, so the estimate alone does not establish the individual’s true value. (4) The posterior median is \(\exp(m_i)\); the posterior arithmetic mean is \(\exp(m_i+v_i/2)\).