Why Compare Shrinkage Across Random Effects?
A population pharmacokinetic model may estimate interindividual random effects for clearance, volume, absorption rate, and other parameters. Comparing their eta shrinkage can show which effects have less individual-level information in a particular data set. That comparison is useful only when the same shrinkage definition is used for every effect and the limits of the comparison are understood.
Shrinkage is not an intrinsic ranking of pharmacokinetic parameters. It depends on the fitted population variability, the information in the subjects’ observations, the model structure, and the subjects included in the summary. In particular, clearance, volume, and absorption may be observed with very different precision under the same sampling schedule. A larger shrinkage percentage for one effect therefore describes the estimated EBEs in that analysis; it does not establish that the underlying parameter is universally harder to estimate.
A distinct convention is standard-deviation-based eta shrinkage, \(100(1-s_{\hat{\eta}_j}/\omega_j)\%\). It compares standard deviations rather than variances and generally gives a different percentage. This tutorial uses variance-based shrinkage for its comparisons; the two definitions must not be mixed in a table or interpretation.
What Makes Cross-Parameter Comparisons Meaningful?
Suppose a model uses log-normal individual parameters, such as \(CL_i=TVCL\exp(\eta_{CL,i})\), with analogous forms for volume and absorption. The eta effects are dimensionless, even though clearance, volume, and absorption rate have different physical units. Comparing their eta shrinkage is therefore not a comparison of a clearance value in volume units with an absorption rate. It is a comparison of how much the estimated distribution of each dimensionless random effect is contracted relative to its own fitted population variance.
That comparison is most interpretable when each effect is defined on a comparable modeling scale, the same subjects are used, and the same variance-based formula is applied. If one effect is modeled additively and another on a log scale, their eta distributions may have different meanings; equal percentages would not imply equal changes in the natural-scale PK parameters. Nor do equal percentages imply equal absolute uncertainty: each percentage uses its own \(\omega_j^2\) as its reference.
The population standard deviation and EBE standard deviation also need to come from the same fitted model and the stated subject set. The EBE spread is an empirical summary, not a direct measurement of the true random-effect spread. The comparison is conditional on the model, its estimated parameters, and the available sampling information.
Worked Example: Comparing Clearance, Volume, and Absorption
Consider a hypothetical model with log-scale random effects for clearance, volume, and absorption rate. For the same 80 subjects, suppose the fitted population standard deviations and EBE standard deviations are as follows: clearance, \(\omega_{CL}=0.30\) and \(s_{\hat{\eta}_{CL}}=0.12\); volume, \(\omega_V=0.25\) and \(s_{\hat{\eta}_V}=0.20\); absorption, \(\omega_{ka}=0.50\) and \(s_{\hat{\eta}_{ka}}=0.35\). Calculate variance-based shrinkage for each.
Clearance: $$ 100\left(1-\frac{0.12^2}{0.30^2}\right) =100\left(1-\frac{0.0144}{0.0900}\right) =100(0.84)=84.0\%. $$
Volume: $$ 100\left(1-\frac{0.20^2}{0.25^2}\right) =100\left(1-\frac{0.0400}{0.0625}\right) =100(0.36)=36.0\%. $$
Absorption: $$ 100\left(1-\frac{0.35^2}{0.50^2}\right) =100\left(1-\frac{0.1225}{0.2500}\right) =100(0.51)=51.0\%. $$
In this hypothetical analysis, clearance has the greatest variance-based eta shrinkage and volume the least. For clearance, the variance of the EBE distribution is 16% of the fitted population variance; for volume it is 64%, and for absorption it is 49%. These percentages compare relative spreads, not absolute parameter uncertainty. For example, the calculation does not show that an individual clearance estimate has a particular error in clearance units.
For contrast, standard-deviation-based shrinkage for clearance would be \(100(1-0.12/0.30)=60.0\%\), not 84.0%. Reporting 60.0% as variance-based shrinkage would use the wrong formula.
Why Information and Sampling Design Matter
A short justification clarifies why shrinkage can differ across effects. Consider an idealized scalar model in which a random effect has prior distribution \(\eta\sim N(0,\omega^2)\), and the subject’s data provide a noisy estimate \(Z=\eta+\epsilon\), where \(\epsilon\sim N(0,\tau^2)\) is independent of \(\eta\). The posterior mean is the EBE in this simplified setting: $$ E(\eta\mid Z)=\frac{\omega^2}{\omega^2+\tau^2}Z. $$
Let \(\lambda=\omega^2/(\omega^2+\tau^2)\). Because \(\operatorname{Var}(Z)=\omega^2+\tau^2\), the variance of the posterior means across repeated subjects is $$ \operatorname{Var}\{E(\eta\mid Z)\} =\lambda^2(\omega^2+\tau^2) =\frac{\omega^4}{\omega^2+\tau^2} =\lambda\omega^2. $$
Thus, in this idealized normal setting, the variance-based shrinkage is \(100(1-\lambda)\%=100\{\tau^2/(\omega^2+\tau^2)\}\%\). Greater observation noise relative to population variance produces more contraction of the EBE distribution. Real PK models are more complex, and their empirical shrinkage is not generally given by this scalar formula. The derivation nevertheless explains why information about a particular parameter, and hence sampling design, can affect its shrinkage.
Worked Example: A Sampling-Design Difference
Suppose two hypothetical designs are evaluated using the same fitted clearance population standard deviation, \(\omega_{CL}=0.30\). Both summaries use 60 subjects. In a sparse design, the EBE standard deviation is 0.24; in a richer design, it is 0.12. Calculate and compare variance-based shrinkage, treating the stated \(\omega_{CL}\) as fixed for this illustration.
For the sparse design: $$ 100\left(1-\frac{0.24^2}{0.30^2}\right) =100\left(1-\frac{0.0576}{0.0900}\right) =36.0\%. $$ For the richer design: $$ 100\left(1-\frac{0.12^2}{0.30^2}\right) =100\left(1-\frac{0.0144}{0.0900}\right) =84.0\%. $$
Under the stated comparison, the richer design has greater clearance shrinkage, despite having more information. This might seem reversed, but the example illustrates an important distinction: the formula \(1-s_{\hat{\eta}}^2/\omega^2\) measures contraction of the EBE variance, so greater shrinkage means a smaller EBE spread. The assumptions in the simplified derivation above also show that, for a fixed \(\omega\), increasing noise increases this shrinkage. Therefore the stated EBE spreads do not support the interpretation that the richer design provides less shrinkage. If the labels “sparse” and “richer” are intended to imply more and less information respectively, the values are inconsistent with that expectation in the idealized setting and should prompt investigation of the model, design comparison, or summaries. The arithmetic itself is correct; the label alone cannot validate the result.
This check is valuable in practice. Do not assume a percentage must move in a particular direction just because a design is described as richer. Verify the EBE spread, \(\omega\), subjects, and computation. Also note that fitting separate models to separate designs can change the estimated \(\omega\), so the fixed-\(\omega\) illustration does not isolate design effects in that situation.
Correlated Random Effects and Joint Interpretation
Random effects for clearance, volume, and absorption may be correlated. Variance-based shrinkage is marginal: each calculation uses one effect’s EBE variance and its own population variance. It does not report the joint precision of the vector of individual effects, nor does it show whether a particular combination of effects is well informed.
Worked Example: Marginal Shrinkage Does Not Describe a Parameter Contrast
Suppose hypothetical clearance and volume etas each have population standard deviation 0.30 and population correlation 0.80. Their population covariance is \(0.80(0.30)(0.30)=0.0720\). Their EBE standard deviations are 0.15 for clearance and 0.24 for volume. The marginal variance-based shrinkages are: $$ \text{Shrinkage}_{CL} =100\left(1-\frac{0.15^2}{0.30^2}\right) =100(1-0.25)=75.0\%, $$ $$ \text{Shrinkage}_{V} =100\left(1-\frac{0.24^2}{0.30^2}\right) =100(1-0.64)=36.0\%. $$
The population variance of the contrast \(\eta_{CL}-\eta_V\) is $$ \operatorname{Var}(\eta_{CL}-\eta_V) =0.30^2+0.30^2-2(0.0720) =0.0360. $$
If the two effects were independent with the same marginal variances, this contrast variance would instead be \(0.30^2+0.30^2=0.180\). Thus, correlation substantially changes the variability of the clearance-versus-volume contrast, even though each marginal shrinkage calculation uses only the corresponding marginal standard deviations. A joint interpretation requires the covariance information and, when relevant, the posterior uncertainty for the contrast; the two marginal percentages alone do not supply it.
Diagnostics, Alternatives, and Limitations
A cross-parameter comparison is a summary, not a diagnostic verdict. Before interpreting it, check that each \(\omega_j\) and EBE standard deviation refer to the same model and subjects, that the calculation squares both standard deviations, and that the effects are defined on interpretable scales. Inspect the number of subjects, unusual EBE values, and whether a few subjects dominate the sample variance. A high percentage can reflect weak individual information, but model misspecification or unstable variance estimates can also affect it.
Finite-sample estimates need not behave ideally. If \(s_{\hat{\eta}_j}>\omega_j\), the formula yields negative shrinkage. Do not automatically replace it with zero: the result can arise from sampling variation, parameter-estimation uncertainty, or a mismatch between the fitted model and the EBE distribution. Investigate the calculation and model before drawing a conclusion. Conversely, a small shrinkage percentage does not prove that every individual EBE is precise.
Other summaries answer related but different questions. Standard-deviation-based shrinkage is an alternative scale summary, not an interchangeable calculation. Per-subject uncertainty measures address uncertainty for individuals rather than the overall EBE spread. The estimated covariance matrix and joint posterior summaries are needed to assess correlated effects or parameter combinations. These approaches complement shrinkage; none makes a marginal shrinkage percentage a universal measure of individual accuracy.
| Summary | What it compares or describes | What it cannot establish alone |
|---|---|---|
| Variance-based eta shrinkage | EBE variance relative to fitted population variance for one effect | Individual accuracy or joint precision of correlated effects |
| Standard-deviation-based eta shrinkage | EBE standard deviation relative to fitted population standard deviation | The same percentage as the variance-based definition |
| Covariance or joint posterior summaries | Dependence and uncertainty for combinations of effects | A simple one-number ranking across parameters |
| Individual uncertainty summaries | Uncertainty for particular subjects and parameters | The population-wide contraction summarized by shrinkage |
Check Your Understanding
Work through each question, then compare your reasoning with the solutions.
- For a hypothetical volume effect, \(\omega_V=0.40\) and \(s_{\hat{\eta}_V}=0.20\). Calculate variance-based shrinkage. What proportion of the fitted population variance is represented by the EBE variance?
- Why can clearance and absorption shrinkage percentages be compared even when clearance and absorption rate have different physical units in a log-normal model?
- Two random effects each have variance-based shrinkage of 50%. Does this mean they have the same absolute EBE uncertainty? Explain.
- What does a negative variance-based shrinkage estimate indicate, and how should it be handled?
Solutions. (1) \(100[1-(0.20^2/0.40^2)]=100(1-0.25)=75.0\%\). The EBE variance is 25.0% of the fitted population variance. (2) In a log-normal model, the etas are dimensionless; each percentage compares an EBE variance with that effect’s own population variance. The comparison is relative, not a comparison of physical units. (3) No. Each has the same ratio of EBE variance to population variance, but the population variances can differ, producing different absolute EBE variances. (4) It means the EBE sample variance exceeds the fitted population variance. Check the formula, data, model, and finite-sample behavior; do not automatically truncate the estimate to zero.