How Sampling Information Controls Eta Shrinkage

Why Sampling Information Changes Eta Estimates

An empirical Bayes estimate (EBE) combines information from an individual’s observations with the population distribution of a random effect. When the observations provide little information about that individual, the EBE is pulled toward the population mean, conventionally zero for an eta. With more informative observations, the individual’s data can exert greater influence.

This tutorial makes that relationship explicit in a simple Gaussian model. It shows how within-subject sampling, residual error, and random-effect variability affect the expected spread of EBEs. The formulas are useful for understanding the mechanism; real pharmacokinetic models may be nonlinear, have multiple correlated random effects, and use more complex residual models.

The formula’s interpretation depends on the comparison being meaningful: the EBEs and the random effects must use the same scale, and \(\omega^2\) must be positive. In the model below, the random effect has mean zero. The formula therefore compares the spread of estimates centered toward zero with the population spread before observing an individual’s data.

A Gaussian Information Model

Suppose the individual’s data can be summarized by an information-equivalent observation \(z\), with \(\eta\sim N(0,\omega^2)\) and \(z\mid\eta\sim N(\eta,1/J)\). Here \(J>0\) is the within-subject information about \(\eta\). A larger \(J\) means lower observation uncertainty about the random effect. This model assumes a normal population distribution, a normal observation error independent of \(\eta\), and a correctly specified value of \(J\).

For a locally linear PK model with independent residual errors of variance \(\sigma^2\), one way to obtain this information is \(J=\sum_j g_j^2/\sigma^2\), where \(g_j\) is the sensitivity of the expected observation at sample \(j\) to \(\eta\). Thus, more samples can increase \(J\), while greater residual variance decreases it. The sensitivity matters too: a sample taken where the prediction barely changes with \(\eta\) contributes little information.

In detail, the terms involving \(\eta\) in the negative twice-log posterior are \(\eta^2/\omega^2+J(z-\eta)^2\). Expanding the second term gives a coefficient of \(\eta^2\) equal to \(\omega^{-2}+J\), and a linear term \(-2Jz\eta\). Completing the square therefore yields:

The factor \(a\) is between zero and one. It is the attenuation of this individual’s data summary toward zero: low information makes \(a\) small, while high information makes \(a\) approach one. In this Gaussian model the posterior mean is also the posterior mode, so either definition of the EBE gives the same estimate.

From Information to Expected Eta Shrinkage

The EBE spread across subjects is not simply the spread of the posterior distribution within one subject. To connect it to \(J\), consider repeated subjects generated under the same model. Marginally, \(z=\eta+e\), where \(e\sim N(0,1/J)\) is independent of \(\eta\). Therefore, \(\operatorname{Var}(z)=\omega^2+1/J\). Since \(\hat{\eta}=az\), the across-subject variance of the EBEs is:

The second equality follows by substituting \(a=\omega^2J/(1+\omega^2J)\) and simplifying. Dividing the EBE standard deviation by \(\omega\) gives \(\sqrt{a}\). Thus the model’s expected variance-based shrinkage is:

This is an expected relationship under the stated model, not a guarantee about the observed EBE standard deviation in every finite sample. Finite cohorts fluctuate, and estimated population parameters add uncertainty. The expression nevertheless shows the direction of the main effects: increasing \(J\) reduces expected shrinkage, while increasing \(\omega^2\) at fixed \(J\) also reduces it.

Reading the Result and Checking Its Assumptions

The key quantity is the dimensionless product \(\omega^2J\), which compares data information with prior precision. If it is small, \(a\) is small and EBEs are strongly pulled toward zero. If it is large, \(a\) is near one and the EBE follows the individual data summary more closely. This provides a compact explanation of why sparse sampling and noisy observations often yield more eta shrinkage.

The EBE’s posterior variance answers a different question from variance-based shrinkage. For example, in the four-observation setting with \(\omega^2=0.25\) and \(J=4\), the posterior variance is \(0.25/[1+(0.25)(4)]=0.125\). That is uncertainty about \(\eta\) conditional on one subject’s data. Variance-based shrinkage instead summarizes the spread of EBEs across subjects relative to the population standard deviation. The two quantities are related in this idealized model, but they are not interchangeable in general.

In a PK model, \(J\) is affected not only by the count of samples but also by their timing, the sensitivity of predicted concentrations to the parameter, dose history, and the residual error model. Repeated samples that carry nearly redundant information may add less than their count suggests. A local linear information approximation can also be inaccurate when the PK predictions are strongly nonlinear over the plausible range of \(\eta\).

Diagnostics, Alternatives, and Limitations

A practical information-based interpretation should be checked against the fitted model and the design. Compare sampling schedules and residual assumptions with the resulting EBE spread, and examine whether observations at the available times are actually sensitive to the random effect of interest. If EBEs remain tightly clustered despite apparently informative observations, possible explanations include a larger-than-appropriate residual variance, weak parameter sensitivity, a misspecified random-effect variance, or model misspecification. Those are prompts for investigation, not proof of a particular cause.

The Gaussian derivation assumes a single normally distributed random effect, a correctly specified normal observation model, and a fixed information value. In nonlinear models, EBEs may not be exactly linear in the data; with correlated random effects, information about one effect can depend on the others; and parameter estimates used to calculate shrinkage are themselves uncertain. The observed sample standard deviation can also differ from its model expectation simply because the cohort is finite.

The formula is one useful summary, not the only way to assess individual information. Posterior standard errors or conditional variances describe uncertainty for individuals; profile-based or simulation-based assessments can be useful when a Gaussian approximation is inadequate. These alternatives answer related but distinct questions. Shrinkage itself does not establish whether the population model fits well, nor does low shrinkage guarantee that every individual estimate is accurate.