Defining Eta Shrinkage From Empirical Bayes Estimates

Why Compare Eta Estimates With Population Variability?

A population pharmacokinetic model describes how subjects may differ from one another, while an empirical Bayes estimate (EBE) summarizes what the data for a particular subject suggest about that subject’s random effect. These are not the same object: the model specifies a population distribution, whereas the EBEs are subject-specific estimates informed by both that distribution and the subject’s observations. Eta shrinkage gives a compact description of how much less variable the EBEs are than the random effects are expected to be in the population.

This comparison is useful when interpreting subject-level diagnostics. If EBEs have little spread, that may reflect limited information in the individual observations and the model’s pull toward the population distribution. It does not, by itself, establish that the population model is correct, or that every subject’s estimate has been pulled by the same amount.

For a normally distributed random effect, the usual single-effect model is \(\eta_i\sim N(0,\omega^2)\), so \(\omega\) is the population standard deviation and \(\omega^2\) is the population variance. The quantity \(\omega\) must be on the same scale as the EBEs. For example, if \(\eta\) is a dimensionless random effect on a log scale, both \(\omega\) and \(s_{\hat{\eta}}\) are on that scale.

The SD-based definition compares standard deviations, not variances. It uses the sample standard deviation across the EBEs, which is centered on their sample mean. For values \(\hat{\eta}_1,\ldots,\hat{\eta}_n\), that standard deviation is $$ s_{\hat{\eta}}=\sqrt{\frac{\sum_{i=1}^{n}(\hat{\eta}_i-\bar{\hat{\eta}})^2}{n-1}}, $$ where \(\bar{\hat{\eta}}\) is the mean of the EBEs. In a fitted model, \(\omega\) is ordinarily a model-estimated population standard deviation; the numerical examples below take it as given so that the definition and arithmetic are clear.

How to Read the Definition

The definition can be justified directly by separating the comparison into two parts. First, \(s_{\hat{\eta}}/\omega\) compares the observed EBE spread with the model’s population spread. This is the fraction of the population standard deviation retained in the EBE distribution. Second, shrinkage is defined as the fraction not retained, so subtracting that ratio from one gives \(1-s_{\hat{\eta}}/\omega\). Multiplication by \(100\%\) expresses the result as a percentage.

Thus, if \(s_{\hat{\eta}}\) equals \(\omega\), the EBE spread equals the population standard deviation and this definition gives zero shrinkage. If \(s_{\hat{\eta}}\) is smaller than \(\omega\), shrinkage is positive. A smaller EBE spread corresponds to a larger shrinkage value. These statements describe the aggregate spread across subjects; they do not give the amount of shrinkage for a particular subject.

The normal random-effects assumption supplies a familiar interpretation for \(\omega\): it is the standard deviation of the modeled population distribution of \(\eta\). The arithmetic definition itself is a comparison of two standard deviations. To interpret it as a meaningful diagnostic, however, the EBE values and \(\omega\) should refer to the same random effect, parameterization, and model scale, and the subjects should be the relevant set for the comparison.

Worked Calculations and Interpretation

A careful calculation keeps the EBE standard deviation at full precision until the final reporting step. Rounding it first and then calculating a percentage can change the last reported digit. Each example should also state whether it reports the retained fraction or shrinkage: these quantities sum to one, so their percentages sum to \(100\%\).

Assumptions, Diagnostics, and Alternatives

The definition requires a population standard deviation and a set of EBEs for the same random effect. The usual normal random-effects model assumes a mean-zero distribution with standard deviation \(\omega\); the calculations also assume that all compared EBEs are expressed on the scale of that random effect. If EBEs have been transformed, or if the model’s random-effect parameterization differs from the one used for the calculation, the comparison is not interpretable as stated.

Several checks help catch errors or misleading summaries:

• Check the model quantity. If the model output gives a variance, take its square root to obtain \(\omega\) before using this SD-based definition. Comparing an EBE standard deviation directly with a variance mixes units and is incorrect.
• Check the EBE set. Confirm that each included value belongs to the intended random effect and subject set. Missing, duplicated, or mismatched records change the sample standard deviation.
• Check centering. The sample SD is calculated around the EBE sample mean. Do not substitute the root mean square around zero without explicitly defining a different quantity. If the EBE mean is noticeably nonzero, report or investigate that feature as well; a spread measure alone does not describe a shift in the estimates.
• Check the reported percentage. A retained fraction of \(0.20\) means \(20\%\) retained and \(80\%\) shrinkage, not \(20\%\) shrinkage. Keep full precision until the final rounding.
• Check extreme results. A negative value can occur when the EBE sample SD exceeds \(\omega\). Verify the calculation, then consider finite-sample variation and whether the subjects, model, and random-effect definition are appropriate for comparison.

Eta shrinkage is a descriptive summary of the EBE distribution relative to the population distribution. It is not a subject-specific uncertainty measure, a test of a model hypothesis, or a direct measure of goodness of fit. In particular, a low EBE spread does not prove that the population variability is small: it may also reflect limited individual information. Likewise, a high shrinkage value does not establish that the population model is misspecified.

The SD-based definition used here should be distinguished from summaries based on comparing variances. A variance-based measure uses a different scale of comparison and should not be substituted into this formula. Another complementary approach is to inspect subject-level estimates and diagnostic plots directly: such inspection can reveal patterns hidden by a single across-subject summary, but it does not replace a clearly defined shrinkage calculation. The appropriate summary depends on the question, and the convention should be stated when reporting results.