Why Plot Empirical Bayes Estimates?
A population PK/PD model represents between-subject differences through random effects. For subject \(i\), a parameter might be written as \(\theta_i=\theta_{\mathrm{typ}}\exp(\eta_i)\), where \(\theta_{\mathrm{typ}}\) is the typical value and \(\eta_i\) is the subject-specific random effect on the log scale. A plot of estimated \(\eta_i\) values can help reveal whether the model leaves systematic subject-level structure unexplained. For example, a trend in estimated random effects against a biologically plausible covariate may motivate closer investigation.
These plots are easy to overread. The plotted values are not direct measurements of each subject’s true random effect. They are estimates informed by both that subject’s observations and the population distribution, and they can be pulled toward zero to different degrees across subjects. A useful interpretation therefore considers the estimates, their uncertainty, the amount of shrinkage, and the model assumptions together.
What an EBE Plot Can and Cannot Show
A common plot places an EBE for one random effect on the vertical axis and a subject-level covariate on the horizontal axis. The horizontal reference at zero is useful because, under the model, random effects have a population distribution centered at zero conditional on the modeled covariates. A sustained pattern away from that reference or a systematic trend across the covariate range can flag a possible mismatch between the model and the data.
That interpretation depends on the model being a reasonable description of the data: the structural and observation models should be adequate for this diagnostic purpose, the assumed random-effect distribution should be plausible, and the fitting procedure should have converged appropriately. The EBE estimates and their uncertainty are also conditional on the fitted population parameters in many common implementations. Thus their uncertainty generally does not include all uncertainty about those population parameters.
For a random effect with population variance \(\omega^2\), the model commonly assumes \(\eta_i\sim N(0,\omega^2)\). With several random effects, the corresponding assumption is often a multivariate normal distribution with mean zero and covariance matrix \(\Omega\). An EBE plot should be read as a model diagnostic under this distributional assumption, not as a direct picture of the true individual effects.
Conditional Uncertainty and a Short Derivation
Many modeling programs provide an approximate conditional standard error or covariance matrix for each subject’s random-effect estimate. These quantities describe uncertainty about the individual random effect conditional on the fitted population model and the subject’s data. A simple approximate interval for one EBE is the estimate plus or minus 1.96 conditional standard errors, when a normal approximation is reasonable. It is an approximate conditional interval, not a 95% probability statement about a fixed true effect in a frequentist interpretation.
The role of information and shrinkage is especially clear in a simplified normal example. Suppose the subject-level information can be summarized by a noisy quantity \(z_i\), with \(z_i\mid\eta_i\sim N(\eta_i,s_i^2)\), and suppose the population model is \(\eta_i\sim N(0,\omega^2)\). These assumptions are for illustration; nonlinear PK/PD models usually require numerical approximations. The conditional density is proportional to the product of the normal likelihood and normal prior:
Collecting the terms involving \(\eta_i\), the coefficient of \(\eta_i^2\) is \(1/s_i^2+1/\omega^2\), and the linear term is \(-2z_i\eta_i/s_i^2\). Completing the square gives a normal conditional distribution with mean \(w_i z_i\) and variance \(v_i\):
When the subject’s data are informative, \(s_i^2\) is small, \(w_i\) is closer to one, and the estimate is closer to \(z_i\). When the data are less informative, \(w_i\) is smaller, the EBE is pulled more strongly toward the population mean of zero, and its conditional uncertainty is larger. This illustrates why EBE plots should be read alongside information and uncertainty rather than as raw individual observations.
Worked Example: Reading Conditional Intervals
Suppose a hypothetical model reports four EBEs for a log-scale random effect and their conditional standard errors. The estimates are \(-0.08,\ 0.10,\ 0.25,\ 0.30\), and the corresponding standard errors are \(0.20,\ 0.18,\ 0.15,\ 0.20\). Use the approximate interval \(\widehat{\eta}_i\pm1.96SE_i\).
For subject 1, \(1.96(0.20)=0.392\), so the interval is \(-0.08\pm0.392=[-0.472,\,0.312]\). For subject 2, \(1.96(0.18)=0.3528\), so \(0.10\pm0.3528=[-0.2528,\,0.4528]\). For subject 3, \(1.96(0.15)=0.294\), so \(0.25\pm0.294=[-0.044,\,0.544]\). For subject 4, \(1.96(0.20)=0.392\), so \(0.30\pm0.392=[-0.092,\,0.692]\). All values are in log units.
All four intervals include zero. That means these approximate conditional intervals do not distinguish each individual effect from zero at the stated interval level, conditional on the fitted model and approximation. It does not establish that every true individual effect is zero, nor does it answer whether the population variance is zero. The relatively wide intervals also warn that the point estimates alone may give an overly definite impression.
Shrinkage: Why the Estimates Cluster
Shrinkage describes the tendency of EBEs to be less dispersed than the population distribution of the underlying random effects. One commonly used summary for a random effect is \(100\{1-SD(\widehat{\eta})/\omega\}\%\), where \(SD(\widehat{\eta})\) is the sample standard deviation of the EBEs and \(\omega\) is the estimated population standard deviation. Software and references may use related conventions, so the calculation used should be stated. The measure is a diagnostic summary, not a direct estimate of the fraction of each person’s random effect that has been removed.
Worked Example: Calculating an EBE Shrinkage Summary
In a hypothetical fit, eight EBEs for one random effect are \(-0.12,-0.08,-0.04,0,0.04,0.08,0.12,0.16\). Suppose the estimated population standard deviation is \(\omega=0.20\). Calculate the sample mean, sample standard deviation, and shrinkage summary.
The sum of the EBEs is \(0.16\), so their sample mean is \(\bar{x}=0.16/8=0.0200\). The sum of their squares is \(0.0704\). The sum of squared deviations from the mean is \(0.0704-8(0.0200)^2=0.0672\). Therefore, \(s=\sqrt{0.0672/7}=\sqrt{0.0096}=0.09798\).
The shrinkage summary is \(100\{1-0.09798/0.20\}=51.0\%\). The EBEs are less dispersed than the estimated population distribution, consistent with substantial shrinkage by this summary. It does not mean that each subject’s random effect was reduced by exactly 51.0%; individual degrees of shrinkage differ with the information available for each subject.
A second warning sign is a systematic difference in how tightly EBEs cluster across groups or covariate values. If subjects in one region have fewer or less informative observations, their estimates may be more strongly pulled toward zero. A plot can then show differences in spread—or a distorted trend—partly because information differs, not because the underlying random effects follow a different covariate relationship.
Worked Example: An Apparent Covariate Trend With Unequal Information
Consider a hypothetical illustration with \(\omega=0.30\), so \(\omega^2=0.09\). Four subjects have centered covariate values \(x=-1,0,1,2\). Their noisy subject-level summaries \(z_i\) are \(-0.10,-0.02,0.16,0.24\). Suppose the first two have \(s_i=0.10\), while the last two have \(s_i=0.30\). Apply the normal calculation above.
For the first two subjects, \(w=0.09/(0.09+0.01)=0.90\), and the conditional standard deviation is \(\sqrt{0.09(0.01)/0.10}=0.09487\). Their EBEs are \(0.90(-0.10)=-0.090\) and \(0.90(-0.02)=-0.018\). For the last two, \(w=0.09/(0.09+0.09)=0.50\), and the conditional standard deviation is \(\sqrt{0.09(0.09)/0.18}=0.2121\). Their EBEs are \(0.50(0.16)=0.080\) and \(0.50(0.24)=0.120\).
The four EBEs increase overall with \(x\), which could attract attention in a plot. But the higher-\(x\) subjects have much wider conditional uncertainty, and all four approximate 95% conditional intervals include zero: the first two are \([-0.276,\,0.096]\) and \([-0.204,\,0.168]\); the last two are approximately \([-0.336,\,0.496]\) and \([-0.296,\,0.536]\). This tiny illustration does not establish a covariate effect. It shows why an apparent line through EBE point estimates should be considered together with the information and uncertainty at each covariate value.
Practical Reading, Comparisons, and Failure Modes
A disciplined reading of an EBE plot starts with the model and the data supporting each estimate. Check whether the fitting procedure converged, whether estimates are extreme or boundary-like, and whether the conditional uncertainty is available and interpretable. Then look for broad patterns: a trend, curvature, a shift from zero, clusters, or changing spread. Ask whether the pattern is supported across the covariate range or is driven by a few subjects, whether information differs systematically, and whether correlated random effects or correlated covariates could contribute. A plot can suggest where to investigate; it does not isolate the explanation.
EBE plots serve a different purpose from several related displays. Population and individual prediction checks compare predictions with observed responses and address how well the model reproduces data at different levels. EBE plots instead inspect the fitted subject-level random-effect estimates and their relationship to covariates. A covariate-stratified visual predictive check, by contrast, assesses simulated-versus-observed behavior within covariate groups. These displays can complement one another, but none should be treated as interchangeable.
For a proposed covariate effect, an EBE-versus-covariate plot is a screening and diagnostic tool, not a replacement for a model-based comparison. EBE uncertainty bars are not ordinarily a complete substitute for estimating and evaluating the covariate effect within the population model; the EBEs share information through estimated population parameters and are subject to shrinkage. Use the plot to formulate a question, then evaluate the candidate model using the appropriate model-building and fit-assessment procedures.
- Overinterpreting a fitted line. A line through a small number of EBEs can look persuasive by chance. Check sample size, uncertainty, and influential subjects before assigning meaning.
- Treating EBEs as observed truth. They are conditional estimates and may be substantially shrunk. Do not interpret their scatter as the direct distribution of the true individual effects.
- Ignoring uneven precision. If uncertainty changes with the covariate, a pattern or apparent lack of pattern may partly reflect different information. Inspect conditional uncertainty and the underlying sampling design.
- Equating no visible trend with no effect. High shrinkage or sparse data can conceal structure. A flat plot is not evidence that a covariate effect is absent.
- Confusing a pointwise interval with a population conclusion. An interval for an individual EBE does not test whether the population random-effect variance or a covariate coefficient is zero.
The central limitation is that these plots inherit the assumptions and weaknesses of the fitted model. Misspecification, sparse observations, parameter correlations, and uncertainty in the estimated population parameters can all affect the apparent pattern. Their value lies in making those questions visible—not in providing a standalone decision rule.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- What does an EBE represent, and why is it not a direct observation of an individual’s true random effect?
- In the normal illustration, what happens to \(w_i\) when the subject-level standard error \(s_i\) increases, holding \(\omega\) fixed?
- Suppose the EBE standard deviation is \(0.12\) and the estimated population standard deviation is \(0.24\). Calculate the shrinkage summary using the convention in this tutorial.
- Why might an EBE-versus-covariate trend be misleading if subjects at high covariate values have less informative data?
- If an approximate conditional interval for one EBE includes zero, what does that establish—and what does it not establish?
Solutions. (1) An EBE is an estimate of an individual random effect conditional on that subject’s data and the fitted population parameters; the population distribution contributes information, often pulling the estimate toward zero. (2) \(w_i=\omega^2/(\omega^2+s_i^2)\), so increasing \(s_i\) decreases \(w_i\) and increases shrinkage toward zero. (3) \(100\{1-0.12/0.24\}=50.0\%\). This summarizes relative dispersion and is not an individual-level shrinkage percentage. (4) Less informative subjects have more strongly shrunk and less precise EBEs; differences in precision can distort the apparent point-estimate pattern or make it harder to see a real one. (5) It says the approximate conditional interval does not distinguish that individual effect from zero at the stated interval level under the fitted model and approximation. It does not prove the true effect is zero, nor does it test a population covariate effect or random-effect variance.