Why Individual PD Estimates Can Mislead
A population PD model represents between-subject differences with random effects. For example, if \(\eta_i\) is the random effect for subject \(i\), a positive value may indicate a subject-specific PD parameter above its population-typical value. Individual empirical Bayes estimates (EBEs) are estimates of these subject-specific effects, conditional on that subject’s data and the fitted population model.
An EBE is not a direct measurement of a subject’s random effect. It combines the information in that subject’s observations with the population distribution assumed by the model. When an individual’s data are sparse or weakly informative, the population distribution has more influence, and the EBE is pulled toward its prior mean. In a model with no covariate effect on the random-effect mean, that mean is commonly zero. Consequently, a plot of EBEs against a covariate can show a weaker relationship than the underlying relationship—or can be difficult to interpret when subjects differ in how informative their data are.
Shrinkage is a property of the estimate and the available information, not evidence that a subject truly has no between-subject deviation. It is strongest when the individual data provide little information about the relevant PD parameter relative to the estimated between-subject variability. It is usually weaker when the individual data are more informative.
A Normal-Model Derivation
A simple normal model makes the source of shrinkage explicit. Suppose a random effect has distribution \(\eta_i\sim N(0,\omega^2)\). For illustration, assume the subject’s data can be summarized by \(z_i\), where \(z_i\mid\eta_i\sim N(\eta_i,s_i^2)\). Here, \(s_i^2\) represents the uncertainty in that subject-level summary. Assume the population parameters \(\omega^2\) and \(s_i^2\) are known for this derivation.
The conditional distribution of \(\eta_i\) given \(z_i\) is normal. Its mean is also its mode, so it is the EBE in this example. Multiplying the prior density by the likelihood gives a conditional density proportional to:
Completing the square in \(\eta_i\) gives the conditional mean and variance:
where \(\lambda_i=\omega^2/(\omega^2+s_i^2)\) is between zero and one. If the individual estimate is precise, \(s_i^2\) is small and \(\lambda_i\) is near one: the conditional estimate stays close to \(z_i\). If the estimate is imprecise, \(\lambda_i\) is small and the EBE is pulled toward zero. This derivation illustrates the principle; nonlinear PD models can produce more complicated conditional distributions and do not generally have this exact formula.
Worked Example: Calculating Shrinkage for One Subject
Suppose a hypothetical PD random effect has \(\omega=0.30\), and a subject’s data provide a summary \(z_i=0.50\) with uncertainty \(s_i=0.40\). Assume the normal model above. The variances are \(\omega^2=0.09\) and \(s_i^2=0.16\), so:
The EBE is:
The subject-level summary is \(0.50\), but the conditional estimate is \(0.18\), closer to the population mean of zero. The conditional variance is \(0.09(0.16)/0.25=0.0576\), giving a conditional standard deviation of \(0.240\). The estimate is therefore both pulled toward zero and still uncertain; reporting \(0.18\) alone would hide that uncertainty.
How Shrinkage Changes Covariate Screening
Suppose a centered continuous covariate \(x_i\) is related to the true random effect through \(\eta_i=\beta x_i+\kappa_i\), where \(\beta\) is the covariate slope and \(\kappa_i\) is residual between-subject variation. Consider screening this relationship with EBEs from a model that does not yet include the covariate. In the simplified normal setting, an EBE behaves like a shrunken version of the subject-level information: \(\widehat{\eta}_i=\lambda_i z_i\).
If all subjects have the same shrinkage factor \(\lambda\), and the errors in \(z_i\) and residual effects \(\kappa_i\) are unrelated to \(x_i\), then the expected EBE-versus-covariate slope is \(\lambda\beta\), not \(\beta\). To see why, write \(z_i=\beta x_i+\kappa_i+\epsilon_i\), with \(E(\kappa_i\mid x_i)=E(\epsilon_i\mid x_i)=0\). It follows that \(E(\widehat{\eta}_i\mid x_i)=\lambda\beta x_i\). Thus, the EBE plot can understate the magnitude of the covariate relationship.
In real population PD models, shrinkage can differ across subjects. Subjects may have different numbers or timing of observations, different response variability, or different information about a particular PD parameter. Then the EBE is not simply a common constant times the information about the random effect. A covariate plot can be attenuated, curved, or dominated by subjects whose individual parameters are estimated more precisely. If information differs systematically with the covariate, the visual relationship may be especially hard to interpret.
Worked Example: Attenuation of an EBE Slope
Consider three hypothetical subjects with centered covariate values \(x=(-1,0,1)\). Suppose their underlying random effects follow \(\eta_i=0.40x_i\), with no residual deviations in this illustration. If all three have shrinkage factor \(\lambda=0.50\), their EBEs are \((-0.20,0,0.20)\).
The ordinary least-squares slope of EBE on \(x\) is:
The true slope in the hypothetical construction is \(0.40\), but the EBE plot’s slope is \(0.20\), half as large because the common shrinkage factor is \(0.50\). This calculation illustrates attenuation; it is not a general method for recovering a covariate effect by dividing an observed EBE slope by an estimated shrinkage factor.
Worked Example: Unequal Information Across Subjects
Now consider four hypothetical subjects with \(x=(-2,-1,1,2)\) and true effects \(\eta_i=0.20x_i\), again with no residual deviations for clarity. Suppose their individual shrinkage factors are \((0.25,0.80,0.80,0.25)\). Their EBEs are therefore \((-0.10,-0.16,0.16,0.10)\).
The slope from these EBEs is:
The underlying slope is \(0.20\), so the EBE slope is substantially smaller. The subjects at \(x=-2\) and \(x=2\) have stronger shrinkage than those nearer zero; consequently, their extreme covariate values contribute less extreme EBEs. In actual analyses, the shrinkage factors are not usually known exactly and the random effects are not arranged without residual variation. The example shows why a single overall shrinkage summary cannot fully describe the behavior of every point in a covariate plot.
Using Shrinkage Diagnostics and Covariate Plots
A common summary of random-effect shrinkage compares the empirical spread of EBEs with the estimated population standard deviation. One convention is:
where \(SD(\widehat{\eta})\) is the sample standard deviation of the EBEs and \(\widehat{\omega}\) is the estimated random-effect standard deviation. Under this convention, values closer to one indicate stronger compression of the EBE distribution. Some software or reports instead use a variance-based calculation, \(1-\operatorname{Var}(\widehat{\eta})/\widehat{\omega}^2\); the convention should be stated because the numerical values differ. Finite-sample estimates can also be negative, so the formula is a diagnostic summary, not a bounded probability.
A useful screening workflow is to first review whether the random effect is sufficiently informed for an EBE plot to be interpretable, then plot the EBE against the candidate covariate with the direction and scale clearly labelled. Look for a pattern, but also inspect whether the spread of EBEs or apparent precision changes across the covariate range. A flat plot with substantial shrinkage is weak evidence against a covariate effect. A visible pattern is a reason to formulate and fit a covariate model, not proof that the covariate belongs in the model.
For formal evaluation, compare the nested population models using the likelihood-based approach appropriate to the covariate hypothesis, including the assumptions and decision criteria for the likelihood-ratio test. That comparison uses the fitted model’s likelihood rather than treating EBEs as observed outcomes. EBEs are useful for exploratory screening and diagnosing individual-level information; they are not a substitute for fitting the covariate effect in the hierarchical model.
| Approach | What it uses | Appropriate interpretation |
|---|---|---|
| EBE-versus-covariate plot | Conditional individual random-effect estimates | Exploratory evidence that may be weakened or distorted by shrinkage |
| Population-model comparison | Likelihoods from models with and without the covariate effect | Formal evidence under the model and test assumptions |
| Subject-level regression on EBEs | EBEs treated as if directly observed responses | Generally not a valid replacement for the population-model comparison |
Common Pitfalls and Limitations
- Calling an EBE the subject’s true parameter. It is a conditional estimate that reflects both individual data and the fitted population distribution.
- Interpreting a weak EBE trend as evidence of no covariate effect. Strong shrinkage can attenuate the trend, so a near-flat plot may be uninformative.
- Correcting the EBE slope by dividing by a shrinkage summary. A single summary does not capture subject-specific shrinkage, uncertainty, or model nonlinearity. Fit the covariate effect in the population model instead.
- Treating a visually strong trend as a confirmed effect. Sampling variation, influential subjects, misspecified random-effects assumptions, or different information across subjects can all affect the plot.
Shrinkage diagnostics also have limits. The empirical spread of EBEs depends on the fitted model, the sample, and the information available for each subject. A low summary shrinkage value does not establish that every individual estimate is precise, and a high value does not mean that all individual information is useless. In nonlinear models, conditional distributions can be asymmetric or multimodal, and a conditional mode may not summarize uncertainty well. These diagnostics should therefore be considered alongside model fit, individual data support, and the formal population-model comparison.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- In the normal model, what happens to the shrinkage factor when the subject-level uncertainty \(s_i^2\) increases while \(\omega^2\) stays fixed?
- A centered covariate has a true random-effect slope of \(0.30\), and all subjects have a common shrinkage factor of \(0.40\). Under the simplified assumptions in this tutorial, what EBE slope would be expected?
- Why can a flat EBE-versus-covariate plot fail to rule out a covariate effect?
- Why is regressing EBEs on a covariate not equivalent to fitting that covariate in the population model?
Solutions. (1) The factor \(\lambda_i=\omega^2/(\omega^2+s_i^2)\) decreases as \(s_i^2\) increases, so the conditional estimate is pulled more strongly toward zero. (2) The expected slope is \(\lambda\beta=0.40(0.30)=0.12\). This is an illustration under a common-shrinkage normal model, not a general correction formula. (3) Shrinkage can compress individual estimates toward zero and attenuate the apparent trend, particularly when individual data are weak. (4) EBEs are estimated conditional on individual data and population assumptions, with different uncertainty and potentially different shrinkage across subjects. A population-model fit estimates the covariate effect within the hierarchical likelihood rather than treating those conditional estimates as directly observed responses.