Residual Shrinkage And Apparent Individual Fit

Why Conditional Residuals Can Look Too Good

A conditional residual compares an observation with a prediction that uses an individual-level parameter estimate, often an empirical Bayes estimate (EBE). That estimate is informed by the subject’s own observations. As a result, the observations help determine the prediction against which they are then compared. This feedback can pull conditional residuals toward zero, making individual fit appear better than the data independently support.

This effect is called residual shrinkage. It is related to, but distinct from, eta shrinkage: eta shrinkage concerns how individual random-effect estimates are pulled toward the population mean, while residual shrinkage concerns how residuals calculated from individual predictions may be reduced or made less variable. A tidy conditional residual plot is therefore not, by itself, strong evidence that the model describes the data well.

The key question is not whether conditional residuals are useful. They can reveal within-subject patterns and unusually large discrepancies. The question is what their apparent size and distribution mean when each subject’s data helped determine the fitted individual prediction.

A Simple Model Shows the Mechanism

Consider a simplified random-intercept model for one subject. After subtracting the population prediction, write the subject’s \(m\) residual-scale observations as \(r_j=\eta+\epsilon_j\), for \(j=1,\ldots,m\). Assume \(\eta\sim N(0,\omega^2)\), the observation errors are independent with \(\epsilon_j\sim N(0,\sigma^2)\), and \(\eta\) is independent of all \(\epsilon_j\). Here \(\eta\) represents a subject-level deviation and \(\epsilon_j\) an observation-level error. The variance components and population prediction are treated as known for this illustration.

Under these assumptions, the empirical Bayes estimate of the random intercept is a weighted version of the subject’s mean residual. Define \(\bar r=m^{-1}\sum_{j=1}^m r_j\). The weight \(\lambda\) increases with the amount of information in the observations:

The conditional residual for observation \(j\) is \(c_j=r_j-\hat{\eta}\). Substituting \(r_j=\eta+\epsilon_j\) and \(\bar r=\eta+\bar\epsilon\), where \(\bar\epsilon=m^{-1}\sum_j\epsilon_j\), gives

This expression makes two points clear. First, the subject-level deviation in the conditional residual is multiplied by \(1-\lambda\): when the individual estimate uses more information, \(\lambda\) is larger and that component is reduced. Second, the observation-level errors do not all disappear. In particular, the residuals retain contrasts between observations within a subject.

The variance calculation quantifies this distinction. Since \(\eta\) is independent of the observation errors, and \(\operatorname{Var}(\epsilon_j-\lambda\bar\epsilon)=\sigma^2(1-2\lambda/m+\lambda^2/m)\), we obtain:

The population-prediction residual \(r_j\), before using an individual estimate, has variance \(\omega^2+\sigma^2\). Conditional residuals are not independent draws from this same marginal distribution: their shared estimate induces dependence, and their subject-level component is reduced. In the limiting case \(m=1\), \(\lambda=\omega^2/(\sigma^2+\omega^2)\), and \(c_1=(1-\lambda)r_1\). With several observations, within-subject contrasts remain, so it would be wrong to conclude that every conditional residual must become arbitrarily close to zero as information increases.

Worked Examples: Residuals at Different Information Levels

Reading Conditional Residual Plots Carefully

The plot’s apparent quality depends on what feature is being inspected. A subject-level offset may be less visible after the EBE adapts to that subject, while a time-related pattern among observations can remain. Consequently, a cloud centered near zero in a conditional residual plot is weaker evidence against a systematic population-level discrepancy than it might first appear. The observations have been used both to estimate the individual prediction and to assess its residuals.

When reviewing a diagnostic, state which prediction and residual definition produced it. A residual against a population prediction and a conditional residual answer different questions: the former retains between-subject deviations, while the latter evaluates observations after individual-level adaptation. If a residual has also been weighted or standardized, its scale and reference distribution depend on that definition; do not assume that all residual types have the same variance or expected distribution.

A useful comparison is to examine residual behavior at more than one level of prediction, while keeping the question for each plot explicit. Population-prediction residuals retain subject-level deviations; conditional residuals show the remaining discrepancies after individual adaptation. Subject-level panels can help reveal whether a tidy overall cloud hides a small number of poorly sampled subjects or a pattern concentrated at particular times. These views are complementary, not interchangeable.

The assumptions in the derivation are deliberately simple: a single normally distributed random intercept, independent normal observation errors with constant variance, and known population prediction and variance components. Real pharmacokinetic models may have nonlinear predictions, multiple random effects, heteroscedastic observation error, or estimated population parameters. The formula above then does not directly give the residual variance. The general lesson still applies: an individual prediction estimated using the observations being checked can make conditional residuals appear more favorable. The size and pattern of that effect must be assessed for the residual type and fitted model actually used.

Diagnostics, Alternatives, and Common Pitfalls

Warning signs include conditional residuals that are markedly less dispersed than population-prediction residuals, a residual plot that looks uniformly centered despite weak information for some subjects, or a strong contrast between the two views. These are prompts for interpretation, not automatic evidence of model failure. Conversely, residual shrinkage does not prove that a model is correct; a misspecified model can still yield conditional residuals that look orderly after adaptation.

The main alternative is not to discard conditional residual plots, but to interpret them beside a view that does not use the same individual adaptation. The population-prediction residual view and subject-level panels provide different information, and sampling-design context helps explain how much weight to put on each. Simulation-based predictive checks are another broad diagnostic family, but they answer a distinct question and should not be treated as a substitute for understanding how conditional residuals were formed.

• Mistake: “The conditional residuals are near zero, so individual fit is independently confirmed.” The observations contributed to the individual predictions. Their residuals can be reduced for that reason alone. Compare the prediction basis and consider the information available per subject.
• Mistake: “More observations make all conditional residuals shrink to zero.” Repeated observations can improve estimation of the subject effect, but within-subject contrasts remain. The nine-observation example retains substantial residual spread.
• Mistake: “Residual shrinkage and eta shrinkage are the same statistic.” They concern related adaptation but describe different quantities. Specify whether the reported quantity concerns random-effect estimates or residuals, and how it was calculated.
• Mistake: “A smaller residual spread proves the model is better.” A change in spread may follow from changing the prediction from population-level to conditional, even without any improvement in independent predictive evidence.

In practice, label conditional plots clearly, note whether observations per subject are sparse or informative, and avoid treating a visually neat cloud as a standalone model-validation result. If individual plots look especially favorable for subjects whose estimates are strongly informed by their own data, that is a reason to qualify the apparent fit, not to dismiss the plots.