Distinguishing Interindividual Variability From Residual Error

Why the Source of Concentration Variation Matters

Concentrations from different people can differ because their pharmacokinetic parameters differ, because measurements fluctuate around each person’s model-predicted concentration, or because both processes operate. These sources are not interchangeable. Interindividual variability changes a subject’s predicted concentration profile through shared subject-specific parameters. Residual error describes the remaining observation-level deviation after that profile is specified.

Repeated observations help clarify the distinction. If one subject’s concentrations are consistently higher than the population prediction, that pattern may reflect a subject effect shared across observations. If one measurement is high and another is low around the same subject-specific prediction, that pattern is more consistent with residual variation. This is a conceptual distinction, not a guarantee that every observed pattern has only one cause.

For a positive pharmacokinetic parameter such as clearance, a common parameterization is \(\mathrm{CL}_i=\mathrm{CL}_{\mathrm{pop}}\exp(\eta_i)\), with \(\eta_i\) having mean zero and variance \(\omega^2\). This makes individual clearance positive and assigns a shared value of \(\eta_i\) to all observations for subject \(i\). Residual error can be additive, proportional, or combined; consequently, its variance need not be constant across predicted concentrations.

Conditional and Marginal Variation

The conditional distribution describes concentrations for a subject whose random effect is held fixed. In the simple model above, assume \(E(\epsilon_{ij}\mid\eta_i)=0\) and \(\mathrm{Var}(\epsilon_{ij}\mid\eta_i)=R_{ij}\). The residual variance \(R_{ij}\) may depend on the prediction or observation. For the basic repeated-measurement calculation, also assume residual errors at distinct observations are conditionally independent and that random effects are independent between subjects. These assumptions make the source of each covariance term explicit; they are not universal properties of concentration data.

The marginal distribution averages over the subject effects as well as residual errors. The law of total variance gives

The first term is variation among the subject-specific predictions. The second is the average conditional residual variance. Thus the marginal variance combines both sources, even though they describe different levels of the model.

For two observations \(j\) and \(k\) on the same subject, the law of total covariance gives

Under conditional independence of residuals at distinct observations, the first term is zero. The second term can remain positive because both predictions depend on the same subject effect. By contrast, observations from different subjects have no shared interindividual effect under the assumed independence of their random effects. This shared-versus-observation-specific pattern is the key mathematical distinction.

A Random-Intercept Illustration

A simple model makes the covariance pattern especially clear. Suppose two measurements are taken under conditions where the population prediction is the same, and write \(C_{ij}=\mu+b_i+\epsilon_{ij}\). Let \(b_i\) have mean zero and variance \(\omega^2\); let each \(\epsilon_{ij}\) have mean zero and variance \(\sigma^2\); and assume the subject effects and residuals are mutually independent, with independent residuals across observations. Here \(b_i\) represents an additive subject-level departure in concentration, not a general pharmacokinetic parameter model.

The covariance result follows because the shared \(b_i\) appears in both measurements, while the independent residuals contribute no cross-observation covariance. Equivalently, subtracting the two observations removes the shared subject effect:

This cancellation is useful for understanding repeated measurements, but it depends on the additive shared-effect model and the residual assumptions. In a nonlinear pharmacokinetic model, a random effect usually changes the predicted concentration through a parameter such as clearance rather than adding the same concentration shift at every time.

Estimating Components From a Balanced Repeated Design

A small method-of-moments calculation illustrates how within-subject and between-subject patterns can be separated. Suppose \(n\) subjects each have \(r\) measurements under the same conditions, with the random-intercept model and independent, equal-variance residuals. Let \(\bar{C}_i\) be subject \(i\)’s mean and \(s^2_{\bar{C}}\) the sample variance of those \(n\) subject means. Then

The reason is that \(\bar{C}_i=\mu+b_i+\bar{\epsilon}_i\). Its between-subject variance is \(\omega^2+\sigma^2/r\), because averaging \(r\) independent residuals with variance \(\sigma^2\) gives variance \(\sigma^2/r\). A pooled within-subject variance estimates \(\sigma^2\). Subtracting the estimated contribution \(\hat{\sigma}^2/r\) from \(s^2_{\bar{C}}\) therefore gives the method-of-moments estimate \(\hat{\omega}^2=s^2_{\bar{C}}-\hat{\sigma}^2/r\). With limited data this estimate can be negative; a negative estimate signals sampling variability or model mismatch, not a physically negative variance.

Nonconstant Residual Variance and Interpretation

The total-variance decomposition still applies when residual variance depends on the prediction. If \(R_{ij}=R\{m_{ij}(\eta_i)\}\), the marginal variance is \(\mathrm{Var}\{m_{ij}(\eta_i)\}+E[R\{m_{ij}(\eta_i)\}]\). The residual contribution is then an average over the distribution of subject predictions, not one common \(\sigma^2\). This matters for proportional or combined error models: a concentration-dependent residual spread should not automatically be interpreted as greater interindividual variability.

Diagnostics, Alternatives, and Limitations

Several features of the data can help assess whether a proposed separation is plausible. Under a model with shared subject effects and conditionally independent residuals, repeated observations from the same subject may show association after accounting for the predicted time course. Subject-level patterns that persist across observations are compatible with interindividual variability; deviations that fluctuate around each subject’s predictions are compatible with residual error. These are diagnostic clues, not definitive labels.

Residual plots should be examined against predictions, time, and other relevant covariates. Increasing spread with predicted concentration can indicate that a constant-variance residual model is inadequate. Trends in residuals over time may indicate a structural mismatch or a dependence pattern rather than a simple independent residual term. Earlier material on serial correlation in within-subject residuals addresses the case where residuals themselves remain correlated; then their covariance also contributes to repeated-measurement covariance.

The random-intercept calculation is a deliberately simple alternative to a full nonlinear mixed-effects model. It is useful for explaining the source of covariance, but it does not capture how random clearance, volume, absorption, or correlated parameters alter the shape of a concentration-time profile. In a nonlinear model, the variance of \(m_{ij}(\eta_i)\) may depend on time and dose, and a single additive \(\omega^2\) on the concentration scale is generally not appropriate. Replicate concentrations, as discussed in the earlier material on using replicate concentrations to identify residual variance, can provide information about residual variation under their design and modeling assumptions; they do not remove the need to model subject-level effects.

A common mistake is to call every deviation from the population prediction “residual error.” That reasoning ignores the possibility that a subject’s prediction differs systematically because of a shared random effect. The correction is to distinguish the prediction conditional on the subject effect from the population-level prediction, and then define residuals relative to the appropriate conditional prediction. Conversely, a persistent pattern is not proof of a random effect: omitted covariates, misspecified structural kinetics, assay problems, or dependence in residuals can also create patterns.

The decomposition relies on assumptions about the structural model, the random-effect distribution, and residual behavior. If residual variance is misspecified, residual dependence is ignored, or the sampling conditions differ substantially among subjects, apparent component estimates can be misleading. Estimates also have uncertainty, especially with few subjects or few repeats. The variance identities clarify what a model attributes to each source; they do not by themselves establish that the model is correct or that the data contain enough information for precise estimation.