Interindividual and Interoccasion Variability in Development Models

Why Separate Sources of Variability?

A population PK model may fit the average concentration-time profile well while giving an incomplete account of how exposures vary. A person can differ persistently from other people, the same person can differ from one dosing occasion to another, and individual measurements can depart from the model prediction. These are distinct sources of variation. Separating them matters when development decisions depend not only on typical exposure but also on how many people, or how many treatment occasions, may have unusually high or low exposure.

Interindividual variability (IIV) describes persistent differences between people. Interoccasion variability (IOV) describes changes within the same person across defined, repeated occasions, such as separate treatment periods or dosing occasions with comparable sampling. Residual error describes departures of observations from the model prediction after the modeled person- and occasion-level effects have been accounted for. It can reflect assay error, unmodeled short-term fluctuations, and model inadequacy; it is not automatically a pure measure of laboratory error.

A Hierarchical Model for the Three Sources

For illustration, suppose clearance for individual \(i\) on occasion \(j\) is modeled as lognormal:

Here \(CL_{\mathrm{typ}}\) is the typical clearance, \(\eta_i\) is the persistent individual effect, and \(\kappa_{ij}\) is the occasion-specific effect. In this basic model, both effects have mean zero, are normally distributed, and are mutually independent; occasion effects are also independent across occasions. Thus \(\omega^2\) is the IIV variance and \(\pi^2\) is the IOV variance, both on the log-clearance scale. The model assumes the same occasion definition and comparable data-generating conditions across occasions. A systematic change between periods should not be mistaken for random IOV.

A simplified observation model on the log-concentration scale might be written as:

where \(\widehat C_{ij\ell}\) is the concentration predicted from the structural model and that individual’s and occasion’s PK parameters, and \(\ell\) indexes observations within an occasion. This example assumes independent, mean-zero residuals with constant variance on the log scale. Real analyses may need a different residual error model; the key distinction remains that residual error applies at the observation level, not as a persistent effect shared across an individual’s observations or an occasion’s observations.

When the log-scale random effects are independent, the variance of log clearance across a randomly selected individual and occasion is \(\omega^2+\pi^2\). For a lognormal quantity with log variance \(v\), its coefficient of variation (CV) is \(\sqrt{\exp(v)-1}\). This converts a variance on the log scale into a relative measure on the original scale; the variance itself is not a percentage.

Interpreting Variance Estimates

A variance estimate is interpretable only with its level, scale, and model in view. For example, \(\omega^2=0.09\) is variance on the log-clearance scale, not “9% IIV.” Its log-scale standard deviation is \(\sqrt{0.09}=0.30\). Under the lognormal model, the corresponding CV is \(\sqrt{\exp(0.09)-1}\), or about 30.7%. This is a relative spread, not a statement that every person’s clearance lies within 30.7% of typical clearance.

Likewise, \(\pi^2\) describes occasion-to-occasion differences within people, conditional on the model’s individual effect. A CV derived from \(\pi^2\) summarizes the relative spread associated with the occasion effect. Residual variance \(\sigma^2\), by contrast, describes observation-level deviations on the specified residual scale. Comparing numerical values of \(\omega^2\), \(\pi^2\), and \(\sigma^2\) is meaningful only if their scales and assumptions are understood. Even when all are log-scale variances, they represent different patterns of dependence across the data.

Why Repeated Occasions and Samples Matter

The data structure determines whether the components can be distinguished. Multiple people provide information about IIV, but estimating IOV requires repeated, meaningfully comparable occasions within at least some people. Multiple samples within an occasion help characterize the concentration-time profile and residual variation; they do not, by themselves, provide repeated occasions. To separate occasion-level effects from residual error, the model needs observations that share the same occasion effect, alongside information about observation-level deviations.

A short variance decomposition shows why this distinction matters. Suppose a log-scale measurement has a person effect, an occasion effect, and independent residual error. The difference between two measurements in the same person and same occasion cancels both shared effects:

The last equality follows because the two residuals are independent and each has variance \(\sigma^2\): the variance of their difference is \(\sigma^2+\sigma^2\). In contrast, occasion means from the same individual vary because of both IOV and the residual contribution to each mean. With \(m\) independent observations per occasion, the residual variance of an occasion mean is \(\sigma^2/m\). Therefore, the variance of occasion means around an individual’s expected level is \(\pi^2+\sigma^2/m\), not just \(\pi^2\). Estimating the components requires the model to account for this distinction.

Consequences for Exposure-Based Decisions

Omitting a variance component can change a development conclusion even when the typical prediction is nearly unchanged. Consider a drug for which AUC is inversely proportional to clearance at a fixed dose, with the log-AUC random effects therefore having the same variance components as log clearance but opposite signs. If IOV is omitted, predicted variability across treatment occasions is reduced. That can understate the proportion of occasions above a safety-relevant exposure threshold or below an efficacy-relevant target. The direction and size of the decision impact depend on the exposure metric, threshold, model, and decision rule; it is not enough to note that the typical exposure remains the same.

Model Comparisons, Diagnostics, and Limitations

The main alternatives answer different questions. An IIV-only model allows persistent differences between people but treats a person’s parameter as stable across occasions. An IIV-plus-IOV model allows both persistent and occasion-specific differences, at the cost of needing repeated occasions and enough information to estimate another variance. A residual-only model, or a model that relies on larger residual error instead of occasion effects, may fit some observations but does not reproduce the shared pattern among observations from the same occasion. Fixed occasion effects are different again: they represent systematic mean shifts associated with particular periods or conditions, whereas IOV represents random occasion-to-occasion deviations. A systematic period shift should not be represented merely by increasing a random variance.

Useful warning signs include an IOV estimate near zero with large uncertainty, unstable estimates across reasonable model specifications, convergence difficulties, or a variance estimate that changes substantially when the sampling design is simplified. These can reflect weak information, not proof that the source is absent. Conversely, a large IOV estimate may be absorbing period effects, data handling problems, or a misspecified residual model. Examine whether deviations are shared by observations within an occasion, whether residuals show patterns across time or concentration, and whether occasions are genuinely comparable. Such checks help distinguish levels of variation; they do not by themselves establish a unique explanation.

Variance components are conditional on the model and study design. Sparse sampling, few repeated occasions, uneven numbers of observations, and changes in dose or clinical state can limit separation of IOV from residual error or systematic changes. On nonlinear PK models, a parameter’s variance does not translate mechanically into the same variance for every concentration or exposure metric. The exposure implications should therefore be evaluated using the fitted structural model and the intended dosing and sampling context, rather than by interpreting a variance component in isolation.

Common Pitfall

Wrong reasoning: “The residual variance is large, so it already includes all within-person variation; adding an occasion effect would count the same variability twice.” This confuses the level of a random effect with the level of an observation error. Residual error is not generally shared across observations, while an occasion effect is shared by observations from that occasion. Correction: specify the hierarchy, use repeated occasion data where possible, and assess whether within-occasion residual patterns and between-occasion shifts support distinct components. If the design cannot distinguish them, report that limitation rather than assigning all variation to whichever component is easier to estimate.