Interoccasion Variability Mistaken for Interindividual Variability

Why Repeat Occasions Matter

A subject’s concentration profile can differ from another subject’s profile because their typical pharmacokinetic parameters differ. But the same subject’s profiles can also differ across dosing occasions: clearance may change with temporary physiology, adherence, concomitant treatment, or measurement conditions. If a model represents only between-subject differences, it may assign some of these occasion-to-occasion changes to interindividual variability. The resulting subject-specific parameter estimates can look more variable than the subjects’ stable underlying differences really are.

Repeated occasions provide the key comparison. Interindividual variability (IIV) describes persistent differences between subjects; interoccasion variability (IOV) describes changes within a subject from one occasion to another. The distinction concerns the level at which the variation occurs, not merely how large the observed variation is. Profiles must be interpreted in the context of their sampling, dosing history, and the definition of an occasion.

Definition: An occasion is a defined period or dosing episode in which a subject’s PK parameter is treated as constant for the model. For a parameter such as clearance, a basic log-normal model with IIV and IOV is $$ \log(CL_{ij})=\log(CL_{\mathrm{typ}})+\eta_i+\kappa_{ij}, $$ where $CL_{ij}$ is subject $i$’s clearance on occasion $j$, $CL_{\mathrm{typ}}$ is typical clearance, $\eta_i$ is the subject-level random effect, and $\kappa_{ij}$ is the occasion-level random effect. In this model, $\eta_i$ is shared across that subject’s occasions, whereas $\kappa_{ij}$ can differ by occasion.

The simple model assumes that $\eta_i$ and $\kappa_{ij}$ have mean zero, are mutually independent, and are independent across subjects. It also assumes that occasion effects are independent across occasions and that each occasion’s effect is constant throughout the profile. Their variances are $\omega^2$ and $\pi^2$, respectively. These assumptions are a starting point, not a guarantee: repeated occasions may be correlated, and a systematic shift over calendar time is not automatically a random IOV effect.

A random occasion effect represents unexplained occasion-to-occasion deviations around the model’s typical value. If all subjects shift similarly between occasion 1 and occasion 2, the pattern may instead call for a fixed occasion effect or a time-varying covariate. The occasion definition must also be meaningful: treating every concentration sample as a separate occasion would not represent a parameter that is stable over a dosing episode.

What the Variance Decomposition Predicts

The model gives a direct way to distinguish the two sources of variability. For the log-clearance on any one occasion, both random effects contribute. Across two different occasions for the same subject, however, only the shared subject effect contributes to their covariance, under the independence assumptions above.

\begin{aligned} \operatorname{Var}\{\log(CL_{ij})\}&=\omega^2+\pi^2,\\ \operatorname{Cov}\{\log(CL_{ij}),\log(CL_{ik})\}&=\omega^2 \quad (j\ne k),\\ \operatorname{Corr}\{\log(CL_{ij}),\log(CL_{ik})\} &=\frac{\omega^2}{\omega^2+\pi^2}. \end{aligned}

This result explains why one occasion per subject cannot separate IIV from IOV using this model: the data show only the combined variance $\omega^2+\pi^2$, with no within-subject repeat comparison to identify the shared component. With repeated occasions, similarities between a subject’s occasions inform IIV, while the additional differences between those occasions inform IOV. In real population analyses, the variance components are estimated using the full concentration data and the structural and residual-error models, not by simply calculating variances of noisy individual estimates.

Worked Example: Reading the Covariance Pattern

Suppose a hypothetical model estimates $\omega^2=0.09$ and $\pi^2=0.16$ for log-clearance. Calculate the variance on one occasion, the covariance between two occasions for the same subject, and their correlation.

Variance on one occasion. Both components contribute: $$ 0.09+0.16=0.25. $$ The standard deviation on the log scale is $\sqrt{0.25}=0.50$.

Covariance across occasions. The occasion effects are independent across occasions, so only the shared subject effect contributes: $$ \operatorname{Cov}=0.09. $$

Correlation across occasions. The two occasion-specific log-clearances have the same variance, so $$ \operatorname{Corr}=\frac{0.09}{0.25}=0.36. $$ Thus, the model predicts a positive but incomplete correlation between a subject’s log-clearance values on different occasions. The occasion component contributes to variation on each occasion but not to covariance across occasions under the stated assumptions.

Recognizing the Pattern in Repeat Profiles

A useful diagnostic starts with repeated concentration profiles from the same subjects. Ask whether the profiles preserve a subject’s relative position across occasions, or whether a subject’s inferred parameter changes substantially from one occasion to another. For example, if a subject’s concentrations are consistently lower across a well-sampled occasion than expected from their other occasions, a higher occasion-specific clearance could explain the difference in a linear PK model, all else equal. The entire profile and the sampling times matter; a single low concentration does not establish a clearance change.

Occasion-specific parameter estimates can help visualize this pattern, but they are not direct measurements of the underlying parameters. With sparse samples, the estimates may be imprecise and pulled toward the population prediction. A striking spread of individual estimates can therefore reflect estimation noise, structural misspecification, or residual error rather than genuine IOV. Interpret these estimates alongside concentration residuals, parameter uncertainty, and how much information each occasion contributes.

Worked Example: Repeat Profiles Reveal Within-Subject Changes

Consider a hypothetical subject given the same dose on two occasions. Under a simple linear model with complete exposure measurement, suppose the dose is 100 mg and the estimated AUC is 20 mg·h/L on occasion 1 and 12.5 mg·h/L on occasion 2. Use $CL=\text{Dose}/\text{AUC}$ to compare the implied clearances.

Occasion 1: $$ CL_1=\frac{100\ \mathrm{mg}}{20\ \mathrm{mg\cdot h/L}}=5.00\ \mathrm{L/h}. $$ Occasion 2: $$ CL_2=\frac{100\ \mathrm{mg}}{12.5\ \mathrm{mg\cdot h/L}}=8.00\ \mathrm{L/h}. $$ The occasion-specific estimates differ by $8.00-5.00=3.00\ \mathrm{L/h}$, and the ratio is $8.00/5.00=1.60$. If the dose, bioavailability, and other relevant assumptions are comparable, this is a candidate within-subject clearance change. It is not, by itself, proof of IOV: differences in dose delivery, bioavailability, measurement, or model fit could also contribute. Repeated profiles from multiple subjects are needed to assess whether such changes recur as occasion-level variation rather than being isolated noise.

Worked Example: A Shared Occasion Shift Is Not Necessarily Random IOV

Suppose three hypothetical subjects have clearance estimates on two occasions as follows: subject A, 5 and 8 L/h; subject B, 10 and 16 L/h; subject C, 20 and 32 L/h. Compare the occasion ratios.

For A, $8/5=1.60$; for B, $16/10=1.60$; and for C, $32/20=1.60$. Every subject has the same 1.60-fold shift from occasion 1 to occasion 2. This pattern is consistent with a systematic occasion difference, such as an occasion-specific change shared by the subjects, rather than evidence on its own for random subject-by-occasion deviations. A fixed effect for occasion may be more appropriate to investigate.

The example also shows why comparing absolute differences alone can mislead: the differences are 3, 6, and 12 L/h, increasing with the subjects’ clearance values. A multiplicative model is naturally examined on the log scale, where each ratio corresponds to the same change, $\log(1.60)$. With only three hypothetical subjects and two occasions, the pattern is suggestive, not a definitive model-selection result.

Model Checks and Failure Modes

A practical evaluation uses the repeat structure explicitly. First check that the occasion labels correspond to the intended dosing episodes and that the relevant dose and sampling records are correctly aligned. Then compare observed profiles and residual patterns by subject and occasion. Look for occasion-wide shifts or shape changes, and ask whether a proposed parameter effect could plausibly produce them. A clearance effect, for example, should be assessed against the profile’s time course rather than inferred from one concentration.

A useful workflow is to compare a model with IIV alone against a model that also allows IOV for a scientifically plausible parameter. Check whether the added component improves the description of repeated profiles and whether its estimate is supported by the data. Inspect uncertainty and stability, as well as whether the apparent improvement is driven by one subject or one occasion. Residual patterns that persist after adding IOV may indicate that the problem lies elsewhere, such as in the structural or residual-error model.

Confirm repeat information.
Verify that subjects contribute more than one informative occasion and that the occasions are defined consistently.

Separate shared and occasion-specific patterns.
Compare within-subject profiles, while distinguishing a common occasion shift from deviations that vary by subject.

Check competing explanations.
Evaluate dose history, covariates, residual patterns, and structural adequacy before attributing a discrepancy to IOV.

Assess support for the added component.
Review the estimated variance, its uncertainty, and whether the model describes repeated profiles more plausibly without relying on a few influential occasions.

The principal alternative is a model with IIV alone. It is simpler and may be adequate when repeat occasions show little supported within-subject change. But if real IOV is omitted, the model can inflate estimated IIV because the marginal variance on one occasion contains both components. Occasion-specific fixed effects are a different alternative when there is a systematic shift shared across subjects. A time-varying covariate or explicit changing-parameter model may be more suitable when the pattern follows a measured or evolving process. These approaches answer different questions; a random IOV term should not be used as a generic remedy for every occasion-related pattern.

Common pitfall: “The subject-specific clearance estimates vary, so the population has high IIV.” That conclusion ignores whether each subject’s estimates also vary across occasions. If within-subject occasion changes are being absorbed by an IIV-only model, the estimated between-subject spread may be misleading. Compare repeated occasions and consider the model’s variance decomposition before interpreting IIV.

Limitations and Interpretation

IOV is difficult to estimate when there are few repeated occasions, sparse sampling within each occasion, or large residual error. If an occasion contains too little information to distinguish clearance from other parameters, its apparent parameter change can be highly uncertain. Likewise, a poor structural model can create occasion-specific residual patterns that an added random effect partly absorbs without identifying the underlying cause. A variance estimate near zero does not prove that biological occasion changes never occur; it can indicate that the available data do not support estimating them.

The basic model also assumes independent occasion effects. If effects persist across adjacent occasions, the covariance between occasions need not equal $\omega^2$; an appropriate correlation structure may be needed. Further, subject-specific occasion changes can be correlated across PK parameters. Modeling one parameter’s IOV in isolation may be inadequate if, for example, clearance and volume change together. These extensions require sufficient repeated information and should be motivated by the observed design and scientific question.

The distinction matters because IIV and IOV have different interpretations. IIV concerns differences among subjects that persist across occasions in the model; IOV concerns within-subject variation across the defined occasions. Treating one as the other can distort the estimated variability and the model’s account of repeat profiles. The most credible interpretation is the one supported by the repeat design, the profile-level diagnostics, and explicit consideration of alternative explanations.

Check Your Understanding

Answer each question, then compare with the solutions below.

In the log-clearance model, which random effect is shared by a subject across occasions, and which may change between occasions?
If $\omega^2=0.04$ and $\pi^2=0.21$, what are the log-clearance variance on one occasion and the covariance across two occasions for the same subject?
Every subject in a small data set has a 1.3-fold higher clearance estimate on occasion 2 than on occasion 1. Why should this pattern prompt consideration of a fixed occasion effect?
Why can one occasion per subject fail to distinguish IIV from IOV in the basic model?
What is one reason an apparent occasion-specific clearance change might not represent true IOV?

Solutions. (1) $\eta_i$ is shared across occasions; $\kappa_{ij}$ is occasion-specific. (2) The variance is $0.04+0.21=0.25$, and the covariance across occasions is $0.04$, because independent occasion effects contribute no cross-occasion covariance. (3) A common proportional shift across subjects suggests a systematic occasion difference; a fixed effect can represent a shared shift, whereas random IOV represents deviations that vary across subject-occasion pairs. (4) With only one occasion, the observed log-clearance variance is the sum $\omega^2+\pi^2$; there is no within-subject repeat covariance to separate the components. (5) Examples include sparse or noisy sampling, a difference in dose delivery or bioavailability, or structural misspecification.