Why Separate Variability by Its Pattern?
Suppose a patient receives the same dose on several occasions under comparable conditions. Concentrations may differ from one patient to another because some patients consistently have higher clearance or lower bioavailability. They may also differ between occasions for the same patient, even when the patient’s underlying tendency is unchanged. These patterns have different implications for the model: one is persistent across occasions within a subject, while the other is specific to an occasion.
In a population PK model, interindividual variability (IIV) represents differences between subjects. Interoccasion variability (IOV) represents occasion-to-occasion differences within a subject. Separating them is not merely a matter of naming random effects. It changes how the model interprets repeated data and predicts concentrations for another occasion.
A Model With Two Levels of Random Effect
For a positive PK parameter, such as clearance, a simple model on the log scale is
where \(\theta_{ij}\) is the parameter for subject \(i\) on occasion \(j\), \(\theta_{\mathrm{typ}}\) is its typical value, \(\eta_i\) is the subject-level effect, and \(\kappa_{ij}\) is the occasion-level effect. The same \(\eta_i\) applies to all occasions for subject \(i\); \(\kappa_{ij}\) may change with \(j\). This multiplicative model keeps \(\theta_{ij}\) positive.
A basic variance-components model assumes \(\eta_i\sim N(0,\omega_{\mathrm{IIV}}^2)\) and \(\kappa_{ij}\sim N(0,\omega_{\mathrm{IOV}}^2)\), with the two effects independent and occasion effects independent across occasions. These are modeling assumptions, not facts guaranteed by repeated dosing. In this simple model, there is no systematic trend in the occasion effects, and their distributions are the same across subjects and occasions. The parameter \(\theta_{\mathrm{typ}}\) is the median parameter value when both effects have mean zero; because the model is lognormal, it need not equal the arithmetic mean.
The structural model maps the parameter for an occasion to its concentration-time profile. For example, the same subject-level clearance effect influences that subject’s profiles on every occasion, while a clearance occasion effect changes the clearance used for that occasion’s profile. Concentration observations also contain residual error, which is not the same as either random effect: an occasion effect is shared by the profile’s parameter value, whereas residual error is attached to observations under the residual model.
Worked Example: Computing Occasion-Specific Clearance
Consider a hypothetical model with typical clearance \(\theta_{\mathrm{typ}}=5.00\) L/h. For one subject, let \(\eta_i=0.20\), with occasion effects \(\kappa_{i1}=-0.10\) and \(\kappa_{i2}=0.15\). Calculate clearance on each occasion.
For occasion 1, the combined log effect is \(0.20-0.10=0.10\), so \(\theta_{i1}=5.00\exp(0.10)=5.00(1.1052)=5.526\) L/h.
For occasion 2, it is \(0.20+0.15=0.35\), so \(\theta_{i2}=5.00\exp(0.35)=5.00(1.4191)=7.095\) L/h.
The subject-level effect is present in both calculations. The difference between occasions comes from the change in \(\kappa_{ij}\); the clearance ratio is \(\theta_{i2}/\theta_{i1}=\exp(0.15-(-0.10))=\exp(0.25)=1.284\). Thus clearance is about 28.4% higher on occasion 2 than on occasion 1 in this hypothetical example.
What Repeated Occasions Reveal
The distinction can be seen directly on the log scale. Define \(Z_{ij}=\log(\theta_{ij}/\theta_{\mathrm{typ}})\). Under the model, \(Z_{ij}=\eta_i+\kappa_{ij}\). Within one subject, the shared \(\eta_i\) shifts all occasions in the same direction; the occasion effects add deviations around that subject’s persistent level.
The model implies a particular covariance pattern. For two different occasions \(j\ne k\) from the same subject,
To justify the covariance result, substitute the random-effects model and use the assumed independence: \(\operatorname{Cov}(\eta_i+\kappa_{ij},\eta_i+\kappa_{ik})\) equals \(\operatorname{Var}(\eta_i)+\operatorname{Cov}(\eta_i,\kappa_{ik})+\operatorname{Cov}(\kappa_{ij},\eta_i)+\operatorname{Cov}(\kappa_{ij},\kappa_{ik})\). The last three terms are zero under the assumptions, leaving \(\omega_{\mathrm{IIV}}^2\). For the same occasion, the variance includes both independent components, giving their sum. If both components are positive, the correlation across two occasions on the log scale is therefore \[ \frac{\omega_{\mathrm{IIV}}^2}{\omega_{\mathrm{IIV}}^2+\omega_{\mathrm{IOV}}^2}. \] Persistent similarity across occasions reflects the shared subject effect; deviations around that persistence reflect occasion variability. This covariance interpretation is specific to the stated model and scale.
Worked Example: Interpreting Variance Components
Suppose a hypothetical log-clearance model estimates \(\omega_{\mathrm{IIV}}^2=0.09\) and \(\omega_{\mathrm{IOV}}^2=0.04\). Find the total log-scale variance for one occasion and the correlation between log clearances on two occasions from the same subject.
The variance for one occasion is \(0.09+0.04=0.13\), and its standard deviation is \(\sqrt{0.13}=0.3606\) on the log scale. The covariance between different occasions for the same subject is \(0.09\), from the shared interindividual effect. Hence their correlation is \(0.09/0.13=0.6923\).
Under this model, the repeated log-clearance values are positively correlated because they share a subject effect. The result does not mean that 69.23% of an individual clearance value is “caused” by IIV; it is a correlation implied by the two variance components on the log scale.
Worked Example: Reading Subject and Occasion Patterns
For illustration, let \(\theta_{\mathrm{typ}}=10.0\) L/h. Two hypothetical subjects have effects \(\eta_A=0.10\) and \(\eta_B=-0.20\). Let subject A’s occasion effects be \(-0.10\) and \(0.10\), and subject B’s be \(0.20\) and \(-0.20\). Calculate the four clearances and identify the two patterns.
For A, the combined effects are \(0.10-0.10=0\) and \(0.10+0.10=0.20\). The clearances are \(10.0\exp(0)=10.00\) L/h and \(10.0\exp(0.20)=12.21\) L/h.
For B, the combined effects are \(-0.20+0.20=0\) and \(-0.20-0.20=-0.40\). The clearances are \(10.0\exp(0)=10.00\) L/h and \(10.0\exp(-0.40)=6.703\) L/h.
The subject effects differ: A has a positive persistent effect and B a negative one. Within each subject, the occasion effects also change the clearance from one occasion to the next. Equal clearance on one occasion does not imply equal subject effects; here the occasion effects happen to offset them on the first occasion.
Assumptions, Diagnostics, and Alternatives
The simple decomposition is most interpretable when occasions are meaningfully defined, the parameter has the same role in the structural model on each occasion, and the random-effect distributions are appropriate approximations. It also assumes the occasion-level effects are independent across occasions and independent of the subject-level effect. If occasions differ systematically, for example through a consistent time trend, a zero-mean exchangeable occasion effect may not capture that structure.
Repeated observations can help diagnose whether a proposed decomposition is plausible. After accounting for the structural model, ask whether a subject’s deviations tend to persist across occasions, whether the remaining within-subject occasion shifts are substantial, and whether residual patterns are being mistaken for parameter shifts. These are model checks, not a way to label every observed change as IOV. A poorly fitting structural model can also produce occasion-like discrepancies.
A key limitation is that a single occasion per subject provides no direct within-subject replication for separating IIV from IOV. With repeated occasions, the shared-versus-changing pattern is available in principle, but the data may still be too sparse or noisy to estimate both components reliably. If the model assigns variability to IOV, it should be supported by repeated information and a defensible structural interpretation.
| Model component | Pattern represented | What it does not represent |
|---|---|---|
| Interindividual effect \(\eta_i\) | A subject’s persistent deviation across that subject’s occasions | A change unique to one occasion |
| Interoccasion effect \(\kappa_{ij}\) | A parameter shift specific to an occasion within a subject | Independent noise at each concentration observation |
| Residual error | Observation-level discrepancy under the specified error model | A shared parameter shift across an occasion |
A model with IIV alone is simpler, but it cannot represent genuine occasion-to-occasion parameter shifts. A model with IOV alone allows occasion changes but omits persistent between-subject differences, which are often scientifically important. Residual error is another alternative for unexplained concentration discrepancies, but it has a different pattern: it does not, by itself, make all observations on one occasion share a parameter deviation. These models should be compared by whether their assumed variability patterns make sense, not just by whether they can reduce discrepancies.
Check Your Understanding
Answer each question, then compare with the solutions below.
- In \(\theta_{ij}=\theta_{\mathrm{typ}}\exp(\eta_i+\kappa_{ij})\), which effect is shared across two occasions for the same subject?
- If \(\omega_{\mathrm{IIV}}^2=0.16\) and \(\omega_{\mathrm{IOV}}^2=0.09\), what is the log-scale variance for one occasion and the covariance between two different occasions for the same subject?
- Why can one occasion per subject not directly distinguish IIV from IOV in this model?
- On a particular occasion, the concentrations are all shifted in a similar direction from the structural prediction. Which model component could represent a shared occasion-level parameter shift, and why is independent residual error a different interpretation?
Solutions. (1) \(\eta_i\) is shared; \(\kappa_{ij}\) can differ by occasion. (2) The variance is \(0.16+0.09=0.25\); the covariance across different occasions is \(0.16\), because only the shared \(\eta_i\) contributes under the independence assumptions. (3) There is no repeated occasion within a subject from which to distinguish a persistent subject effect from an occasion-specific effect; both contribute to the single observed occasion’s parameter deviation. (4) An interoccasion effect can shift the parameter used for the whole occasion’s profile. Independent residual errors apply to individual observations and do not, by themselves, impose a shared shift across that occasion.