Separating Interoccasion Variability From Residual Error

Why the Timing of Variation Matters

A concentration can depart from its model-predicted value for more than one reason. A patient's pharmacokinetic parameter may differ on one dosing occasion from that patient's value on another occasion. Alternatively, a single measurement may be noisy even when the underlying occasion-specific profile is unchanged. These explanations have different implications: an occasion-level change can affect several concentrations together, whereas residual error is attached to individual observations.

The key evidence is therefore not simply that concentrations vary. It is whether deviations from the predicted profile persist across observations from the same occasion and whether they recur across occasions. The distinction is clearest when each subject contributes multiple observations on multiple occasions, with comparable sampling times or a model that accounts for their differences.

Definition: Interoccasion variability (IOV) is variation in a pharmacokinetic parameter or parameter-related effect from one occasion to another within the same subject. Residual error is observation-level variation remaining after the model's population, subject, occasion, and measured-covariate effects have been accounted for. IOV is shared by observations on the same occasion through the model; residual errors need not be shared.

A Model for Separating the Sources

Let $i$ index subjects, $j$ occasions within subject, and $k$ observations within an occasion. Write $y_{ijk}$ for a transformed observed concentration, such as log concentration, and $m_{ijk}$ for the model-predicted transformed concentration before the occasion effect and residual error. A useful local representation is

y_{ijk}=m_{ijk}+a_{ijk}\kappa_{ij}+\epsilon_{ijk}.

Here $\kappa_{ij}$ is a mean-zero occasion effect with variance $\omega^2_{\mathrm{IOV}}$, and $\epsilon_{ijk}$ is a mean-zero residual error with variance $\sigma^2_{ijk}$. The coefficient $a_{ijk}$ describes how the occasion effect changes the predicted concentration at that time. It can be one for a simple occasion-level shift, or it can vary with time when the effect arises through a particular parameter, such as clearance. In a full population model, subject effects are also included; when comparing occasions within one subject, those stable subject effects are part of the shared baseline and are not themselves IOV.

The basic separation relies on assumptions that should be checked against the design and model. Occasion effects are assumed to be independent across occasions, conditional on the subject effects and covariates. Residual errors are assumed to have mean zero conditional on the model and to be independent across observations for the simplest covariance argument below. Their variances may depend on predicted concentration if the chosen residual model allows that. The mean model must adequately describe systematic changes over time, dose, and occasion; otherwise, an omitted trend or a misspecified concentration-time curve can look like either source of variation.

Key result: For two distinct observations $k$ and $l$ on the same occasion, if their residual errors are independent, then $$ \operatorname{Cov}(y_{ijk}-m_{ijk},y_{ijl}-m_{ijl})=a_{ijk}a_{ijl}\omega^2_{\mathrm{IOV}}. $$ For observations from independent occasions, the occasion-effect covariance is zero. The residual variance contributes to the variance of each individual observation, but not to the covariance between distinct observations under the independence assumption.

This result explains the diagnostic contrast. A shared occasion effect tends to create a coherent pattern across an occasion's profile. Residual noise tends to create point-to-point scatter around that profile. The pattern need not be a constant vertical shift: a clearance effect, for example, can change the profile shape in a predictable, time-dependent way. Thus the relevant question is whether the deviations align with the concentration-time signature expected from the affected parameter.

Deriving the Covariance Contrast

Set $r_{ijk}=y_{ijk}-m_{ijk}=a_{ijk}\kappa_{ij}+\epsilon_{ijk}$. Assume $E(\kappa_{ij})=E(\epsilon_{ijk})=0$, that the occasion effect is independent of residual errors, and that distinct residual errors are independent. For two distinct samples on the same occasion, expanding the covariance gives

\begin{aligned} \operatorname{Cov}(r_{ijk},r_{ijl}) &=\operatorname{Cov}(a_{ijk}\kappa_{ij}+\epsilon_{ijk}, a_{ijl}\kappa_{ij}+\epsilon_{ijl})\\ &=a_{ijk}a_{ijl}\operatorname{Var}(\kappa_{ij}) +\operatorname{Cov}(\epsilon_{ijk},\epsilon_{ijl})\\ &=a_{ijk}a_{ijl}\omega^2_{\mathrm{IOV}}. \end{aligned}

The cross terms are zero by independence, and the residual covariance is zero for distinct observations by assumption. For two observations on different occasions, the shared-effect term also vanishes because the occasion effects are independent. For a single observation, both sources contribute to the variance:

\operatorname{Var}(r_{ijk})=a_{ijk}^2\omega^2_{\mathrm{IOV}}+\sigma^2_{ijk}.

These equations are not a universal recipe for estimating components from any data table. They state what the model predicts under its assumptions. If residuals are serially correlated, for example, distinct residuals can have nonzero covariance, so same-occasion correlation is no longer unique evidence of IOV. Likewise, if occasions are not independent or the mean model omits an occasion-related trend, the covariance interpretation changes.

Worked Example: A Repeated-Dose Profile With Persistent Deviations

Consider hypothetical log-concentration residuals from one subject after repeated doses on three occasions. Each occasion has samples at the same four post-dose times. The residual is $r=\log(C_{\mathrm{observed}}/C_{\mathrm{predicted}})$, so a positive value means the observed concentration is above prediction on the log scale. Assume for this descriptive illustration that the mean profile is adequate and the within-occasion residuals are independent.

Occasion	Sample 1	Sample 2	Sample 3	Sample 4	Occasion mean
1	0.10	0.08	0.12	0.07	0.0925
2	-0.11	-0.08	-0.12	-0.09	-0.1000
3	0.02	0.05	0.00	0.04	0.0275

The grand mean is $(0.0925-0.1000+0.0275)/3=0.00667$, close to zero. Within each occasion, the four residuals are relatively similar, while the occasion means differ substantially. For example, occasion 1 is mostly above prediction and occasion 2 is mostly below it. That persistence is qualitatively consistent with an occasion-level effect rather than independent pointwise noise alone.

A simple within-occasion variance calculation quantifies the scatter. The sums of squared deviations from the three occasion means are $0.001475$, $0.001000$, and $0.001475$. Therefore the pooled within-occasion variance is

s^2_{\mathrm{within}}= \frac{0.001475+0.001000+0.001475}{3(4-1)} =\frac{0.003950}{9}=0.000439.

The sample variance of the three occasion means is

s^2_{\mathrm{means}}= \frac{(0.0925-0.00667)^2+(-0.1000-0.00667)^2+(0.0275-0.00667)^2}{3-1} \approx 0.00959.

With four equally precise samples per occasion, independent within-occasion noise contributes approximately $s^2_{\mathrm{within}}/4=0.0001097$ to the variance of an occasion mean. Subtracting this contribution gives a descriptive between-occasion component of approximately $0.00959-0.0001097=0.00948$ on the squared log-residual scale. This calculation is only illustrative: three occasions from one subject do not establish a population IOV estimate, and uncertainty in the mean model and variance estimates is not represented. The central evidence is the grouping of residuals by occasion, not the precision of this small-sample component calculation.

Worked Comparisons of Variation Patterns

Worked Example: Translating Variance Components Into Covariance

Suppose a hypothetical log-concentration model has a constant occasion loading $a=1$, occasion-effect variance $\omega^2_{\mathrm{IOV}}=0.0400$, and independent residual variance $\sigma^2=0.0100$. Both variances are on the squared log-concentration scale.

The variance of one residual is the sum of the two components:

\operatorname{Var}(r)=0.0400+0.0100=0.0500.

For two distinct samples from the same occasion, their covariance is $1\times1\times0.0400=0.0400$. Their correlation is therefore $0.0400/\sqrt{0.0500\times0.0500}=0.800$. For samples from different independent occasions, the occasion-effect covariance is zero; under the assumed independent residual model, their covariance is also zero. Thus a large share of the single-sample variance appears as shared within-occasion variation, rather than independent noise.

This is a model-based implication, not a claim that every observed pair will have correlation exactly 0.800. The value depends on the stated variance assumptions, and a different residual covariance structure or time-varying loading would change the calculation.

Worked Example: When One Sample Per Occasion Is Not Enough

Suppose a study records only one concentration per occasion, and the residual is modeled as $r_{ij}=\kappa_{ij}+\epsilon_{ij}$ with loading one. If $\omega^2_{\mathrm{IOV}}=0.0400$ and $\sigma^2=0.0100$, the observed residual variance is $0.0500$. But that single variance alone does not reveal how much came from IOV and how much from residual error.

Indeed, the same $0.0500$ total variance could arise from $\omega^2_{\mathrm{IOV}}=0.0200$ and $\sigma^2=0.0300$, or from $\omega^2_{\mathrm{IOV}}=0$ and $\sigma^2=0.0500$. With one observation on each occasion, there is no within-occasion covariance to distinguish a shared occasion effect from observation-level noise. Replicated samples within occasions, informative structural differences in the parameter effect, or other design information are needed to separate the components. Repeated occasions alone do not automatically solve the problem when each occasion supplies only one residual and the model gives both sources the same observational signature.

Design, Diagnostics, and Alternative Explanations

A practical assessment begins with residuals aligned by subject, occasion, and comparable post-dose time. Plotting each occasion's profile can reveal whether deviations are coherent across samples, whether they follow a plausible parameter-specific shape, and whether the apparent shift is actually a time trend. The same subject should contribute enough observations within an occasion to reveal within-occasion scatter, and enough occasions to reveal recurring between-occasion differences. Unequal occasion counts and missing samples can still be informative, but they change the amount of evidence contributed by each subject and occasion.

Useful diagnostic questions include:

Do several observations on the same occasion deviate in a coordinated way, beyond what the residual model predicts?
Are deviations tied to occasion after accounting for dose, sampling time, adherence, co-medication, and other measured predictors?
Does a parameter-specific IOV effect generate the expected change across the concentration-time profile, rather than merely fitting isolated residuals?
Do within-occasion residuals show serial correlation or changing variance that could mimic shared occasion effects?
Are apparent occasion shifts actually long-term trends, assay changes, or misspecification of the structural concentration-time model?

The main alternatives answer different questions. A fixed occasion effect estimates systematic differences for specified occasions, such as a known formulation or study period; it does not by itself describe a distribution of occasion-to-occasion variation in the target population. A random occasion effect represents occasion deviations as draws from a distribution and is appropriate when occasions are viewed as exchangeable realizations from that distribution. A more flexible residual covariance model can represent within-occasion correlation without attributing it to parameter variation. Choosing among these requires scientific knowledge and diagnostics, not just a visual pattern or a reduction in a fit statistic.

The distinction also has limits. Sparse sampling may leave IOV weakly identified, especially when every occasion has only one observation or when the occasion effect and residual error have nearly indistinguishable patterns. A wrong residual variance model can move variation into the IOV component, and a misspecified mean model can create structured residuals that an occasion effect then absorbs. Conversely, an effect too small relative to residual noise may be difficult to detect even with repeated profiles. The model's estimated components should therefore be interpreted alongside design information, profile diagnostics, and plausible biological mechanisms.

Common pitfall: “Concentrations from the same occasion are similar, so the occasion effect must be large.” Similarity alone is not enough. The residual model may include serial correlation, sampling times may not be comparable, or the structural model may miss a trend. The correction is to ask whether the observed within-occasion covariance exceeds what the specified residual model predicts, and whether its profile shape is consistent with the proposed parameter effect.

Check Your Understanding

Answer each question, then compare with the solutions below.

Under the independent-residual model, which component creates covariance between two distinct samples from the same occasion?
With constant occasion loading one, $\omega^2_{\mathrm{IOV}}=0.0300$, and $\sigma^2=0.0200$, calculate the single-observation variance and same-occasion covariance.
Why can one concentration per occasion fail to separate IOV from residual error?
Name one way serially correlated residuals could mislead an assessment of IOV.
What is the distinction between a fixed occasion effect and a random occasion effect?

Solutions. (1) The shared occasion effect creates covariance, equal to $a_{ijk}a_{ijl}\omega^2_{\mathrm{IOV}}$; with unit loadings it is $\omega^2_{\mathrm{IOV}}$. (2) The single-observation variance is $0.0300+0.0200=0.0500$, and the covariance of distinct same-occasion samples is $0.0300$. (3) With one sample per occasion there is no within-occasion covariance to identify the shared component separately; only the sum of the variance components is observed under this simple model. (4) Correlated residuals can make same-occasion observations move together even without IOV, falsely suggesting a shared occasion effect. (5) A fixed effect estimates differences for specified occasions; a random effect treats occasion deviations as draws from a population distribution of occasion effects.