Occasion-Specific Effects in Repeated Individual PK Data

Why Separate Occasions?

Repeated concentration data from one person can be more informative about that person’s pharmacokinetics than a single occasion. But the occasions may differ for reasons that do not represent a lasting change in the person’s PK: absorption conditions, temporary physiology, or other occasion-level variability can shift observed concentrations. If every shift is attributed to a stable individual parameter, the posterior for that parameter can become too certain or be pulled in the wrong direction.

The key modelling distinction is between a stable individual effect, shared across the person’s occasions, and an occasion-specific effect, which varies from one occasion to another. This tutorial focuses on representing both in the individual likelihood and on updating the stable parameter without treating temporary deviations as persistent. An occasion effect is not automatically evidence of a time trend.

A Hierarchical Model for Repeated PK Data

Let \(\theta_i\) denote a stable individual parameter for person \(i\), such as log clearance, and let \(\eta_{ij}\) denote an occasion-specific deviation for occasion \(j\). Write the concentration data from occasion \(j\) as \(y_{ij}\), and let \(d_{ij}\) represent the recorded dose history and sampling times used to calculate the PK model prediction. A general concentration model is

The distribution on the right is determined by the PK structural model and residual error model. For example, an occasion deviation might enter additively on the log-clearance scale, so that occasion \(j\)’s log clearance is \(\theta_i+\eta_{ij}\). Other scientifically justified structures are possible; the deviation must be attached to a defined parameter or model component rather than added mechanically to concentrations.

A common starting assumption is \(\eta_{ij}\sim N(0,\tau^2)\), independently across occasions conditional on \(\theta_i\), with occasion effects independent of residual errors. The mean-zero assumption makes \(\theta_i\) the centre of the person’s occasion-specific values on the modeled scale. The variance \(\tau^2\) describes variation between occasions, not residual measurement error. The individual parameter has a prior distribution, such as \(\theta_i\sim N(\mu,\omega^2)\), possibly informed by a population model. These assumptions should be checked against the setting and data; they are not guaranteed by having repeated measurements.

For all the person’s occasions, the stable-parameter posterior is proportional to the prior times a likelihood in which each occasion effect is integrated out:

Here \(y_i\) denotes the person’s data across occasions. This product assumes conditional independence of the occasion data given the stable parameter and the occasion effects. When that assumption is inappropriate—for example, if deviations are correlated across occasions—the dependence structure must be represented rather than ignored. In a nonlinear PK model, the integrals may require numerical methods such as posterior sampling. The essential point is that each occasion contributes information about \(\theta_i\) while retaining uncertainty about its own deviation.

A Gaussian Update and Its Justification

A simple model makes the effect of occasion variability transparent. Suppose each occasion supplies a log-scale summary \(z_{ij}\) that is approximately an unbiased observation of \(\theta_i+\eta_{ij}\), with residual error \(\epsilon_{ij}\):

Assume the occasion deviations and residual errors are mutually independent, and that \(\theta_i\sim N(\mu,\omega^2)\). Integrating out \(\eta_{ij}\) gives \(z_{ij}\mid\theta_i\sim N(\theta_i,\tau^2+\sigma^2)\). Thus the occasion effect adds uncertainty to the information an occasion provides about the stable parameter. The posterior is normal, with the following mean and variance for \(J\) occasions:

To justify the result, the prior density contributes a quadratic term \(-(\theta_i-\mu)^2/(2\omega^2)\) to the log posterior. Each integrated occasion likelihood contributes \(-(z_{ij}-\theta_i)^2/[2(\tau^2+\sigma^2)]\). Collecting the coefficients of \(\theta_i^2\) gives posterior precision \(1/\omega^2+J/(\tau^2+\sigma^2)\); collecting the linear terms gives the numerator in the posterior mean. Completing the square yields the normal distribution stated above. The formula applies to this Gaussian summary model, not automatically to an arbitrary nonlinear concentration likelihood.

What an Occasion Effect Means in the Posterior

An occasion effect can be estimated conditionally as well as integrated out. In the Gaussian example, if \(\theta_i\) is temporarily treated as fixed, then the conditional posterior for \(\eta_{ij}\) given \(z_{ij}\) and \(\theta_i\) is normal. Its mean is \(\tau^2(\tau^2+\sigma^2)^{-1}(z_{ij}-\theta_i)\), and its variance is \(\tau^2\sigma^2/(\tau^2+\sigma^2)\). This conditional result describes how the model allocates an occasion’s deviation between the occasion effect and residual error. When \(\theta_i\) is uncertain, the joint posterior must also account for that uncertainty and the dependence between \(\theta_i\) and \(\eta_{ij}\).

Checking the Model and Comparing Approaches

The model’s assumptions concern both the PK structure and the repeated-occasion hierarchy. Check that dose histories and sample times are correctly represented, that the chosen occasion effect acts on a scientifically meaningful scale, and that the residual error model is suitable for the concentrations. A normal, mean-zero occasion distribution may be a useful approximation, but strong skewness, outliers, or dependence between occasions can make it inadequate. Sparse sampling can also leave stable parameters and occasion effects weakly identified; posterior correlations, shrinkage, and prior-data conflict are relevant diagnostics in that situation.

A useful diagnostic in the Gaussian summary model is the standardized difference between two occasion summaries. Under the model, \(z_{i1}-z_{i2}\) has mean zero and variance \(2(\tau^2+\sigma^2)\). Differences much larger than this scale indicate that the assumed combined variability may not explain the observed between-occasion discrepancy. In a full PK analysis, posterior predictive checks can assess whether simulated data reproduce the observed patterns across occasions.

The main alternatives clarify what the occasion model is doing. A model with no occasion effects sets \(\tau^2=0\), attributing all between-occasion differences to residual error and stable-parameter uncertainty. This may be appropriate if occasion variability is negligible, but it can make the stable posterior too precise when real occasion variation is present. At the other extreme, estimating a separate unrelated parameter for every occasion avoids pooling, but discards the shared information that motivates a stable individual parameter. The hierarchical model partially pools: each occasion informs the shared \(\theta_i\), while its own \(\eta_{ij}\) absorbs modeled occasion variation. A time-varying parameter model instead represents systematic evolution across time; a random occasion deviation alone does not specify such evolution.

Common Pitfalls and Limitations

• Pooling as if all observations were exchangeable residuals. This treats occasion shifts as measurement noise and can overstate precision. Include an occasion component when the scientific model and data support it.
• Calling every occasion deviation a lasting parameter change. A mean-zero occasion effect is a transient deviation under the model; it does not imply a trend or permanent change.
• Adding an effect without defining its scale. A deviation on log clearance has a different meaning from an additive concentration error. State which model component it modifies.
• Assuming the data can identify every variance component. With few occasions or sparse samples, \(\tau^2\), residual variance, and the stable parameter may be difficult to distinguish. Prior sensitivity and posterior dependence should be examined.
• Interpreting a posterior occasion effect causally. The effect summarizes variation under the model. It does not identify whether adherence, physiology, sampling, or another factor caused that variation.

Occasion-specific effects are most useful when repeated data plausibly contain both a persistent individual signal and occasion-level variation, and when there are enough data or informative prior assumptions to distinguish them. If the actual process has serial dependence, a changing mean, or an incorrect dose record, an independent mean-zero occasion effect may not be an adequate description. The model’s posterior is only as relevant as the assumptions and data representation that produce it.