Separating Between-Occasion and Between-Patient Variability

Why the Source of Variability Matters

Repeated concentration measurements from one patient need not agree perfectly, even when the dose, sampling time, and clinical conditions appear unchanged. Some differences reflect a persistent characteristic of that patient, such as a consistently higher exposure than the population average. Other differences are temporary: a particular dosing occasion may produce concentrations above or below that patient’s usual level. Treating both sources as one undifferentiated patient effect can make forecasts for a new occasion too confident or can make a temporary deviation look like a permanent patient characteristic.

The distinction matters when using earlier observations to forecast a future dosing occasion. Evidence repeated across occasions is informative about a persistent patient effect. An unusually high result on just one occasion may instead reflect that occasion’s deviation. Even if the patient effect were known exactly, a future occasion can still differ from the patient’s usual level.

A Model That Separates the Components

To make the distinction precise, consider a fixed dose regimen and fixed sampling design. Let \(Z_{ik}\) be a log-concentration summary for patient \(i\) on occasion \(k\), after accounting for the expected effect of the dose, sampling time, and recorded covariates. The summary could be a transformed concentration at a specified sampling time or a suitable occasion-level summary. Use the model

Here \(\mu\) is the population typical value on the log scale. Assume \(\eta_i\sim N(0,\omega^2)\), \(\kappa_{ik}\sim N(0,\pi^2)\), and \(\epsilon_{ik}\sim N(0,\sigma^2)\). The three terms have mean zero, are mutually independent, and are independent across patients; occasion effects and residual errors are also independent across occasions. The residual \(\epsilon_{ik}\) represents remaining measurement-level variation in this simplified summary model. These assumptions are a working model, not facts guaranteed by repeated measurement. In this tutorial, forecast calculations isolate the random effects and do not develop propagation of assay or residual error.

Because the response is on the log scale, differences in \(\eta_i\) and \(\kappa_{ik}\) correspond to multiplicative differences on the concentration scale. The model assumes the same variance components across patients and occasions, and it assumes that occasion deviations do not persist from one occasion to the next. The dose history and sampling conditions must be represented adequately in the expected value; otherwise a systematic change can be mistaken for a random occasion effect.

Why Repeated Occasions Separate Persistent and Temporary Variation

The covariance result follows directly by centering each observation at \(\mu\):

All cross-covariances involving independent terms are zero. The variance of the difference between two occasions for the same patient gives another view:

The patient effect cancels in this within-patient contrast. In practice, variance components are inferred from many patients and repeated occasions, rather than calculated exactly from a single pair. The covariance and contrast identities explain which features of the data inform each component. With only one occasion per patient, the total variance alone generally cannot identify all three components without additional information or assumptions.

Updating a Patient Effect for a Future Occasion

Suppose the population mean \(\mu\) and variance components are treated as known for an individual forecast. Let \(Y_{ik}\) be an occasion summary with known measurement-error variance \(v\), so that after integrating over the occasion effect and summary error,

Assume the summaries are conditionally independent across occasions given \(\eta_i\). With \(K\) observed occasions, their mean is \(\bar{Y}_i\). The prior is \(\eta_i\sim N(0,\omega^2)\). Combining this normal prior with the normal likelihood gives a normal posterior. Its mean and variance are

The posterior mean is a weighted version of the patient’s average deviation from \(\mu\). The weight is less than one because observed deviations include occasion variation and summary error as well as the persistent patient effect. More independent occasions reduce the contribution of temporary deviations to the average. A future occasion has a new independent effect \(\kappa_{i,\mathrm{new}}\); it is not the occasion effect estimated from a past occasion.

Forecasting, Diagnostics, and Alternatives

A forecast for the same already-observed occasion is not the same target as a forecast for a new occasion. If the target is the same occasion, its \(\kappa_{ik}\) is shared with observations from that occasion and may be informed by them. If the target is a future occasion, its random effect is a new draw under the independent-occasion model. For a new occasion, the predictive mean on the log scale is \(\mu+E(\eta_i\mid\mathbf{Y}_i)\), while the variance of the latent occasion summary is \(\operatorname{Var}(\eta_i\mid\mathbf{Y}_i)+\pi^2\). This expression deliberately excludes residual and assay error.

The model should be questioned when observed patterns conflict with its assumptions. Useful checks include whether within-patient differences vary systematically with time, dose history, or sampling conditions; whether occasion deviations appear correlated across successive occasions; and whether the spread differs substantially across patients or across the range of predicted concentrations. A systematic shift after a dose change or clinical change is not automatically random occasion variability. If such structure is ignored, the estimated \(\pi^2\) may absorb a predictable change and forecasts can be misleading.

The random-effects decomposition is not the only approach. A model with only \(\eta_i\) assumes no additional occasion-level variation beyond residual error; it may understate uncertainty for a future occasion when genuine occasion differences exist. A model with only an occasion effect but no \(\eta_i\) cannot represent persistent differences between patients. Occasion-specific fixed effects can be suitable when occasions represent a small set of scientifically meaningful conditions, but they do not by themselves describe a distribution of new occasion deviations. Covariates or a time-varying model may be preferable when occasion changes are systematic or correlated rather than independent random departures.

Separation also depends on the data. Many patients with repeated occasions are more informative about between-patient covariance than a small number of patients with many observations each. Repeated observations within occasions can help distinguish occasion shifts from measurement-level error, but identifying variance components remains difficult if the design has too few occasions or the components are very small. A fitted variance near zero can reflect limited information as well as a genuinely negligible component.

Common Mistake

Wrong reasoning: “This patient’s concentration was high on one occasion, so the patient effect must be equally high, and I should carry that level into the next occasion.” This assigns the entire observed deviation to \(\eta_i\), despite the model’s separate \(\kappa_{ik}\) and \(\epsilon_{ik}\) terms.

Correction: Update the persistent effect using the population prior and the uncertainty from occasion variation and measurement error. Then forecast a new occasion with a fresh occasion deviation. Repeatedly high or low results across occasions provide stronger evidence for a persistent patient difference than a single extreme occasion, while still not eliminating uncertainty about a future occasion.