Occasion-Specific Residual Variance

Why Allow Residual Scales to Differ?

Measurements from different dosing occasions or assay conditions need not have the same unexplained scatter around the model-predicted concentration. For example, one occasion may involve a less precise assay, or the measurement process may be more variable under a particular condition. A model that forces one residual scale across all observations can then misrepresent the uncertainty attached to concentrations from some occasions.

This tutorial concerns the residual distribution conditional on the model’s predicted mean and any included individual effects. It does not define a different typical PK profile for each occasion. The occasion label selects a residual scale; the PK mean structure remains the same unless it separately includes occasion-related predictors or effects.

The model is conditional on the included individual effects \(b_i\). A common assumption is that, after conditioning on those effects and the mean model, residuals are independent across observations and have mean zero. The simple model above also assumes normal residuals and additive error on the concentration scale. These assumptions are not automatic: the chosen scale must suit the data representation, and serial dependence or heavy tails may require other residual models.

A useful extension allows the residual variance to depend on a known function \(h_{ijl}\) of the predicted mean or another specified quantity: \( \operatorname{Var}(Y_{ijl}\mid b_i)=\sigma_j^2h_{ijl}^2 \). The occasion-specific parameter \(\sigma_j\) is then a scale multiplier, while \(h_{ijl}\) describes the observation-level scaling. For the additive model, \(h_{ijl}=1\). The interpretation of \(\sigma_j\) depends on this choice: it is a concentration-scale standard deviation when \(h=1\), but a multiplier of the specified variance function otherwise.

Estimating a Scale for Each Occasion

First consider the case in which the conditional means \(m_{ijl}\) are known and the additive normal model applies. Let \(n_j\) be the number of observations on occasion \(j\), and let \(e_{ijl}=y_{ijl}-m_{ijl}\) be the residual. For each occasion, the likelihood contribution is the product of normal densities with common variance \(\sigma_j^2\). Maximizing that likelihood gives a direct estimate of the occasion-specific variance.

Write \(RSS_j=\sum_{i,l}e_{ijl}^2\). Differentiating the log likelihood with respect to \(\sigma_j^2\), setting the derivative to zero, and solving gives the maximum-likelihood estimate. The second derivative is negative at this solution when \(RSS_j>0\), confirming a maximum.

This is a likelihood estimate, not a universal unbiased estimate. In this derivation the means are treated as known. In an actual population PK analysis, mean parameters and random-effect distributions are usually estimated jointly with residual scales; individual random effects are also integrated over or otherwise handled by the fitting method. The simple residual sum-of-squares calculation is therefore useful for understanding the scale estimate, but it is not generally a substitute for fitting the full population model.

Scale Functions and Unequal Occasion Counts

If the model specifies \(\operatorname{Var}(Y_{ijl}\mid b_i)=\sigma_j^2h_{ijl}^2\), define a scaled residual \(u_{ijl}=e_{ijl}/h_{ijl}\), for positive known \(h_{ijl}\). Under the normal model, these scaled residuals have variance \(\sigma_j^2\). The likelihood derivation therefore applies to them: the maximum-likelihood estimate of \(\sigma_j^2\) is their mean squared value for that occasion. This estimates the occasion scale multiplier, not the unscaled variance of every concentration.

Unequal numbers of observations per occasion do not prevent separate scale estimation, but they do affect precision. An occasion with few observations has less information about its scale. A pooled estimate can conceal this difference by averaging squared residuals across occasions, with more observations contributing more terms.

Keeping Mean and Variance Questions Distinct

A variance model changes the assumed distribution around the conditional mean; it does not, by definition, add an occasion-specific shift to that mean. For example, increasing \(\sigma_j\) makes larger deviations from the same predicted concentration more compatible with the model. It does not raise the predicted concentration. An occasion effect in the mean, a random occasion effect, and an occasion-specific residual scale answer different questions.

Although the variance parameters do not change the definition of the PK mean model, joint estimation can change the fitted parameter estimates. The likelihood weights observations according to their modeled uncertainty. If occasions differ in scale, fitting a common scale can give an inappropriate balance to their contributions, and the mean parameters may adjust as a result. Therefore, “variance-only” describes the model component being added; it does not guarantee that all estimated PK parameters remain numerically identical after refitting.

Diagnostics, Assumptions, and Limitations

After fitting, examine residuals by occasion, time after dose, predicted concentration, and relevant assay condition. Under a fitted variance model, scaled residuals should not show a systematic increase or decrease in spread across the occasions or across values of the modeled scale function. Also check for systematic residual patterns in the mean: a consistent positive or negative residual on one occasion points to a possible mean-model problem, not merely a different scale. The residual assumptions should be assessed conditional on the model structure, rather than inferred from raw concentrations alone.

Warning signs include an estimated scale near zero, very unstable scale estimates on sparsely observed occasions, persistent time patterns, or residual dependence not represented by the model. A few large residuals can dominate a squared-residual estimate; a heavy-tailed residual model may be more suitable if outlying observations are the substantive concern. A scale difference can also be an artefact of a misspecified mean: if predictions systematically miss one occasion, the apparent residual scatter may be inflated.

The main alternative is a pooled residual scale, which is more parsimonious but assumes equal residual variance across occasions. A separate scale for every occasion is more flexible but requires enough information at each occasion and can overfit sparse data. A mean shift or occasion random effect is appropriate only when the scientific and data structure support variation in the predicted level, not simply because residual spread differs. These alternatives can be considered as distinct model assumptions; choosing among them requires attention to the design, diagnostics, and the full fitted model.

The simple likelihood calculation above assumes known means and independent normal residuals. With estimated population parameters, random effects, censoring, or correlated residuals, the likelihood and uncertainty calculations must reflect those features. Separate occasion scales are also not identifiable with useful precision if an occasion has too little information, and an occasion label that is confounded with another design feature cannot reveal which feature accounts for a difference in scatter. Estimates should therefore be reported with units, the variance-function form, and the occasion definition.