Residual Error Models and Posterior Updating

Why the Residual-Error Model Changes an Individual Posterior

An individual PK posterior combines information from a patient’s concentration observations with a prior distribution for that patient’s PK parameters. The likelihood determines how strongly each observation contributes. That contribution depends not only on the distance between the observed and predicted concentration, but also on the residual-error model: an observation with a larger assumed residual variance is generally less precise and receives less weight.

This tutorial compares additive, proportional, and combined residual-error models. The distinction matters when concentrations cover a wide range. A constant absolute error may be plausible near the assay’s lower range, while the variability of an observation may increase with its predicted concentration. The chosen model therefore changes both the likelihood’s shape and, in general, the resulting posterior.

For the models below, assume that observations are conditionally independent given \(\theta\), that the PK predictions are positive, and that the residuals are normally distributed. Treat the residual-error parameters as fixed and known for the individual update. These assumptions are modeling choices, not automatic properties of concentration data. In particular, independence is conditional on the PK parameters; it does not claim that observations are unconditionally independent after accounting for the shared uncertainty in \(\theta\).

Three Common Error Models

In an additive model, the residual standard deviation is constant on the concentration scale. Write its value as \(\sigma_a\), in the same units as concentration. In a proportional model, residual standard deviation is proportional to the prediction: \(\sigma_p f_j(\theta)\), where \(\sigma_p\) is a dimensionless coefficient of variation. In a combined model, the additive and proportional components contribute to the variance.

Here \(N(\mu,\sigma^2)\) uses variance as its second argument. The combined variance is the sum of the component variances; this convention corresponds to independent additive and proportional error components. It is not the square of the sum of their standard deviations. With positive predictions, the proportional model describes constant relative variability, while the additive model describes constant absolute variability. The combined model allows both components to matter.

For any of these models, denote the conditional variance at observation \(j\) by \(v_j(\theta)\). The likelihood contribution is

The residual’s squared distance is scaled by its variance, but the variance also appears in the normalizing factor. Equivalently, the negative twice log likelihood, omitting a constant that does not depend on \(\theta\), is

The ratio \(1/v_j(\theta)\) acts as a precision weight in the squared-residual term. However, when the variance depends on the prediction, it is not correct to compare candidate parameter values using only weighted squared residuals: the \(\log v_j(\theta)\) term is part of the likelihood too.

From Likelihood to Posterior

Let \(p(\theta)\) be the individual prior density, perhaps obtained from a population PK model and conditioned on patient covariates. For conditionally independent observations, the posterior density is proportional to the prior density multiplied by the likelihood contributions:

This is the Bayesian updating principle used in individual PK estimation. The residual model controls the likelihood factors. If a model assigns a small variance to an observation, a substantial residual is penalized strongly; if it assigns a larger variance, that same residual is less surprising. In a proportional or combined model, changing \(\theta\) can change both the predicted concentration and the variance. The resulting posterior need not be a simple normal distribution, even if the prior is normal.

How Additive Error Produces a Precision-Weighted Update

A simple derivation shows how residual variance translates into posterior precision. Suppose a scalar parameter \(\theta\) has prior \(\theta\sim N(\mu_0,\tau^2)\), and observations follow the additive-error model \(y_j=\theta+\epsilon_j\), with independent \(\epsilon_j\sim N(0,\sigma_a^2)\). This is a simplified measurement model in which the prediction equals \(\theta\); it illustrates the weighting, rather than representing a general nonlinear PK model.

The prior contributes a squared deviation \((\theta-\mu_0)^2/\tau^2\) to negative twice the log density. Each observation contributes \((y_j-\theta)^2/\sigma_a^2\). Collecting the terms involving \(\theta\) gives a quadratic with posterior precision equal to prior precision plus observation precision. Completing the square yields:

The result makes the role of residual variance explicit: reducing \(\sigma_a^2\) increases the data’s precision and pulls the posterior mean more strongly toward the observations. For a general PK model, \(f_j(\theta)\) is usually nonlinear and may depend on several parameters, so this closed-form expression need not apply. The likelihood still follows the same variance-weighting principle, but the posterior may require numerical approximation or sampling.

Prediction-Dependent Variance and Posterior Shape

With proportional or combined error, the variance depends on \(f_j(\theta)\). Consequently, a candidate parameter value changes the residual and the scale against which that residual is judged. The following calculation illustrates why both parts of the likelihood must be retained.

Diagnostics, Alternatives, and Limitations

A residual-error model should be assessed against the scale and pattern of the observations, not selected only because it is familiar. Useful checks include plotting residuals against predicted concentrations, examining standardized residuals \(r_j/\sqrt{v_j(\theta)}\), and checking whether their spread changes systematically across the concentration range. Under the stated model, standardized residuals should not show a pronounced trend in spread against prediction. These checks are descriptive: a small individual data set may provide little power to distinguish the models, and parameter estimation itself affects the residual pattern.

The three structures make different assumptions. Additive error is often a reasonable description when absolute assay variability is approximately constant. Proportional error is appropriate when relative variability is more nearly constant, but it implies that the residual variance approaches zero as the predicted concentration approaches zero. Combined error avoids relying on only one scale over the entire range, while adding parameters that need adequate data to estimate. If the data contradict the normality or independence assumptions, changing from additive to proportional variance does not by itself resolve those problems.

A practical comparison is to fit or evaluate the same PK model under each plausible residual structure, then inspect predictive and residual behavior as well as the individual posterior. The choice is not merely a technical adjustment to standard errors: it can change which observations dominate, the posterior location, and its uncertainty. In nonlinear models, a different error structure can also change the posterior’s shape.

No one of these models is universally correct. The normal additive model can assign positive probability to negative concentrations, particularly when predictions are near zero. The proportional model can be problematic at or near zero predictions and may overstate precision at low concentrations. The combined model can be more flexible, but its parameters may be weakly identified with sparse individual observations. In practice, residual-error parameters are often informed by population data; treating them as fixed in the update ignores uncertainty in those parameters. These limitations should be considered when interpreting an individual posterior rather than hidden by a seemingly precise numerical result.