Propagating Assay and Residual Error Into Forecasts

Why Error Propagation Matters in Individual Forecasts

A measured concentration is not usually an exact readout of a patient’s underlying pharmacokinetic state. It may differ from the model-predicted concentration because of assay imprecision and because the observation model allows residual variability. Those sources affect how much a concentration should influence an individual fit. They also affect how wide a forecast should be.

The key distinction is between uncertainty about the patient’s parameters and variation in a concentration around the model prediction. A patient’s parameter posterior reflects the information in the observed concentrations, including their measurement error. A future concentration then has additional uncertainty from future observation error. The introductory material on posterior parameter uncertainty and predictive uncertainty distinguishes these two kinds of uncertainty; here the focus is how assay and residual error enter their calculations.

This definition requires care. The variances add because the two error terms are assumed independent and expressed on the same scale. If they are correlated, their covariance must also be included: \(\operatorname{Var}(\epsilon_{\mathrm{res}}+\epsilon_{\mathrm{assay}})=\sigma_{\mathrm{res}}^2+\sigma_{\mathrm{assay}}^2+2\operatorname{Cov}(\epsilon_{\mathrm{res}},\epsilon_{\mathrm{assay}})\). If the observation model is proportional, combined error is not generally a constant variance in concentration units. The specified error model, rather than a convenient default, determines the likelihood.

From Error Variance to Concentration Likelihood

Under the additive normal model, conditional on \(\theta\), the observation has distribution \(y_j\mid\theta\sim N(f_j(\theta),v_j)\), where the second argument is the variance. Its likelihood contribution is the normal density evaluated at the observed concentration:

The density has units inverse to the concentration unit. It is not a probability that the exact continuous value occurred. When observations are conditionally independent given the individual parameters and model, the full likelihood is the product of their density contributions. In practical Bayesian fitting, this likelihood is combined with the individual’s parameter prior to obtain a posterior distribution.

Increasing the total variance makes the likelihood less sensitive to a fixed residual: the exponential penalty for that residual is smaller. But the normalizing factor changes too, so likelihoods should be calculated from the full density when comparing parameter values under a model. With a common variance across candidate values, the normalizing factor cancels in a likelihood ratio; it does not necessarily cancel when variances depend on the prediction.

How Error Changes the Posterior

The likelihood’s influence on an individual posterior depends on its precision, the reciprocal of its variance. To see this directly, consider a single unknown parameter \(\theta\) with a normal prior and a measurement whose model prediction is \(\theta\). Assume a known total observation variance \(v\). The prior and likelihood are

Multiplying the prior density by the likelihood gives a posterior density proportional to

Collect the terms involving \(\theta\). The coefficient of \(\theta^2\) is \(1/\tau^2+1/v\), and the linear term is \(-2\theta(\mu_0/\tau^2+y/v)\). Completing the square therefore gives a normal posterior with variance equal to the reciprocal of the combined precision and mean equal to that variance times the precision-weighted sum:

This derivation shows why adding assay variance can weaken an update: it increases \(v\), decreases the observation precision \(1/v\), and reduces the measurement’s weight relative to the prior. In a nonlinear population PK model, the posterior generally is not available in this closed form, but the likelihood still reflects the observation model’s total error variance.

Propagating Posterior and Observation Uncertainty Into a Forecast

Let \(C^*=g(\theta)\) denote a future model-predicted typical concentration, conditional on the patient’s parameters, and let \(Y_{\mathrm{new}}\) be a future measured concentration. The posterior uncertainty in \(C^*\) comes from uncertainty in \(\theta\). A measured future concentration also contains its own residual and assay errors. If these future errors are independent of \(\theta\), have mean zero, and have constant additive variances, then the law of total variance gives

The first term describes uncertainty in the patient-specific typical forecast. The second describes dispersion of a future measurement around that typical forecast. If the variance depends on \(\theta\) or on the predicted concentration, use its posterior expectation; if errors are correlated with parameter uncertainty or with other measurements, the covariance structure must also be accounted for.

For a linear forecast \(g(\theta)=a+b\theta\), with fixed \(a\) and \(b\), the posterior mean is \(a+b\mu_{\mathrm{post}}\) and the posterior variance is \(b^2V\). If the posterior for \(\theta\) is normal and future errors are normal with constant variance, the measured predictive distribution is also normal. This gives an exact interval in this special case. For nonlinear \(g\), a normal interval based only on a linear approximation may be inaccurate, particularly when posterior uncertainty is large or the transformation is strongly curved.

Concentration-Dependent Error and Model Choice

Assay precision is often described by a coefficient of variation, and residual variability may also increase with concentration. A simple proportional error model may specify a standard deviation proportional to \(f_j(\theta)\); an additive-plus-proportional model may use a variance such as \(\sigma_a^2+\sigma_p^2 f_j(\theta)^2\). These are model choices, not interchangeable descriptions. If independent assay and residual errors are both specified on the concentration scale, their variances can be added at that concentration. If the model is defined on a transformed scale, combine errors on that scale instead.

Diagnostics, Alternatives, and Common Pitfalls

Before using an error model for patient forecasting, check that it matches the units and scale of the concentration data. Review assay validation information and the fitted residual-error specification. Compare standardized residuals, \(\{y_j-f_j(\theta)\}/\sqrt{v_j}\), across concentrations and sampling times: systematic changes in their spread can indicate that a constant additive variance is unsuitable. A pattern in residuals may instead reflect model misspecification, so simply inflating the variance is not a general remedy.

A direct variance sum is appropriate for independent additive errors on the same scale. An alternative is to specify a joint error distribution with a covariance matrix when measurements share a calibration effect or other correlated error. A log-scale model is another alternative when multiplicative error is more appropriate; it requires a likelihood for the transformed data and a careful distinction between the mean on the log scale and the mean on the concentration scale. These approaches encode different assumptions and should not be mixed casually.

• Double counting assay error. If the fitted residual variance already includes assay imprecision, adding the assay variance again exaggerates uncertainty. Establish what the residual model represents before combining terms.
• Adding standard deviations instead of variances. For independent errors, add variances and then take the square root: \(s_{\mathrm{total}}=\sqrt{s_1^2+s_2^2}\), not \(s_1+s_2\).
• Mixing scales. A variance in log concentration cannot be added directly to a variance in mg/L squared. Express the components on the same modeled scale.
• Using only parameter uncertainty for a measured forecast. This omits future observation variability. Conversely, adding observation error to an interval intended only for the typical concentration changes the target of the interval.
• Assuming all uncertainty is captured. The calculations rely on a specified structural model, dosing history, and error distribution. Uncertainty in residual-error parameters or model structure is not included unless it is explicitly represented and propagated.