Combining Parameter, Residual, and Between-Patient Uncertainty

Why a Prediction Needs More Than One Uncertainty Layer

A QSP model can be uncertain in several distinct ways. Its parameters may not be precisely known; patients may differ even when they share the same parameter values; and an individual’s measured outcome may depart from the model’s latent expected outcome. These sources are not interchangeable. Whether each belongs in a predictive distribution depends on the clinical quantity being predicted.

For example, a prediction for one future patient’s measured biomarker generally includes uncertainty in the model parameters, variability among patients, and measurement or residual error. A prediction for the average latent response in a hypothetical population has a different target: individual residual error does not describe uncertainty in that population mean. The calculation should begin by defining the target, not by adding every available variance.

The parameter distribution represents uncertainty about which model parameter values are compatible with the available evidence. It is not itself a distribution of different biological parameter values across patients. Patient variability is represented separately, often through a distribution for \(B\). Residual error is also distinct: it may represent measurement error, unexplained within-patient variation, or both, depending on the observation model.

A Hierarchical Model and Its Variance Decomposition

A useful general representation is \(Y=m(\Theta,B)+\varepsilon\), where \(m(\Theta,B)\) is the conditional expected outcome for specified parameter values and patient effect, and \(\varepsilon\) has conditional mean zero. Assume first that the predictive distribution is formed by drawing \(\Theta\) from an uncertainty distribution, then drawing \(B\) from its patient distribution conditional on \(\Theta\), then drawing residual error conditional on \(\Theta\) and \(B\). The distributions need not be independent: for example, the distribution of patient effects may depend on parameter values.

Write \(v(\Theta,B)=\operatorname{Var}(Y\mid\Theta,B)\). Applying the law of total variance first across parameter values and then across patients gives the following decomposition:

The first term is the contribution from parameter uncertainty in the parameter-specific mean prediction. The second is between-patient variability in the latent expected outcome. The third is residual variance for an observed outcome. This decomposition assumes that residual error has conditional mean zero; if it has a nonzero conditional mean, that mean must be included in the conditional expectation rather than treated as zero-mean error.

This is a variance identity, not a claim that the predictive distribution is normal. Even if each conditional distribution is normal, mixing over uncertain parameters or patient effects can produce skewed or multimodal predictions. In such cases, predictive quantiles from the full distribution are usually more informative than an interval constructed from the total variance alone.

Worked Example: One Patient’s Latent and Observed Biomarker

Worked Example: A Threshold Probability Under Parameter Uncertainty

Worked Example: Predicting a Cohort Mean

Constructing the Prediction and Checking It

A practical simulation should follow the hierarchy represented in the model. For each predictive replicate, draw a parameter set, then generate the patient-level quantities relevant to the target, and finally generate residual error only when predicting an observed outcome. Preserve any dependence specified by the model: for example, draw correlated parameters jointly rather than independently, and use the appropriate conditional patient-effect distribution.

Useful diagnostics include checking that conditional means and variances match the model, confirming that each uncertainty source is included exactly once, and comparing simulated component variances with analytic values in simple cases. For simulated intervals, assess Monte Carlo precision by repeating the calculation with more replicates or independent runs. Unstable tail quantiles can indicate too few replicates, while unexpected shifts in the mean can signal coding errors or an incorrect weighting of parameter draws.

A common mistake is to treat the standard error for a cohort average as though it were the uncertainty of one patient’s outcome. Dividing every variance by the cohort size incorrectly shrinks uncertainty in a common parameter draw. The reverse mistake is to include full individual patient and residual variances in a cohort mean without accounting for averaging. The correct calculation follows the sampling structure: shared parameter uncertainty remains, while independent patient-level contributions shrink with cohort size.

Another pitfall is to add standard deviations rather than variances. For independent additive components, variances add; standard deviations are obtained by taking the square root of the total variance. If components are dependent, covariance terms also matter, so simply summing variances is not justified. Finally, a normal interval based on the total variance can misrepresent skewed or multimodal predictions. In that case, summarize the simulated or otherwise calculated predictive distribution directly rather than assuming a normal shape.