Decomposing Uncertainty in Cross-Species Predictions

Why a Cross-Species Prediction Needs an Uncertainty Budget

A translational prediction is not just a central estimate. It is an estimate tied to a target, such as a population geometric mean exposure at a specified dose or the exposure of a future individual. Different sources of uncertainty affect those targets in different ways. Combining them without distinction can produce an interval that is too wide, too narrow, or simply about the wrong quantity.

Here, consider a hypothetical prediction of human AUC from preclinical information and a translational model. AUC is measured in mg·h/L. The numerical examples use log AUC, which is useful when errors are approximately multiplicative: a fixed log-scale difference corresponds to a fixed ratio on the original scale. The target is always stated explicitly, because uncertainty in a human population mean is not the same as variation in AUC among individual people.

What the Components Mean

The labels in an uncertainty budget are useful only when their meanings are operationally clear. In particular, sources that arise from the same data must not be counted twice.

• Parameter uncertainty is uncertainty in the values of fitted model parameters, conditional on the chosen model structure and the data and analysis used to estimate them. It propagates into the prediction because different plausible parameter values can imply different predictions.
• Sampling uncertainty is uncertainty arising because the observed source-species sample is finite and could have differed under another sample from the same sampling process. For a practical budget, this must be defined separately from parameter uncertainty—for example, as uncertainty in a source-species sample summary not already included in the fitted-parameter covariance.
• Measurement uncertainty arises from imperfect measurement of inputs, such as assay error in observed concentrations or uncertainty in a recorded covariate. It affects the prediction through the model’s dependence on those inputs.
• Interindividual variability is real variation among people in the target population, such as differences in clearance. It is variability, not uncertainty about the population mean. It belongs in a prediction for a future individual, not automatically in an interval for the population mean.
• Structural uncertainty reflects uncertainty about the form of the model—for example, whether a plausible relationship is linear or curved on the chosen scale. It concerns alternative model structures, not just alternative parameter values within one structure.
• Model discrepancy is the difference between the selected model’s predictions and the underlying system, even if the model parameters were known exactly. It represents inadequacy not captured by the chosen model, and should be informed by relevant data or a defensible calibration rather than added as an arbitrary safety margin.

The boundary between components depends on the analysis. For example, if the uncertainty in a sample summary has already been propagated into parameter uncertainty, adding it again as a separate sampling term double-counts it. A defensible budget states what each term includes and excludes.

Combining Contributions on a Common Scale

Let \(Z\) be the log of an exposure divided by one stated exposure unit, so the logarithm is dimensionless. A local approximation to a prediction can be written as a central log prediction plus contributions from uncertainty sources. If those contributions have mean zero, are independent, and have standard deviations \(s_1,\ldots,s_k\), their variances add. The result is an approximate standard deviation for the prediction on the log scale.

The variance rule follows by expanding the variance of a sum. For two contributions \(U\) and \(V\), \(\operatorname{Var}(U+V)=\operatorname{Var}(U)+\operatorname{Var}(V)+2\operatorname{Cov}(U,V)\). Repeating this for \(k\) contributions gives a variance sum plus all pairwise covariance terms. Independence makes each covariance zero, yielding the displayed rule. Thus, independence is an assumption to assess, not a consequence of putting terms in separate rows.

A normal approximation on the log scale gives a range of the form \(\hat{z}\pm1.96\,SE_{\text{total}}\) for a central 95% interval under the stated model and assumptions. Exponentiating the endpoints gives an interval on the original scale. Its interpretation depends on the target: an interval for a population geometric mean reflects uncertainty in that mean; a predictive interval for a future individual must also include interindividual variability. These are model-based intervals, not guarantees that the model is correct.

From a Mean Prediction to an Individual Prediction

A future individual’s exposure differs from the population geometric mean partly because people vary, even if the population mean were known. If log-scale interindividual variability is approximately normal with standard deviation \(\sigma_{\mathrm{IIV}}\), and it is independent of the other contributions, then its variance is added for an individual prediction. If it is correlated with other terms, the covariance must also be considered.

Dependence, Diagnostics, and Alternative Summaries

The independence calculation can be misleading when contributions share data or mechanisms. For example, uncertainty in a source-species summary and uncertainty in fitted parameters may move together. If two log-scale contributions have standard deviations \(s_1\) and \(s_2\) and correlation \(\rho\), their combined variance is \(s_1^2+s_2^2+2\rho s_1s_2\). Positive dependence increases the combined variance relative to an independence calculation; negative dependence decreases it.

Useful diagnostics focus on whether the budget’s pieces are credible and appropriately separated:

• Check that every term refers to the same prediction target, scale, dose, and time horizon.
• Trace each component back to data, model comparisons, or an explicit elicitation. A number with no supporting rationale is not made reliable by appearing in a variance sum.
• Check for double counting, especially when parameter uncertainty already includes finite-sample uncertainty or measurement error.
• Assess dependence when components use shared data or represent linked mechanisms. If dependence is unknown, report sensitivity to plausible covariance assumptions rather than silently setting every covariance to zero.
• Compare the resulting interval with observed prediction errors or external validation where available. Poor coverage or systematic residual patterns can indicate that model discrepancy is underestimated or that the assumed model is inadequate.

A variance budget is not the only approach. A local, analytic budget is transparent and efficient when contributions are approximately additive on the chosen scale and covariance can be specified. Simulation can instead draw jointly from uncertain inputs and models, preserving nonlinear transformations and specified dependence; its output is only as credible as those distributions and model alternatives. A scenario analysis reports predictions under several plausible structures without forcing them into a single normal variance. It can be easier to interpret when there are only a few meaningful alternatives, but it does not by itself assign probabilities to them. These approaches can complement one another.

Common Mistakes and Practical Limits

A frequent error is to add standard deviations directly. For independent contributions, variances—not standard deviations—add: \(0.12\) and \(0.08\) combine to \(\sqrt{0.12^2+0.08^2}=0.1442\), not \(0.20\). Another is to include interindividual variability in an interval for a population mean, or to omit it when the target is an individual prediction. Both errors change the question the interval answers.

A further limitation is that a compact budget can make uncertain judgments look precise. Structural uncertainty and model discrepancy are especially difficult to estimate from sparse preclinical data. Nor does a wide interval necessarily protect against an omitted mechanism: uncertainty quantification describes sources that have been represented, and cannot automatically reveal ones that were overlooked. The budget should therefore be accompanied by a clear target, assumptions, component definitions, and sensitivity checks. It supports translational reasoning; it does not establish that animal-to-human translation is valid.