Attributing Uncertainty to Model Components

Why Attribute Predictive Uncertainty?

An integrated prediction can depend on several linked sources of uncertainty. For example, a prediction of a clinical response rate at a proposed dose may depend on PBPK inputs and parameters, population PK parameters, exposure-response parameters, and the random sampling of participants in a future trial. Knowing the prediction’s overall uncertainty is useful, but knowing where that uncertainty comes from helps determine which evidence or model component could most usefully be improved.

Attribution is not a property of a model output alone. It depends on the prediction being studied, the uncertainty distributions assigned to its inputs, and the attribution rule. A decomposition for a trial response rate may differ from one for an exposure quantile, even when both use the same integrated evidence chain. The prediction target, its units, and the included sources of uncertainty should therefore be stated before contributions are calculated.

Let \(Y\) denote the predictive quantity of interest. For a clinical prediction, \(Y\) might be the response percentage observed in a future trial at a specified dose and follow-up time. Represent uncertain quantities from PBPK, population PK, and exposure-response components by \(X_B\), \(X_P\), and \(X_E\), respectively. Let \(X_T\) represent trial-sampling randomness. The integrated predictive distribution describes uncertainty in \(Y\) as these sources vary according to their specified distributions and dependence structure.

A predictive variance can combine different kinds of uncertainty. Parameter and model uncertainty describe incomplete knowledge; trial-sampling variation describes how a future trial’s result varies even if its generating characteristics were known. Earlier material on separating variability from knowledge uncertainty distinguishes these concepts. For the purpose of attribution, include each source only if it belongs to the stated predictive target, and label its interpretation correctly.

Variance Decomposition and Its Assumptions

The simplest attribution applies when the component contributions are independent and additive. Suppose a local representation of the prediction is \(Y=\mu+U_B+U_P+U_E+U_T\), where \(\mu\) is the predictive mean and each \(U_j\) is a mean-zero contribution from one component. If the four contributions are mutually independent and have finite variances, all cross-covariances are zero. Therefore:

In this special case, a component’s variance share is its variance divided by the total predictive variance. The shares sum to one. This calculation requires more than a set of separately estimated standard errors: contributions must be defined on the same output scale, independent, additive, and complete for the target being decomposed.

Integrated models often violate additivity or independence. For example, a PBPK-derived exposure input may be correlated with a population PK parameter carried across a model handoff, or the effect of exposure-response uncertainty may depend on which exposure range the PK model predicts. Then the variance of a sum includes covariance terms, and interactions can contribute additional variance. Treating every component as independent can misallocate uncertainty or yield shares that do not sum to one.

A useful variance-based method for independent uncertain inputs is the functional ANOVA decomposition. In this approach, the model output is divided into a mean, individual input effects, and interaction effects. The first-order effect for component \(j\) is the variance of the conditional expectation of \(Y\) given that component, \(\operatorname{Var}(E[Y\mid X_j])\). It measures variation explained by that component alone. Higher-order effects capture variation that arises from combinations of components. The first-order shares need not sum to one when interactions are present.

For independent inputs, the total-effect index for component \(j\) includes every variance term involving that component, including interactions. In variance terms, it can be calculated as \(E[\operatorname{Var}(Y\mid X_{-j})]/\operatorname{Var}(Y)\), where \(X_{-j}\) denotes all inputs except \(X_j\). Total-effect indices can sum to more than one because an interaction is counted for each component involved. This is not an arithmetic error; the indices answer a different question from an exclusive partition.

Interactions and Dependence Change the Attribution

The following example shows why first-order effects alone can understate a component’s relevance. It uses independent, standardized component scores to isolate interaction variance. The scores are not clinical units; the output is measured in standardized squared units.

With dependent components, even the first-order conditional expectation can contain information about other components. For example, if PBPK and population PK inputs are correlated, learning one may help predict the other. A variance allocation based on the order in which inputs are entered can then assign shared predictive information differently depending on that order.

One order-invariant option is a Shapley allocation. For a set of component blocks \(S\), define \(v(S)=\operatorname{Var}(E[Y\mid X_S])\), with \(v(\varnothing)=0\). The Shapley contribution for component \(j\) is the average increase in \(v(S)\) when \(j\) is added, averaged over all possible orders in which components can enter. When all relevant inputs and random sources are included, the contributions sum to \(\operatorname{Var}(Y)\). For dependent inputs, this allocates shared information across components; it does not identify a uniquely “true” physical source of uncertainty.

Estimating Contributions by Simulation

For a nonlinear PBPK-to-population-PK-to-exposure-response chain, analytic variance formulas may be impractical. A simulation-based analysis can propagate uncertainty through the integrated model and estimate the required conditional expectations or variance components. It should use the joint distributions and covariance structures carried across model handoffs, rather than independently sampling parameters that are known to be dependent.

A sequential conditional-variance decomposition is another simulation-friendly choice. For an ordering \(X_1,\ldots,X_K\), define \(V_k=\operatorname{Var}(E[Y\mid X_1,\ldots,X_k])\), with \(V_0=0\). If all sources determining \(Y\) are included, \(V_K=\operatorname{Var}(Y)\), and the ordered increments \(V_k-V_{k-1}\) sum to the total variance. These increments are nonnegative for nested conditioning sets, but their allocation depends on the chosen order, particularly when components are dependent. The order should be stated and justified; averaging over orders gives the Shapley approach.

Simulation estimates can be noisy, especially when estimating small interaction contributions or conditioning on rare input combinations. Use enough draws to assess Monte Carlo error, and repeat the calculation with increased simulation effort to check stability. For independent-input Sobol estimation, use a design appropriate to the indices being estimated; naive random permutations or one-at-a-time perturbations do not automatically estimate valid variance shares.

Diagnostics, Comparisons, and Limitations

An attribution should be examined alongside the integrated model’s predictive checks and interface diagnostics. If component contributions fail to sum to total variance under a method that promises a complete allocation, investigate omitted sources, inconsistent variance definitions, Monte Carlo error, or an implementation error. If first-order Sobol shares do not sum to one, interactions are one possible explanation, not necessarily a failure. For dependent inputs, independently resampling each block can destroy covariance and alter both the predictive variance and its allocation.

The principal alternatives answer different questions. Independent additive variance shares are transparent and easy to explain, but apply only under strong conditions. First-order Sobol indices measure each component’s standalone contribution under independent inputs; total-effect indices include interactions but overlap. Sequential conditional increments form a complete allocation for a stated ordering, but are order-dependent. Shapley effects average over orders and provide an additive allocation under dependence, at higher computational cost and with a shared-information interpretation.

No attribution method can compensate for an incomplete uncertainty model. If uncertainty about model discrepancy, structural choices, or data reuse is omitted, the reported shares describe only the represented sources. Nor does a large share guarantee that reducing that component’s uncertainty is feasible or valuable: the decision relevance of further evidence also depends on the prediction’s purpose and how uncertainty affects the decision. Attributions are conditional summaries of a specified model and should be reported with the prediction target, uncertainty definitions, dependence assumptions, and method.

• Common mistake: treating all component variances as independent. Adding marginal variances while ignoring covariance and interactions can misstate total variance. Preserve dependence or use an attribution method that explicitly accounts for it.
• Common mistake: calling first-order shares a complete partition. Main effects omit interaction variance. Report interaction contributions or use total-effect indices, with the caveat that total-effect indices overlap.
• Common mistake: interpreting a Shapley share as causal responsibility. It is an allocation of predictive information under a specified joint distribution, not a causal effect or an intrinsic property of a model component.