Attributing Predictive Variance to Uncertain Inputs

Why Attribute Predictive Variance?

A QSP model can produce a clinical prediction that varies across a parameter ensemble. Knowing the total predictive variance describes the spread, but not which uncertain inputs account for that spread. Attribution can help identify which parameter groups drive uncertainty in a specified prediction, and where additional evidence might be most informative. It does not establish that a parameter causes a clinical outcome, nor does it measure how much the outcome would change under an intervention on that parameter.

The attribution depends on the predictive target and the uncertainty distribution used for the inputs. Let \(Y=f(\Theta)\) be a scalar prediction, where \(\Theta=(\Theta_1,\ldots,\Theta_d)\) is a random vector of uncertain model inputs and \(f\) is the model mapping. Assume \(Y\) has finite variance. The target might be a biomarker change at a specified visit; the variance considered here is across the specified input distribution. The prediction target and that distribution must be held fixed when comparing attributions.

The groups can be individual parameters or scientifically meaningful blocks, such as receptor-binding parameters and intracellular turnover parameters. Grouping matters: an attribution to a block does not identify which member of the block is responsible. Parameter correlations must also be represented in the joint input distribution rather than discarded for convenience.

Conditional Variance and Sequential Decomposition

For any input group \(A\), the conditional expectation \(E[Y\mid A]\) is the mean prediction among inputs with that group fixed at its observed value, while the other inputs retain their conditional distribution. The law of total variance gives an exact two-part decomposition:

The first term is variation in the conditional mean prediction as \(A\) varies; the second is the average remaining variation after conditioning on \(A\). This is an identity for any joint distribution with finite variance. It does not require independent inputs. However, the first term is not automatically an exclusive share: if inputs are correlated, conditioning on one group can reveal information about another.

For several groups, choose an ordering \(A_1,\ldots,A_m\), and let \(S_k=(A_1,\ldots,A_k)\), with \(S_0\) empty. Define sequential increments as the increase in variance of the conditional mean when each group is added. The final residual is the average variance remaining conditional on all groups.

Because \(S_{k-1}\) is contained in \(S_k\), the conditional-expectation variance cannot decrease, so \(C_k\geq 0\). Telescoping the increments and applying the law of total variance gives \(\operatorname{Var}(Y)=\sum_{k=1}^{m}C_k+R\). If the prediction is fully determined by the included groups, \(R=0\). Sequential increments depend on group order: information shared by correlated groups is credited to whichever group enters earlier.

Independent Inputs: Variance Components and Sensitivity Indices

When input groups are mutually independent and \(Y\) is square-integrable, a functional ANOVA decomposition separates variation into orthogonal components: a mean, individual-group effects, pairwise interactions, and higher-order interactions. Orthogonality means that distinct components have zero covariance, so their variances add to \(V\). For example, in the preceding model, the components are \(A\), \(B\), and \(AB\); each has variance \(1\), giving \(V=3\).

The first-order index for group \(i\) is the fraction of total variance explained by its conditional mean alone. The total-effect index is the fraction associated with all variance components that involve that group, including interactions. For independent inputs these standard definitions are:

Here \(\Theta_{-i}\) denotes all input groups other than \(i\), and \(V>0\) is required for the ratios. \(S_i\) measures the individual-group component; \(S_{T_i}\) includes interactions involving that group. A difference between them indicates interaction contributions. First-order indices sum to one only if there are no interactions; total-effect indices can sum to more than one because interactions are counted for each participating group.

An additive independent-input model provides a useful contrast. If \(Y=2A+B\), \(A\) and \(B\) are independent, and their variances are \(1\) and \(4\), then \(V=4(1)+4=8\). The variance contributions are \(4\) from \(A\) and \(4\) from \(B\); each first-order and total-effect index is \(4/8=0.5\). The arithmetic relies on independence: without it, the covariance term in \(\operatorname{Var}(2A+B)\) cannot be omitted.

Correlated Inputs: Shared Information Is Not an Exclusive Share

With correlated inputs, the classical orthogonal variance-component decomposition generally does not apply. The conditional quantity \(\operatorname{Var}(E[Y\mid\Theta_i])\) remains defined, but two groups can each carry information about the same variation in \(Y\). Their normalized conditional contributions may therefore sum to more than one. This is not a calculation error; it signals overlap, not two separate pieces of variance.

Shapley effects can be defined for groups using their joint distribution, but their values depend on the selected groups and the conditional distributions used. They provide an additive allocation of the variance in this setting; they do not create a unique, assumption-free answer to “which parameter caused the variance.” Report the allocation rule, input dependence, and grouping alongside the numbers.

Conditional Simulation and Diagnostics

In a QSP application, conditional expectations and variances are often estimated by simulation. For a chosen subset \(S\), sample values of \(\Theta_S\) from their marginal distribution, then sample the remaining inputs from their conditional distribution given \(\Theta_S\). Run the model for those completed input vectors. Repeating this procedure estimates \(E[Y\mid\Theta_S]\), its variance across the outer draws, and, when needed, the conditional residual variance.

The conditional sampling step is essential when inputs are correlated. Pairing a fixed value of one parameter with independently drawn values of the others generally breaks the joint distribution and estimates a different quantity. If conditional distributions are difficult to sample, the simulation method must preserve the intended joint distribution by another validated approach. For independent inputs, pick-freeze estimators or variance-based sampling designs are common alternatives; they are not automatically valid for dependent inputs.

A decomposition that fails to reconcile with \(V\) may indicate an estimator or implementation problem, a residual term that was omitted, or Monte Carlo error. Large changes in estimated shares across batches suggest insufficient simulation, especially when conditional expectations are noisy. Other warning signs include implausible results after breaking input correlations, unstable rankings under modest changes in grouping, and reporting a sequential allocation without stating its order.

Attribution also has substantive limits. It describes variation under a chosen uncertainty distribution, so changing parameter ranges or correlations can change the result even when the QSP equations remain fixed. A low share does not prove an input is biologically unimportant: its effect may be small over the selected range, masked by other inputs, or expressed in a different prediction target. Conversely, a high share is not evidence that reducing uncertainty in that parameter will necessarily improve a decision. Variance attribution is a diagnostic of predictive uncertainty, not a substitute for scientific interpretation.

Common Pitfall

Wrong reasoning: “The conditional indices for two correlated parameters are 0.8 each, so each parameter independently explains 80% of the prediction variance.” This double-counts shared information and treats overlapping conditional summaries as additive.

Correction: State that the two conditional contributions overlap, then choose an allocation rule suitable for the question. For example, a Shapley allocation splits the shared contribution by averaging over input orderings. Alternatively, report the conditional contributions as overlapping measures without claiming they partition the variance. In either case, preserve and describe the input correlation structure.