Why Group Parameters by Physiological Process?
PBPK parameters often enter the model in related sets: several tissue-to-plasma partition coefficients may describe distribution, while other parameters govern clearance or absorption. Ranking each parameter separately can obscure the combined influence of a process. It can also understate a process whose parameters matter mainly through interactions. Grouped sensitivity addresses a different question from an individual-parameter ranking: how much does the model output vary when a defined block of inputs is involved?
The block must be specified before interpreting its influence. It may represent a physiological process, a tissue family, or another scientifically justified collection of parameters. Grouping does not establish that the parameters have a single biological mechanism, nor does a large index identify which parameter within the block is responsible. It summarizes the block’s joint contribution for a specified output and input distribution.
The indices below are variance-based and assume that the blocks being compared are mutually independent. Dependence among parameters inside a block is allowed if that block is sampled from its joint distribution. The model must produce finite output variance, and the same output definition, dose, and physiological scenario must be used across the simulations being compared. The earlier material on Sobol indices provides the individual-input setting; grouping treats a whole block as one joint input.
Two Complementary Grouped Indices
The grouped first-order index measures how much of the output variance is explained by the block’s conditional mean. It includes effects that depend only on variables in the block, including interactions among variables inside it. The grouped total-order index measures the variance contribution of every model component involving at least one variable from the block. Consequently, it includes interactions between the block and variables outside it.
For independent blocks, the variance decomposition gives \(0\leq S_G\leq T_G\leq 1\) at the population level. A substantial gap \(T_G-S_G\) indicates that the block participates in interactions with other blocks. It does not, by itself, identify the interacting process or establish a biological mechanism. When the block is additive and does not interact with other blocks, its first-order and total-order indices are equal.
Estimating Grouped Indices by Pick-Freeze Simulation
A practical estimator uses paired model evaluations in which selected blocks are held fixed and the others are replaced. Let \(A\) and \(B\) be independent draws from the input distribution, with independent draws for each block. Evaluate \(Y_A=f(A)\) and \(Y_B=f(B)\). Construct two hybrid inputs: \(C^G\) takes the block \(G\) from \(A\) and the complement from \(B\); \(D^G\) takes the block \(G\) from \(B\) and the complement from \(A\). Write the corresponding outputs as \(Y_{C^G}\) and \(Y_{D^G}\).
These are population identities; simulation estimates replace expectations, covariances, and variances with sample quantities from many paired evaluations. The hybrid construction is essential: for the first-order index, the two evaluations share the block; for the total-order index, they share its complement. Reusing the wrong hybrid reverses the question being estimated.
The justification follows from conditional expectation and variance. Given \(X_G\), the two outputs \(Y_A\) and \(Y_{C^G}\) have independent complements but share \(X_G\). Their conditional means are both \(\operatorname{E}[Y\mid X_G]\). Thus
For the total-order construction, \(Y_A\) and \(Y_{D^G}\) share \(X_{-G}\), while their block values are independently drawn. Conditional on \(X_{-G}\), each has the same conditional variance and mean. Therefore
Taking expectations and dividing by \(2\operatorname{Var}(Y)\) yields \(T_G\). This derivation also explains the required sampling: the blocks must be independent for these standard pick-freeze identities to represent the stated variance decomposition.
Worked Example: An Additive Distribution Block
Suppose a hypothetical PBPK model produces AUC \(Y\), measured in \(\mathrm{mg\cdot h/L}\), according to \(Y=10+2U+3V+4W\). The standardized inputs \(U,V,W\) are independent and each is equally likely to be \(-1\) or \(1\). Let \(G=(U,V)\) represent two distribution-related parameters, and let \(W\) represent an absorption parameter. This deliberately simple model permits exact calculation and illustrates population indices rather than finite-sample estimates.
Each input has mean zero and variance one. The output mean is \(10\), and independence makes the three centered terms uncorrelated. Therefore
The conditional mean given the distribution block is \(\operatorname{E}[Y\mid U,V]=10+2U+3V\). Its variance is \(2^2+3^2=13\), so \(S_G=13/29=0.4483\). Because the model is additive, the total-order numerator is also \(13\), giving \(T_G=13/29=0.4483\). Thus the distribution block accounts for 44.83% of output variance under this specified input distribution, with no interaction contribution from outside the block.
The individual first-order contributions are \(S_U=4/29=0.1379\) and \(S_V=9/29=0.3103\). Here they sum to the group index because there are no interactions. That equality is not generally true: in models with interactions, individual total-order indices can overlap and should not be added to obtain a group index.
Interactions and Individual Effects Are Not the Same Summary
A group total-order index can be large even when the group first-order index is small. This occurs when the group’s influence appears mainly through interactions with outside inputs. The distinction matters when deciding whether to retain, measure, or simplify a whole process block; it also prevents a group result from being misreported as a ranking of its members.
Worked Example: A Block That Interacts With Another Process
Consider the hypothetical output \(Y=10+2U+3V+4W+6UW\), in \(\mathrm{mg/L}\). Here \(U,V,W\) are independent and equally likely to be \(-1\) or \(1\). Treat \(G=(U,V)\) as a clearance-related block and \(W\) as an outside process input. The product term \(UW\) represents a model interaction; it is not a claim about a particular biological mechanism.
The centered terms \(2U,3V,4W,6UW\) have zero pairwise covariance under these inputs. For example, \(\operatorname{E}[U(UW)]=\operatorname{E}[W]=0\). The total variance is consequently
Conditioning on \(U,V\) averages out \(W\), including the \(UW\) term, so \(\operatorname{E}[Y\mid U,V]=10+2U+3V\). Hence \(S_G=(4+9)/65=13/65=0.2000\). For the total-order index, the terms involving at least one variable from \(G\) are \(2U,3V,\) and \(6UW\). Thus \(T_G=(4+9+36)/65=49/65=0.7538\). The difference, \(36/65=0.5538\), is the interaction contribution associated with \(G\) in this model.
For comparison, \(U\)'s individual first-order index is \(4/65\), while its total-order index is \((4+36)/65=40/65\). The interaction is counted in both the group total-order index and \(U\)'s individual total-order index; that overlap is not an arithmetic error. It reflects different summaries of the same model variance.
Dependence Within and Between Blocks
Physiological parameters can be correlated. Standard grouped Sobol indices remain interpretable when dependent parameters are kept together in one jointly sampled block and that block is independent of the other blocks. If dependence crosses the proposed block boundaries, the usual pick-freeze construction no longer preserves the specified joint distribution. The resulting indices need not have the standard variance-attribution interpretation.
Worked Example: Keep Correlated Parameters in One Block
Suppose \(U\) and \(V\) are correlated standardized deviations for two tissue-partition parameters, with means zero, variances one, and correlation \(0.8\). Let \(W\) be an independent standardized clearance input with mean zero and variance one. Assume the hypothetical AUC output is \(Y=12+U+V+W\), in \(\mathrm{mg\cdot h/L}\). Use \(G=(U,V)\) as a joint block. The exact joint distribution inside \(G\) is retained; only its stated moments are needed for this linear calculation.
Since \(\operatorname{Cov}(U,V)=0.8\), the variance of the block contribution is \(1+1+2(0.8)=3.6\). The independent \(W\) adds variance \(1\), giving \(\operatorname{Var}(Y)=4.6\;(\mathrm{mg\cdot h/L})^2\). With no interaction between blocks, the grouped first-order and total-order indices agree:
The joint tissue block accounts for 78.26% of the output variance in this example. Treating \(U\) and \(V\) as independent would discard their covariance and incorrectly compute their combined variance contribution as \(2\), rather than \(3.6\). Grouping preserves the dependence relevant to the process-level question.
Choosing Blocks, Checking Estimates, and Comparing Approaches
Define blocks using the scientific question and the model’s parameterization, not by inspecting which grouping produces the most convenient result. Record the parameters in each block, their units and transformations, the input distribution, and the output endpoint. A block’s index is conditional on all these choices: changing the dose, time point, physiological scenario, or parameter distribution can change the result.
In a simulation study, use paired hybrid evaluations for each sampled input pair and estimate the output variance from the same design. Repeat or increase the simulation size to assess stability. Check that model failures, solver tolerances, or discontinuities are not dominating apparent variation. Small or negative estimated indices can occur from finite-sample error; they are not population evidence of negative variance contributions. Large violations of the theoretical ordering \(S_G\leq T_G\), or estimates that shift substantially across repeated designs, warrant investigation of sampling variability, implementation, and model evaluation.
| Approach | Question it answers | Main limitation for process-level interpretation |
|---|---|---|
| One-at-a-time perturbation | How does the output change when one input is varied at a time around a chosen setting? | Can miss interactions and depends on the reference setting. |
| Individual Sobol indices | What variance contribution is associated with an individual input? | Does not directly summarize a physiological block; individual total-order indices can overlap. |
| Grouped Sobol indices | What variance contribution involves a specified input block? | Requires a defensible block definition and, for standard attribution, independence across blocks. |
| Dependence-aware attribution | How should influence be allocated when inputs across proposed groups are dependent? | Attribution depends on the chosen dependence-aware method and its interpretation; it is not the standard independent-block Sobol decomposition. |
When dependence crosses blocks, possible responses include redefining a joint block or using a dependence-aware sensitivity method whose estimand matches the scientific question. A joint block may be too broad to identify which process matters, so report its member parameters and consider scientifically justified subdivisions that preserve the dependence structure. Grouped sensitivity also does not establish parameter identifiability, causal effects, or predictive adequacy; it describes model-output variance under the chosen assumptions and input distribution.
Common Pitfall
- Wrong reasoning: “The clearance block has a total-order index of 0.75, so its parameters explain 75% of the variance independently, and their individual indices should add to 0.75.”
- Correction: The group total-order index includes interactions between any member of the block and outside inputs. Individual total-order indices can count shared interaction contributions more than once. Report the group index as a joint block result, and use appropriate additional analyses if the goal is to identify influential members.
- Wrong reasoning: “The parameters are correlated, but ordinary Sobol indices can be computed as though they were independent.”
- Correction: Preserve dependence within a jointly sampled block, and ensure the blocks are independent for standard grouped indices. If dependence crosses blocks, redefine the blocks or choose a dependence-aware estimand rather than interpreting ordinary pick-freeze indices as a valid decomposition.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- For independent blocks, what does \(T_G-S_G\) indicate?
- A model has \(S_G=0.18\) and \(T_G=0.62\). What is the correct process-level interpretation?
- Two parameters within a proposed block are correlated, but the block is independent of all other blocks. Can standard grouped indices be used if the pair is sampled jointly?
- Why should individual total-order indices generally not be summed to recover a group total-order index?
Solutions. (1) It measures the variance contribution involving the block through interactions with inputs outside it; it is not a separate causal effect. (2) Effects confined to the block account for 18% of output variance, while all effects involving the block account for 62%; the difference, 44 percentage points, indicates interaction contribution under the specified model and input distribution. (3) Yes. Within-block dependence may be preserved by joint sampling, provided the blocks are mutually independent and the other assumptions hold. (4) Interactions involving multiple parameters can be included in more than one individual total-order index, so summing those indices can double-count variance rather than produce the group index.