Global Sensitivity with Correlated Parameters

Why Correlated Inputs Change the Question

Global sensitivity analysis asks how uncertainty in model inputs relates to uncertainty in an output over specified input ranges or distributions. In PBPK applications, some physiological quantities and drug properties may be associated: for example, organ size and blood flow may covary across individuals, and estimates of related drug properties may share information. If a sensitivity analysis varies such inputs independently when their joint distribution is not independent, it explores combinations that may be implausible and answers a different question from the one posed by the intended population or uncertainty model.

The classical Sobol indices are based on a variance decomposition that assumes independent inputs. With dependence, the variance of the output can still be studied, but attribution is no longer automatic: a parameter may carry information about another correlated parameter, so their apparent contributions can overlap. The choice of method should therefore follow the scientific question: sensitivity to observing a parameter, allocation of shared explained variance, or sensitivity to a physically meaningful parameter group.

Definition: Let $Y=f(X_1,\ldots,X_d)$ be a model output with finite, nonzero variance under a specified joint distribution of the inputs. Inputs are independent if their joint distribution factors into the product of their marginal distributions. A classical Sobol decomposition and its usual interpretation as a partition of output variance require this independence.

For independent inputs, the model output can be decomposed into components for individual inputs, pairs, and higher-order combinations. These components are orthogonal in the variance calculation, so their variances add to $\operatorname{Var}(Y)$. A first-order Sobol index measures the variance attributed to the component for one input alone; a total-effect index includes that input's contribution across interaction components. The standard pick-freeze estimators used for these indices rely on being able to replace or combine an input while keeping the other inputs independent.

S_i=\frac{\operatorname{Var}(\operatorname{E}[Y\mid X_i])}{\operatorname{Var}(Y)}, \qquad S_{T_i}=\frac{\operatorname{E}[\operatorname{Var}(Y\mid X_{-i})]}{\operatorname{Var}(Y)}

Here $X_{-i}$ denotes all inputs other than $X_i$. Under input independence, $S_i$ is the first-order index and $S_{T_i}$ is the total-effect index. When inputs are dependent, the conditional-variance quantities in these expressions can still be calculated, but they do not generally form the classical Sobol decomposition. In particular, a conditional expectation given $X_i$ can reflect information about other inputs that are correlated with $X_i$.

A Dependence Example and Its Consequence

The overlap can be seen with a simple analytic derivation. Suppose $X_1$ and $X_2$ are standard normal inputs with correlation $\rho$, and let $Y=X_1+X_2$. The model has finite variance. For this bivariate normal distribution, $\operatorname{E}[X_2\mid X_1]=\rho X_1$, so the conditional expectation of the output is $(1+\rho)X_1$. The corresponding result for conditioning on $X_2$ is symmetric.

\operatorname{Var}(Y)=2+2\rho, \qquad \operatorname{Var}(\operatorname{E}[Y\mid X_1])=(1+\rho)^2

Thus, the two conditional explained-variance fractions are each $(1+\rho)/2$. If $\rho=0.8$, each is $0.9$, and their sum is $1.8$. This does not mean that 180% of the output variance exists. It means that either input alone is informative about much of the output because it also carries information about the other input. The fractions overlap; they are not an exclusive partition.

Worked Example: Correlated Inputs and Overlapping Conditional Fractions

Consider a hypothetical PBPK output represented by $Y=X_1+X_2$, where $X_1$ and $X_2$ are standardized uncertain inputs with a bivariate normal distribution and correlation $0.8$. The output is measured in a chosen endpoint unit, and each input is measured in its own standard-deviation units.

The model and joint distribution are specified, and the input moments are finite. The covariance is $0.8$, so $ \operatorname{Var}(Y)=1+1+2(0.8)=3.6 $ endpoint units squared.

For $X_1$, the conditional expectation is $ \operatorname{E}[Y\mid X_1]=(1+0.8)X_1=1.8X_1 $. Therefore, its conditional explained variance is $1.8^2\operatorname{Var}(X_1)=3.24$, and the normalized fraction is $3.24/3.6=0.900$. By symmetry, the fraction for $X_2$ is also $0.900$.

The two fractions sum to $1.80$, not $1$. Each describes how much output variation is explained by conditioning on that input alone; neither is an exclusive share of the total. Reporting these as ordinary first-order Sobol indices would give them a variance-partition interpretation that they do not have under this dependent input distribution.

Dependence-Aware Attributions

One useful quantity for dependent inputs is the conditional explained-variance fraction $ \operatorname{Var}(\operatorname{E}[Y\mid X_i])/\operatorname{Var}(Y) $. It answers: how much does knowing $X_i$ reduce uncertainty about the mean of $Y$, relative to not knowing it? This is a well-defined conditional association measure under a specified joint distribution. It is not a unique effect with the influence of correlated inputs removed, and fractions across inputs need not sum to one.

A conditional residual fraction can also be defined as $ \operatorname{E}[\operatorname{Var}(Y\mid X_{-i})]/\operatorname{Var}(Y) $. It measures residual output variance when all inputs except $X_i$ are known. With dependent inputs, that residual is determined by the conditional distribution of $X_i$ given the others. If $X_i$ is nearly predictable from $X_{-i}$, little uncertainty may remain in it, even if the model responds strongly to changes in $X_i$ considered on its own. Its interpretation is therefore conditional, not a general ranking of intrinsic model influence.

When a single allocation across inputs is needed, Shapley effects provide one defensible choice. For any subset $U$ of inputs, define the normalized value $c(U)$ as the fraction of output variance explained by conditioning on the inputs in $U$. The Shapley effect for input $i$ is its average marginal increase in this value over all possible orders in which inputs could be added.

Definition: For $d$ inputs, let $c(U)=\operatorname{Var}(\operatorname{E}[Y\mid X_U])/\operatorname{Var}(Y)$, with $c(\varnothing)=0$, where $X_U$ denotes the inputs in subset $U$. The Shapley effect for input $i$ is $$ \phi_i=\sum_{U\subseteq \{1,\ldots,d\}\setminus\{i\}} \frac{|U|!(d-|U|-1)!}{d!} \left[c(U\cup\{i\})-c(U)\right]. $$

For a finite-variance output, $c(\{1,\ldots,d\})=1$, since conditioning on all inputs determines $Y=f(X)$. The marginal increments telescope along each ordering of the inputs from the empty set to the full set. Averaging these increments over all orderings therefore gives $ \sum_{i=1}^{d}\phi_i=1 $. This is the reason Shapley effects allocate the total explained variance rather than reporting overlapping fractions. Under dependence, the allocation shares information across correlated inputs by averaging over the order in which each is considered. That is a defined convention, not proof that the shared information belongs uniquely to each parameter.

Worked Example: Allocating Shared Information with Shapley Effects

Use the hypothetical model and joint distribution from the previous example: $Y=X_1+X_2$, with standard normal inputs correlated at $0.8$. The total output variance is $3.6$. The explained-variance values for the two singleton sets are $c(\{1\})=0.9$ and $c(\{2\})=0.9$; for both inputs together, $c(\{1,2\})=1$.

With two inputs, each possible order has weight $1/2$. For input $X_1$, the marginal contribution is $0.9$ when it enters first and $1-0.9=0.1$ when it enters after $X_2$. Hence $ \phi_1=(0.9+0.1)/2=0.5 $. Symmetry gives $ \phi_2=0.5 $, and $0.5+0.5=1$.

The Shapley result assigns equal shares of explained variance in this symmetric example. It does not say that the inputs have independent effects, nor that changing either input alone would cause a 50% change in the output. It reports a variance allocation under the stated joint distribution and allocation rule.

Preserving the Joint Input Distribution

Before calculating any dependence-aware index, specify the joint distribution that represents the uncertainty of interest. Pairwise correlations alone may be insufficient: different joint distributions can have the same marginal distributions and correlations but different tail behaviour or nonlinear dependence. If inputs have structural constraints, the joint model should respect them rather than generate invalid combinations and discard them without accounting for the resulting change in distribution.

For an approximately multivariate normal pair of standardized inputs with correlation $\rho$, a conditional draw can be generated as $X_2=\rho X_1+\sqrt{1-\rho^2}Z$, where $Z$ is an independent standard normal variable. This construction preserves the specified bivariate normal dependence. Other settings may require a suitable multivariate distribution, copula, conditional model, or a mechanistic relationship among inputs. Marginal distributions and dependence should be checked in generated samples, including constraints and relevant tail combinations.

Worked Example: Generating a Conditional PBPK Input Draw

Suppose two standardized physiological inputs $X_1$ and $X_2$ are assigned a bivariate normal distribution with correlation $0.6$. For a conditional draw with $X_1=1.0$, generate an independent standard normal draw $Z=-0.5$, then calculate $X_2$ from the conditional construction.

The conditional mean of $X_2$ is $0.6(1.0)=0.6$, and the conditional standard deviation is $ \sqrt{1-0.6^2}=\sqrt{0.64}=0.8 $. Thus $X_2=0.6+0.8(-0.5)=0.2$. The resulting pair $(1.0,0.2)$ is one conditional draw consistent with the specified normal dependence; it is not a claim that every value of $X_2$ must rise with $X_1$.

If a hypothetical local output representation is $Y=2X_1+X_2$, with coefficients in endpoint units per standard-deviation unit, this draw gives $Y=2(1.0)+0.2=2.2$ endpoint units. The sampling construction preserves the joint distribution used to define sensitivity. Independently drawing $X_2$ after fixing $X_1=1.0$ would instead break the specified conditional relationship.

Choosing and Checking an Analysis

Use classical Sobol indices when independent input distributions are a defensible representation of the uncertainty and the aim is a variance partition under that representation. If dependence is scientifically meaningful, choose a method whose estimand matches the question. Conditional explained-variance fractions describe information associated with individual inputs. Shapley effects allocate explained variance across inputs, including shared information. A physically meaningful group of related quantities can sometimes be treated together, but that changes the reporting unit from individual parameters to the group.

Approach	Question it answers	Main caution
Classical Sobol indices	How is output variance partitioned across independent inputs and their interactions?	The standard interpretation and sampling design rely on independent inputs.
Conditional explained-variance fractions	How much output-mean variation is associated with knowing one input?	Fractions can overlap and need not sum to one.
Shapley effects	How can explained variance be allocated across inputs under a joint distribution?	The allocation shares dependence-related information; it is not a unique causal attribution.
Joint-input grouping	How important is a set of related inputs considered together?	Individual contributions within the group are not distinguished by that result.

A common alternative is to transform dependent inputs into independent latent variables and then calculate Sobol indices on those variables. This can be useful for analyzing the transformed model, but the resulting indices attribute variation to the chosen latent coordinates, not automatically to the original physiological or drug parameters. For transformations that depend on an ordering, changing that ordering can change the attribution. Report the transformation and interpret the indices in those coordinates.

Diagnostics should check both the joint input sample and the sensitivity calculation. Confirm that simulated marginals and dependence match the intended model; inspect whether constraints are violated or important regions are poorly sampled; and verify that the PBPK output is finite and stable over sampled combinations. For Monte Carlo estimates, assess numerical stability by increasing sample size or using independent replicate runs. For conditional or Shapley estimates, the conditional distributions and conditional expectations also need adequate sampling. An apparently precise estimate cannot repair a misspecified joint distribution.

Common Pitfalls and Limits

Using independent pick-freeze hybrids for dependent inputs. Replacing one coordinate while retaining the others can create combinations inconsistent with the intended joint distribution. Use a design and estimator that preserve or correctly condition on the dependence.
Calling overlapping conditional fractions Sobol partitions. When inputs are correlated, two inputs may each explain the same output variation. State that these are conditional association measures and do not add to a variance partition.
Treating correlation as a complete dependence model. A correlation matrix does not specify all forms of dependence. State and justify the full joint distribution or conditional model used.
Interpreting an attribution as causation. Sensitivity indices summarize model behaviour under an uncertainty distribution. They do not establish that changing a physiological quantity causes the observed association in real patients.
Assuming one dependence-aware index answers every question. Conditional indices, Shapley effects, grouped analyses, and latent-variable analyses have different targets. Choose and report the target before ranking inputs.

All variance-based approaches depend on the output and input distributions chosen. They can be difficult to estimate when the PBPK model is computationally expensive, when conditional sampling is challenging, or when rare but influential regions matter. Sensitivity rankings can also change when the population, endpoint, or uncertainty model changes. A defensible result is therefore conditional on the stated model, joint input distribution, and output definition.

Check Your Understanding

Answer each question using the joint distribution and sensitivity target stated in the problem.

For independent inputs, what property allows classical Sobol component variances to add to the total output variance?
In the correlated example with $\rho=0.8$, why do the two conditional explained-variance fractions of $0.9$ not contradict the definition of output variance?
What does a Shapley effect allocate when inputs are dependent, and what does it not establish?
Give one reason that sampling a parameter independently of its correlated partner may be inappropriate for a PBPK analysis.
Why do Sobol indices calculated after transforming to independent latent variables not automatically rank the original PBPK parameters?

Solutions. (1) Independence makes the variance components in the ANOVA decomposition orthogonal, so their variances add. (2) The fractions overlap because either input carries information about the other; they are conditional explained-variance measures, not exclusive shares. (3) A Shapley effect allocates explained output variance by averaging each input's marginal contribution over input orderings. It does not establish a unique physical or causal ownership of shared information. (4) Independent sampling can generate combinations inconsistent with the intended joint population distribution or with a physiological constraint, changing the uncertainty being analyzed. (5) The indices refer to the chosen latent coordinates, and the transformation may depend on its construction or ordering; attribution to original parameters requires an additional interpretation and is not automatic.