Global Sensitivity Analysis With Uncertain Inputs

Why Sensitivity Must Be Assessed Across the Input Space

A local sensitivity describes how a prediction changes near one chosen input value. That can be useful when the model is smooth and the chosen point is representative. Translational predictions, however, often depend on inputs that are uncertain over substantial ranges: potency translation, physiological scaling, turnover rates, or parameters informed by sparse preclinical evidence. A local derivative at one nominal value does not reveal whether an input matters elsewhere in that range, whether its effect is nonlinear, or whether its influence depends on another input.

Global sensitivity analysis evaluates how variation in uncertain inputs across a specified plausible region or probability distribution is associated with variation in a model output. The output must be named and defined first: examples include predicted human exposure at a specified time, a biomarker response at a specified dose, or a derived dose needed to reach a target. Sensitivity results apply to that output and to the input distributions and dependencies used in the analysis. They are not universal rankings of biological importance.

Definition: Let \(Y=f(X_1,\ldots,X_k)\) be a model output evaluated at uncertain inputs with a specified joint distribution. Global sensitivity analysis summarizes how variation in \(Y\), over that joint input distribution, is associated with variation in one or more inputs. The analysis requires a defined output, specified input distributions and dependencies, and a model that can be evaluated over the region being studied.

This definition makes the uncertainty specification part of the scientific question. A narrow potency distribution and a broad potency distribution can give different sensitivity results even if the model is unchanged. Global sensitivity does not establish that an input is biologically causal, nor does a low sensitivity score show that the input is known accurately. It describes the model’s response to the uncertainty represented in the analysis.

Variance-Based Indices

For the standard variance-based indices, assume the inputs \(X_1,\ldots,X_k\) are mutually independent, the model output has finite, nonzero variance, and the model is evaluated over the specified input distributions. Independence matters: it permits the output variance to be partitioned into non-overlapping contributions from individual inputs and their interactions.

Definition: Let \(V=\operatorname{Var}(Y)\). The first-order index for input \(X_i\) is \(S_i=\operatorname{Var}(\operatorname{E}[Y\mid X_i])/V\). It measures the share of output variance associated with \(X_i\) alone, averaged over the other inputs. The total-order index is \(T_i=\operatorname{E}[\operatorname{Var}(Y\mid X_{-i})]/V\), where \(X_{-i}\) denotes all inputs except \(X_i\). It measures the share associated with \(X_i\), including interactions involving that input.

The law of total variance gives a useful justification. For any input \(X_i\),

\operatorname{Var}(Y) = \operatorname{Var}\!\left(\operatorname{E}[Y\mid X_i]\right) + \operatorname{E}\!\left[\operatorname{Var}(Y\mid X_i)\right].

Dividing by \(V\) shows that the first-order index is the fraction of variance explained by the conditional mean given \(X_i\). For mutually independent inputs, the variance can be decomposed into contributions from individual inputs and combinations of inputs. Thus first-order indices do not include interactions, while total-order indices include every variance component in which the input appears. A total-order index can exceed its first-order index when interactions matter. The sum of first-order indices can be below one when the model has interactions.

In practice, indices are often estimated by Monte Carlo sampling. A common design generates two independent input matrices, \(A\) and \(B\), from the product of the specified input distributions. For input \(i\), a hybrid matrix replaces the \(i\)th column of \(A\) with the corresponding column of \(B\). Evaluating the model on these matrices supports estimators of the first-order and total-order indices. Their numerical values are estimates, not exact quantities: Monte Carlo error can be substantial if the sample is too small, the output is noisy, or the model has sharp nonlinear behavior.

Worked Example: Decomposing an Interaction Model

Suppose two uncertain, independent inputs \(X_1\) and \(X_2\) are each uniformly distributed from 0 to 1. A hypothetical translational endpoint \(Y\), measured in response units, is defined by

$$ Y=2.25+2.5(X_1-0.5)+3.5(X_2-0.5) +3(X_1-0.5)(X_2-0.5). $$

The constant is consistent with the centered model: at \(X_1=X_2=0.5\), both centered terms and the interaction are zero, so the prediction is \(2.25\) response units. Set \(A=X_1-0.5\) and \(B=X_2-0.5\). Each is centered and has variance \(1/12\). Independence means the three variable terms \(2.5A\), \(3.5B\), and \(3AB\) are mutually uncorrelated. For example, \(\operatorname{E}[A\cdot AB]=\operatorname{E}[A^2]\operatorname{E}[B]=0\). Therefore their variances add:

V =2.5^2\left(\frac{1}{12}\right) +3.5^2\left(\frac{1}{12}\right) +3^2\left(\frac{1}{12}\right)^2 =\frac{25}{48}+\frac{49}{48}+\frac{3}{48} =\frac{77}{48} \approx 1.604\ \text{response units}^2.

The first-order indices are the individual contributions divided by total variance:

S_1=\frac{25}{77}\approx0.3247,\qquad S_2=\frac{49}{77}\approx0.6364.

The interaction contribution is \(3/77\approx0.0390\), so \(S_1+S_2=74/77\), not one. The total-order indices include that interaction:

T_1=\frac{25+3}{77}=\frac{28}{77}\approx0.3636,\qquad T_2=\frac{49+3}{77}=\frac{52}{77}\approx0.6753.

For these particular uniform input distributions, the model assigns the largest share of output variance to \(X_2\) alone, and a smaller but nonzero share to the interaction. This does not mean that changing \(X_2\) always changes the endpoint more than changing \(X_1\) at every point. It is a distribution-dependent summary across the full stated ranges.

Rank-Based Global Sensitivity

Variance-based indices are not the only option. A rank-based method can be useful when the relationship is nonlinear but broadly monotonic, or when a scale-free measure of association is wanted. Spearman’s rank correlation, for example, is the ordinary correlation between the ranks of an input and the ranks of the output. Its value is between \(-1\) and \(1\): a value near \(1\) indicates strong increasing monotone association in the sampled data, and a value near \(-1\) indicates strong decreasing monotone association.

A rank correlation is not a fraction of output variance. It can be affected by the other inputs, especially if they are dependent, and a single correlation does not describe interaction structure. It is therefore important to state whether the analysis uses the raw input–output rank correlation or a method that adjusts for other inputs. With independent inputs, a raw rank correlation across joint simulation draws is a useful screening summary, but it remains a measure of monotone association rather than a variance decomposition.

Worked Example: Ranking a Nonlinear Monotone Relationship

Consider a hypothetical model in which \(X\) is a dimensionless potency multiplier and the output \(Y\) is a biomarker change in units:

Y=\frac{10X}{0.3+X}.

At six illustrative values \(X=0.1,0.2,0.3,0.4,0.5,0.6\), the corresponding outputs are approximately \(2.500,4.000,5.000,5.714,6.250,6.667\) units. Both input and output ranks are \(1,2,3,4,5,6\), so every rank difference \(d_j\) is zero. The sample Spearman correlation is

r_s=1-\frac{6\sum_{j=1}^{n}d_j^2}{n(n^2-1)} =1-\frac{6(0)}{6(6^2-1)} =1.

This indicates a perfectly increasing rank relationship in these six constructed points, despite the model being nonlinear. It does not imply that the output increases by a constant amount per unit change in \(X\), nor does it quantify the fraction of output variance attributable to \(X\). A sensitivity analysis over a plausible distribution would use many draws and would report the sampling design and resulting estimate, rather than treating six illustrative points as a stable study result.

Rank methods can miss important effects that are not monotonic. For example, if \(X\sim U(0,1)\) and \(Y=(X-0.5)^2\), the output changes systematically with \(X\), but symmetrically around \(0.5\). Its population rank association is zero by symmetry. Yet the output has nonzero variance: \(\operatorname{E}[Y]=1/12\), \(\operatorname{E}[Y^2]=1/80\), and \(\operatorname{Var}(Y)=1/80-(1/12)^2=1/180\). A near-zero rank correlation is therefore not evidence that the input has no effect.

Dependent Inputs and the Choice of Method

Standard Sobol indices rely on independent inputs. In translational work, however, two inputs may be correlated because they share data, are jointly estimated, or are linked by a biological or physiological constraint. Independently sampling such inputs changes the joint distribution and can create combinations that are implausible. Applying ordinary independent-input Sobol interpretations to dependent inputs can also allocate shared variation ambiguously.

Worked Example: Why Correlation Changes the Variance

Let two standardized translational inputs \(X_1\) and \(X_2\) each have mean zero and variance one, with correlation \(0.8\). A hypothetical log-exposure endpoint is \(Y=X_1+X_2\). Its variance under the stated dependent input distribution is

\operatorname{Var}(Y) =\operatorname{Var}(X_1)+\operatorname{Var}(X_2) +2\operatorname{Cov}(X_1,X_2) =1+1+2(0.8) =3.6.

If the two inputs were incorrectly treated as independent, the covariance term would be set to zero and the implied variance would be \(2\), not \(3.6\). Indices computed under that independence assumption would describe a different uncertainty model. Even dividing each marginal variance by the true total, \(1/3.6\approx0.2778\), would leave the covariance contribution unallocated; those ratios are not standard Sobol indices for this dependent setting.

When dependence is scientifically important, preserve the joint distribution in the sampling and use a dependence-aware attribution method, such as conditional sensitivity measures or a Shapley-effect allocation. These methods answer related but not identical attribution questions, so the chosen definition should be stated. If dependence is weak and an independent approximation is used for a specific purpose, report that approximation and examine how it changes the conclusions.

The main alternatives have complementary strengths. Variance-based first- and total-order indices describe contributions to output variance and explicitly reveal interactions under independence. Rank correlations are relatively easy to interpret for monotonic trends and can be useful for screening, but they neither partition variance nor detect every nonlinear pattern. Local derivatives describe behavior near a selected point and can be efficient when uncertainty is small, but they do not summarize the full input distribution. None of these measures alone demonstrates predictive validity or biological causation.

Diagnostics, Limitations, and Practical Workflow

A sensitivity ranking is only as credible as its model evaluations and uncertainty specification. Check that all sampled input combinations are scientifically plausible, that the model is defined throughout the sampled domain, and that the output has meaningful variability. If the output variance is zero or nearly zero, variance-based indices are undefined or numerically unstable. If a small number of extreme predictions dominate the variance, inspect those runs and confirm that they are not coding errors or unintended parameter combinations.

Monte Carlo estimates should be checked for stability. Increase the number of samples and compare estimates, particularly for total-order indices, which can require substantial simulation. Repeat the analysis with independent random seeds or use suitable uncertainty estimates for the indices. A ranking that changes markedly with sample size should not be presented as a settled ordering. For rank-based analyses, inspect scatterplots and rank plots: curvature, U-shaped patterns, clusters, or ties can make a single correlation misleading.

A disciplined analysis proceeds as follows:

Define the prediction.
Name the model output, its units, and any dose, time, or population conditions attached to it.

Specify the uncertainty.
Assign defensible distributions and bounds to inputs, and represent dependence when it is material. Explain the evidence supporting these choices.

Choose the sensitivity measure.
Use variance indices for variance contributions and interactions under independence; use a rank-based measure for monotonic association, with its limitations stated.

Check numerical and scientific behavior.
Assess convergence, inspect influential simulations, and verify that sampled inputs and resulting predictions are plausible within the model’s intended use.

A common mistake is to read a high sensitivity index as proof that an input is poorly known or biologically decisive. The index instead indicates that, given the specified uncertainty distribution, the model output varies substantially with that input or with interactions involving it. A narrow input distribution can yield a small index even when the model is highly responsive to that input per unit change. Conversely, a broad distribution can make an input prominent even when its local effect is modest. Sensitivity, input uncertainty, and scientific importance should therefore be reported separately.

Key takeaway: Global sensitivity results are conditional on the output, model, joint input distribution, and sensitivity definition. Use variance-based indices to quantify variance contributions and interactions when inputs are independent; use rank-based methods for monotone association, not as substitutes for variance decomposition. Diagnose dependence, nonlinearity, and Monte Carlo stability before interpreting a ranking.

Check Your Understanding

Answer each question, then compare with the solutions below.

In the interaction model from the first worked example, why is \(T_1\) greater than \(S_1\), and by how much?
What does a Spearman correlation of \(1\) establish, and what does it not establish?
For \(Y=(X-0.5)^2\) with \(X\sim U(0,1)\), give the variance and explain why a rank correlation of zero would be misleading as a no-effect conclusion.
If two unit-variance inputs have correlation \(0.5\) and \(Y=X_1+X_2\), calculate \(\operatorname{Var}(Y)\). Why are independent-input Sobol indices inappropriate for that joint distribution?

Solutions. (1) The interaction variance is included in the total-order index but not the first-order index. Thus \(T_1-S_1=3/77\approx0.0390\), the interaction’s share of total variance. (2) It establishes a perfectly increasing rank ordering in the analyzed observations; it does not establish a linear relationship, a variance share, causation, or stability beyond the sampled data. (3) The variance is \(1/80-(1/12)^2=1/180\). The symmetric U-shaped response has nonzero output variance even though its increasing and decreasing rank trends cancel. (4) \(\operatorname{Var}(Y)=1+1+2(0.5)=3\). Independent-input indices omit the covariance and represent a different joint distribution, so they do not correctly describe this dependent-input model.)