Rankings Depend on the Uncertainty Being Represented
A global sensitivity ranking is not an intrinsic ordering of model parameters. It describes how uncertainty in a specified output is apportioned across uncertain inputs under particular input distributions. Widen one parameter’s plausible range, or change the shape of its distribution, and its contribution to output variance can change even though the equations of the QSP model do not.
This matters when parameters have more than one defensible uncertainty description—for example, when comparing broad expert-informed ranges with narrower ranges informed by calibration. A ranking that persists across those alternatives is more robust to the uncertainty assumptions examined. A ranking that changes is still informative: it shows that the conclusion depends on those assumptions. Neither result, by itself, proves that a parameter is or is not mechanistically important.
The comparison is meaningful only when the output, model implementation, parameter definitions and transformations are held fixed. The input distributions must also represent scientifically plausible uncertainty, not merely convenient mathematical choices. The introductory material on defining input distributions establishes that these distributions are part of the global sensitivity question; here the focus is on comparing conclusions across alternatives.
What Is Being Compared?
Let \(Y=f(X_1,\ldots,X_k)\) be a scalar model output, with finite, positive variance. For independent inputs, the first-order Sobol index for input \(X_i\) is the fraction of output variance explained by varying \(X_i\) alone:
The total-order index includes the contribution of \(X_i\) both alone and through its interactions with other inputs. One equivalent definition is:
where \(X_{-i}\) denotes all inputs other than \(X_i\). Both indices are properties of the model output under the selected input distribution. First-order indices need not sum to one when interactions contribute; total-order indices can sum to more than one because interactions can be counted for multiple inputs. In an additive model with independent inputs, the first-order and total-order indices coincide.
These definitions make the role of the distribution explicit: the expectations and variances are taken with respect to it. Even for a fixed \(f\), changing input variances, supports or shapes can change the conditional expectations, the output variance, or both.
Why the Ranking Can Change
For an additive linear model \(Y=\sum_i \beta_i X_i\), with independent inputs and finite variances, each input contributes \(\beta_i^2\operatorname{Var}(X_i)\) to output variance. Thus:
This expression gives a direct justification. Independence makes the covariance terms between distinct inputs zero, so the variance of the sum is the sum of the individual variances. Conditioning on \(X_i\) leaves the term \(\beta_iX_i\) variable and makes the other terms constant in the conditional expectation. Its variance is therefore \(\beta_i^2\operatorname{Var}(X_i)\), yielding the stated ratio. The ranking can reverse solely because the input variances change.
In nonlinear models, distribution shape can matter as well as variance. Conditional expectations of nonlinear functions depend on more than the input means and variances. For example, if \(Y=X^2+B\), the contribution of \(X\) depends on \(\operatorname{Var}(X^2)\), which involves the fourth moment of \(X\). Two distributions with the same mean and variance can therefore produce different indices.
Worked Example: A Linear Model With Reversed Rankings
Consider the fixed model \(Y=2U+V\), where \(U\) and \(V\) are independent, dimensionless deviations in log-parameter space. The output is in normalized response units. Compare two hypothetical uncertainty scenarios; all variances below are in squared deviation units.
In scenario 1, let \(\operatorname{Var}(U)=1\) and \(\operatorname{Var}(V)=4\). The output variance is:
Therefore \(S_U=S_{T_U}=4/8=0.500\), while \(S_V=S_{T_V}=4/8=0.500\). The contributions tie.
In scenario 2, let \(\operatorname{Var}(U)=4\) and \(\operatorname{Var}(V)=1\), with the model unchanged. Then:
The ranking changes from a tie to \(U\) dominating. This is not evidence that the model structure changed: its coefficients and equations are identical. The uncertainty assigned to \(U\) and \(V\) changed, and the resulting output-variance contributions changed with it.
Distribution Shape Can Matter Too
To isolate a shape effect, compare distributions with the same mean and variance while keeping the model and other inputs fixed. In the next example, the inputs are independent and the model is additive in \(A^2\) and \(B\). Since \(A\) is transformed nonlinearly, its output contribution depends on the distribution of \(A^2\), not just on the variance of \(A\).
Worked Example: Same Mean and Variance, Different Shape
Let \(Y=A^2+B\), with \(A\) independent of \(B\). In both candidate distributions, \(E[A]=1\) and \(\operatorname{Var}(A)=0.25\). For \(B\), use a two-point distribution taking values \(1-\sqrt{0.6}\) and \(1+\sqrt{0.6}\) with equal probability. Thus \(E[B]=1\) and \(\operatorname{Var}(B)=0.6\); both possible values are positive.
Under distribution 1, \(A\) is equally likely to be \(0.5\) or \(1.5\). Then \(A^2\) is equally likely to be \(0.25\) or \(2.25\), so \(E[A^2]=1.25\) and \(\operatorname{Var}(A^2)=1.000\). Because the model is additive and the inputs are independent:
Under distribution 2, \(A\) is zero with probability \(0.2\) and \(1.25\) with probability \(0.8\). Its mean is \(0.2(0)+0.8(1.25)=1\), and its variance is \(0.2(0-1)^2+0.8(1.25-1)^2=0.250\), as required. Now \(E[A^2]=0.8(1.25^2)=1.25\), and:
It follows that \(\operatorname{Var}(Y)=0.390625+0.600=0.990625\), giving \(S_A=S_{T_A}=0.3943\) and \(S_B=S_{T_B}=0.6057\). The ranking reverses even though the two distributions for \(A\) have the same mean and variance. The changed shape alters the spread of the nonlinear response \(A^2\).
A Practical Comparison Workflow
A useful comparison separates the scientific question from the numerical procedure. The goal is to ask which ranking conclusions survive reasonable alternatives, not to select the distribution that produces a preferred ordering. Keep the output definition and model fixed, estimate the same indices under each defensible distribution, and record exactly what differs between scenarios.
Name the output, model version, parameter definitions, transformations and sensitivity measure. Confirm that the output variance is nonzero under every scenario.
State whether each alternative changes a parameter’s range, variance, shape, calibration basis or dependence with other parameters. Preserve plausible bounds and units.
Use the same estimator and comparable computational effort for every scenario. Check Monte Carlo uncertainty before treating small rank differences as meaningful.
Report the indices as well as ranks. Note which inputs remain influential, which exchange rank, and which differences are small relative to estimation error.
Describe conclusions as applying under the stated distributions. Do not attribute a rank reversal to changed biology unless the model structure or evidence about the mechanism also changed.
Worked Example: Ranking a Saturating Response Under Two Ranges
Consider the fixed hypothetical model \(Y=A/(1+A)+B\), with normalized response units. Inputs \(A\) and \(B\) are independent. In both scenarios, \(B\) takes values \(1-\sqrt{0.03}\) and \(1+\sqrt{0.03}\) with equal probability, so \(\operatorname{Var}(B)=0.030\). Compare two equally likely values for \(A\), both with mean \(2\).
For the narrower range, \(A\) takes values \(1\) and \(3\). The transformed values are \(0.5\) and \(0.75\). Their mean is \(0.625\), so \(\operatorname{Var}(A/(1+A))=((0.75-0.5)/2)^2=0.015625\). Additivity and independence give:
For the broader range, \(A\) takes values \(0.5\) and \(3.5\), again with mean \(2\). The transformed values are \(1/3\) and \(7/9\), whose mean is \(5/9\). Their variance is \(((7/9-1/3)/2)^2=4/81=0.04938\). Therefore:
The ranking reverses under the broader range, although the model and the mean of \(A\) are unchanged. Because \(A\) enters a saturating function, changing its range changes the variability of its response. The result is conditional on these ranges; it does not establish that \(A\) is universally more important than \(B\).
Diagnostics, Alternatives, and Limitations
Before interpreting a changed ranking, check that it is not a numerical or design artifact. Global sensitivity estimates can have Monte Carlo error; rankings near a tie may be unstable even when the index estimates are individually reasonable. Use the earlier material on diagnosing Monte Carlo error to assess uncertainty in the estimates, and avoid presenting a fine ordering when its differences are not resolved. Also check that the model output is computed consistently, the distributions are sampled as specified, and each scenario has a meaningful, non-negligible output variance.
A comparison that changes several assumptions at once cannot identify which change caused a ranking shift. When scientifically useful, vary one feature at a time—for example, alter a range while preserving distribution shape—or use a structured set of scenarios and state which features differ. Such comparisons still describe the chosen scenarios; they do not create a distribution-free ranking.
Independent-input Sobol indices require independent inputs. If calibrated parameter uncertainty is dependent, independently sampling marginal distributions changes the joint uncertainty and may create implausible parameter combinations. Use a method suited to dependence, such as the Shapley effects introduced in the earlier material on correlated QSP parameters, and state the joint distribution being analyzed.
Alternative approaches answer related but different questions. Morris elementary effects can screen parameters with relatively modest computation, while derivative-based measures describe local slopes or their distribution-weighted summaries. Sobol indices partition output variance under specified distributions; Shapley effects allocate contributions when inputs are correlated. None supplies a universal ranking independent of the question, output, model and uncertainty representation.
A common mistake is to call the top-ranked parameter “the most important biological mechanism.” The ranking may instead reflect the selected range, the output’s nonlinear response over that range, or an interaction. The correction is to report the result conditionally: for example, “Under the stated independent input distributions, parameter \(A\) has the largest total-order index for this output.” Then report whether that conclusion persists across the plausible alternatives examined.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- In an additive model with independent inputs, what determines each input’s share of output variance?
- Can two input distributions with the same mean and variance produce different sensitivity indices in a nonlinear model? Why?
- A parameter ranks first under a broad range but third under a narrow calibrated range. What can be concluded, and what cannot?
- Why is independent-input Sobol analysis inappropriate when calibrated parameters have meaningful dependence?
Solutions. (1) The contribution is \(\beta_i^2\operatorname{Var}(X_i)\), divided by total output variance. (2) Yes. Nonlinear transformations can depend on higher moments or on how probability mass is distributed across the input range. (3) The ranking depends on the uncertainty specification over the alternatives tested; this alone does not show that model structure or biological importance changed. (4) Independent sampling discards the dependence and can create an incorrect joint uncertainty representation; use a dependence-aware method such as Shapley effects when appropriate.