Choosing Global Sensitivity Targets for QSP Outputs

Why the Target Must Be Chosen First

A QSP model can produce many outcomes from the same simulated system: a biomarker concentration at a specified time, a peak response, an integrated exposure, or a cell population at the end of a regimen. A global sensitivity analysis cannot rank parameters in the abstract. It asks how variation in a specified output, under a specified distribution of parameters, is associated with variation in that output.

This is different from asking how a simulated trajectory changes when one parameter is nudged near a chosen parameter vector. That local question concerns a derivative at a particular point. A global variance question concerns output variation across a population of parameter vectors. Both can be useful, but they answer different questions and can give different rankings.

Definition: Let \(g(\boldsymbol{\theta};d)\) denote a deterministic QSP simulator, where \(\boldsymbol{\theta}\) is the parameter vector and \(d\) collects fixed design choices such as the dosing regimen, initial conditions, and simulation time grid. A scalar output target is \(Y=h(g(\boldsymbol{\theta};d))\), where \(h\) specifies exactly which quantity is extracted or calculated from the simulated output. A global sensitivity target also specifies a joint probability distribution \(P_{\boldsymbol{\theta}}\) for the parameters. Thus, the target is the output \(Y\) considered over draws from that distribution, with the design choices held fixed.

For example, “the sensitivity of the model’s concentration” is underspecified. It needs a biological quantity, a time or time summary, a regimen and other fixed design conditions, units, and a distribution for the uncertain parameters. A statement such as “variance attribution for plasma drug concentration at 24 hours under regimen R, over parameter distribution \(P_{\boldsymbol{\theta}}\)” is much closer to a reproducible target.

Specify the Output Before Comparing Parameters

The output map \(h\) determines what aspect of model behaviour is being studied. For a trajectory \(C(t)\), possible scalar targets include \(C(24\text{ h})\), the maximum concentration over a stated interval, or the area under the curve over that interval. Those are distinct random variables across parameter draws. Their units and distributions can differ, and so can the parameters that explain most of their variation.

If the scientific question concerns a whole time course, a single scalar target may not be adequate. One can define separate targets at selected times or a prespecified functional summary, but each choice changes what is being analyzed. Combining several outputs into one score also requires a stated rule, such as a weighting scheme and any scaling of the component outputs. Without that rule, a single ranking across outputs is not well-defined.

Worked Example: Peak Concentration Versus Integrated Exposure

Consider two hypothetical parameter settings for a model of concentration after one dose. The simulated concentrations are in ng/mL at 0, 2, 4, and 8 hours:

Parameter setting	0 h	2 h	4 h	8 h
A	0	10	4	0
B	0	5	5	5

For this comparison, define peak concentration as the largest recorded value and define exposure as the trapezoidal area from 0 to 8 hours. Setting A has peak concentration \(10\) ng/mL; setting B has peak concentration \(5\) ng/mL.

For setting A, the trapezoidal area is \(2(0+10)/2 + 2(10+4)/2 + 4(4+0)/2 = 10+14+8=32\) ng·h/mL. For setting B, it is \(2(0+5)/2 + 2(5+5)/2 + 4(5+5)/2 = 5+10+20=35\) ng·h/mL.

Thus A has the higher peak, while B has the higher integrated exposure. These two simulated curves do not themselves establish a global sensitivity ranking: that requires a parameter distribution and enough draws to assess output variation. They do show why the output must be selected to match the scientific question before parameter importance is interpreted.

Variance Attribution Is a Distribution-Level Question

Once a scalar output and parameter distribution are specified, the simulator induces a distribution for \(Y\). If \(Y\) has finite variance, its total output variance is \(\operatorname{Var}(Y)\), calculated across parameter draws under \(P_{\boldsymbol{\theta}}\). A variance-based sensitivity measure describes some aspect of how that output variance relates to variation in one or more parameters. It is therefore conditional on both the chosen output and the chosen parameter distribution.

For independent inputs, a common first-order variance target for parameter \(\theta_i\) is the fraction \( \operatorname{Var}(\operatorname{E}[Y\mid\theta_i])/\operatorname{Var}(Y)\), provided \(\operatorname{Var}(Y)>0\). It measures the fraction of total output variance associated with variation in the conditional mean as \(\theta_i\) varies, averaging over the other inputs. This is not the same as a derivative, nor is it necessarily the full share of variation involving \(\theta_i\) when interactions are present. The definition of a sensitivity measure should be stated alongside the output target.

Worked Example: One Model, Two Variance Targets

Let \(X\) and \(Z\) be independent hypothetical parameter deviations with \(X=1\) or \(-1\), each with probability \(1/2\), and \(Z=2\) or \(-2\), each with probability \(1/2\). Their means are zero and their variances are \(1\) and \(4\), respectively. Suppose two possible scalar model outputs are \(Y_1=X+Z\) and \(Y_2=X+0.5Z\), in their respective output units.

Because the inputs are independent and centered, \(\operatorname{Var}(Y_1)=1+4=5\). Also, \(\operatorname{E}[Y_1\mid X]=X\), so \(\operatorname{Var}(\operatorname{E}[Y_1\mid X])=1\). The first-order variance fraction for \(X\) is \(1/5=0.20\). For \(Z\), the conditional mean is \(Z\), giving \(4/5=0.80\).

For \(Y_2\), the total variance is \(1+(0.5)^2(4)=2\). Its conditional means are \(\operatorname{E}[Y_2\mid X]=X\) and \(\operatorname{E}[Y_2\mid Z]=0.5Z\). The corresponding variances are \(1\) and \(0.25(4)=1\), so each first-order fraction is \(1/2=0.50\).

The input distributions are unchanged, yet the parameter ranking differs because the output map differs. This simple additive example has no interaction term, so the two first-order fractions sum to one for each output. In a nonlinear model with interactions, first-order fractions need not account for all output variance.

Why This Is Not Sensitivity Along One Trajectory

A local sensitivity commonly describes the derivative of an output with respect to a parameter at a fixed parameter vector. For a scalar target \(Y=f(\theta)\), that quantity is \(f'(\theta_0)\) at a chosen value \(\theta_0\). It describes the infinitesimal change near that point, in the units of output per unit of parameter. A global variance target instead summarizes variation across a distribution of parameter values. A large local derivative need not imply that a parameter accounts for a large fraction of output variance: the parameter may vary very little under the chosen distribution, or the response may change slope elsewhere.

A short calculation makes the distinction concrete. Let \(\theta\) be uniformly distributed on \([-1,1]\) and let the output be \(Y=\theta^2\). At \(\theta_0=0.2\), the local derivative is \(2\theta_0=0.4\). Across the full distribution, however, \(E[Y]=E[\theta^2]=1/3\) and \(E[Y^2]=E[\theta^4]=1/5\). Therefore

\operatorname{Var}(Y)=E[Y^2]-E[Y]^2 =\frac{1}{5}-\left(\frac{1}{3}\right)^2 =\frac{4}{45}\approx 0.08889.

The derivative \(0.4\) is a local rate at one point; the variance \(4/45\) describes output spread over the stated distribution. They are not competing estimates of the same quantity. For this one-input example, the first-order variance fraction is one, since the entire random output is determined by \(\theta\), even though the local derivative at \(\theta=0\) is zero.

Justification for additive variance attribution: If \(X\) and \(Z\) are independent and \(Y=aX+bZ+c\), then \(\operatorname{Var}(Y)=\operatorname{Var}(aX+bZ)=a^2\operatorname{Var}(X)+b^2\operatorname{Var}(Z)+2ab\operatorname{Cov}(X,Z)\). Independence gives \(\operatorname{Cov}(X,Z)=0\), so the variance is the sum of the two scaled variances. Also, \(E[Y\mid X]=aX+bE[Z]+c\), whose variance is \(a^2\operatorname{Var}(X)\). These facts justify the first-order fractions calculated in the additive example. If inputs are dependent or the output includes interactions, that simple allocation does not apply without further choices about the sensitivity measure.

Assumptions, Diagnostics, and Limitations

A useful variance-based target requires a well-defined simulator over the parameter values assigned positive probability, a specified joint input distribution, and an output with finite, nonzero variance. Numerical failures, discontinuities caused by solver settings, or outputs undefined for part of the distribution can make the resulting variance target misleading or unavailable. If the output variance is zero, a fraction with \(\operatorname{Var}(Y)\) in its denominator is undefined.

Check that the fixed conditions really are fixed for the question, that the output definition and units are unambiguous, and that the input distribution represents the variation you intend to study. A posterior distribution from calibration, a prior distribution, and a deliberately chosen uncertainty distribution represent different sources or meanings of variation; substituting one for another changes the target. Distribution construction and sensitivity-estimator diagnostics require separate attention.

Also inspect the simulated outputs across the intended parameter range. Failed or implausible trajectories, extreme outputs, and a variance dominated by a small number of draws may indicate that the chosen target or input range needs reconsideration. If parameters are correlated, independent-input variance decompositions cannot be interpreted as though each parameter varied independently. The dependence structure is part of the joint distribution and must be respected by the chosen sensitivity method.

Local derivatives remain useful when the question is about response near a calibrated parameter vector or for gradient-based tasks. Global variance attribution is useful when the question concerns output variability over a specified parameter distribution. Neither replaces the other, and neither is universally preferable. A single ranking can also obscure output-specific effects: parameters important for a biomarker at one time may be less important for an integrated response or a different biological endpoint.

Common Pitfall

Wrong reasoning: “Parameter A has the largest derivative in one nominal simulation, so it is the most important parameter globally.”

Correction: The derivative describes a local change at that nominal parameter vector. A global ranking requires a stated output and parameter distribution, then a sensitivity measure evaluated over that distribution. The ranking can change when the output, distribution, or measure changes.

Key takeaway: Before calculating global sensitivity, write down the scalar output \(Y\), the fixed simulation conditions, the joint distribution \(P_{\boldsymbol{\theta}}\), and the variance-based measure of interest. This defines the question; it does not yet determine the numerical answer.

Check Your Understanding

Answer each question, then compare with the solutions below.

A model produces a cytokine trajectory. Name three details needed to turn “sensitivity of cytokine response” into a precise scalar output target.
Why can a global sensitivity ranking change when the parameter distribution changes, even if the simulator is unchanged?
For \(Y=3X+Z\), let independent centered inputs have variances \(\operatorname{Var}(X)=1\) and \(\operatorname{Var}(Z)=4\). What are the total variance and first-order variance fractions?
At one parameter value a derivative is zero. Does that imply the parameter has no global influence? Explain.

Solutions. (1) State the cytokine quantity, such as its value at a specified time or an explicitly defined summary; state the regimen and other fixed simulation conditions; and state the output units. The joint parameter distribution is also required to define the global target. (2) The distribution determines which parameter values are represented and how much each varies, changing the induced output distribution and its variance attribution. (3) Independence gives \(\operatorname{Var}(Y)=3^2(1)+1^2(4)=13\). The first-order fractions are \(9/13\approx0.6923\) for \(X\) and \(4/13\approx0.3077\) for \(Z\). (4) No. A zero derivative at one point gives only local information there; the response may vary elsewhere over the parameter distribution.