Global Sensitivity Analysis

What global sensitivity analysis does

Unlike one-way or local sensitivity analysis, it can detect nonlinear effects, interactions, and influence that changes across the parameter space.

This page explains variance-based Sobol indices, interaction effects, Morris screening, dependence between inputs, sampling, convergence, and computational shortcuts. It also shows how GSA complements probabilistic sensitivity analysis in health-economic models without replacing structural scenarios or value-of-information analysis.

Global and local sensitivity answer different questions

Local sensitivity measures the slope of an output near one selected input vector. Global sensitivity examines the output over the complete uncertainty space defined for the analysis.

For model output $Y=f(\mathbf{X})$, a local derivative is:

$$\left. \frac{\partial f}{\partial X_i} \right|_{\mathbf{X}=\mathbf{x}_0}$$

The derivative can be informative near $\mathbf{x}_0$ but can miss thresholds, curvature, interactions, and reversals elsewhere. GSA requires more model evaluations but produces influence measures that are tied to the full input specification rather than one base case.

The uncertainty space defines the analysis

Before calculating sensitivity indices, the analyst must define which inputs vary, their distributions or ranges, and their dependence. These choices determine which regions of the model are explored and therefore determine the resulting rankings.

• Parameter uncertainty can be represented using distributions derived from estimation, evidence synthesis, or justified elicitation.
• Decision scenarios can represent alternative structural assumptions, data sources, extrapolation functions, or policy rules when those choices can be encoded coherently.
• Heterogeneity should be distinguished from uncertainty when variation reflects real differences between people rather than imperfect knowledge.
• Numerical settings should be tested separately when grid size, simulation population, solver tolerance, or random error can affect the output.

A narrow distribution makes an input appear less influential even if the model is mathematically very sensitive to it. GSA ranks influence under the declared uncertainty model, not as an intrinsic property of a parameter name.

Variance decomposition under independent inputs

Classical Sobol analysis decomposes output variance into contributions from individual inputs and their interactions. The standard functional decomposition assumes mutually independent inputs and a square-integrable model output.

For $k$ inputs:

$$f(\mathbf{X}) =f_0 +\sum_i f_i(X_i) +\sum_{i<j}f_{ij}(X_i,X_j) +\cdots +f_{1\ldots k}(X_1,\ldots,X_k)$$

The corresponding variance decomposition is:

$$V(Y) =\sum_i V_i +\sum_{i<j}V_{ij} +\cdots +V_{1\ldots k}$$

The components are orthogonal under the independence assumption. If inputs are dependent, the same decomposition and interpretation do not carry over unchanged.

First-order Sobol indices

The first-order index measures the proportion of output variance attributable to one input acting alone. It averages over uncertainty in all other inputs.

$$S_i =\frac{V_{X_i}\left(E[Y\mid X_i]\right)}{V(Y)}$$

A large $S_i$ means that learning the value of $X_i$ would substantially reduce uncertainty in the conditional mean output under the specified model. A small $S_i$ does not establish that the input is unimportant, because its influence may operate mainly through interactions.

Total-effect Sobol indices

The total-effect index includes the first-order contribution of input $X_i$ and every interaction involving that input. It measures the output variance remaining because $X_i$ is still uncertain after all other inputs are fixed.

$$S_{T_i} =\frac{E_{\mathbf{X}_{-i}} \left[V_{X_i}(Y\mid\mathbf{X}_{-i})\right]} {V(Y)}$$

An equivalent form is:

$$S_{T_i} =1- \frac{V_{\mathbf{X}_{-i}} \left[E_{X_i}(Y\mid\mathbf{X}_{-i})\right]} {V(Y)}$$

For independent inputs, $S_{T_i}\geq S_i$. The difference $S_{T_i}-S_i$ indicates the total interaction involvement of $X_i$, although it cannot by itself identify which other inputs create those interactions.

Second-order and higher-order interactions

A second-order index measures the variance attributable specifically to the interaction between two inputs after their main effects are removed. Higher-order indices extend the idea to larger groups.

$$S_{ij} =\frac{ V_{X_i,X_j} \left(E[Y\mid X_i,X_j]\right) -V_i-V_j }{V(Y)}$$

When inputs are independent and all orders are included, the Sobol indices sum to one. Estimating every interaction quickly becomes computationally expensive, so first-order and total-effect indices are often the most useful starting point.

How to read the sums of the indices

For an additive model with independent inputs, first-order and total-effect indices are equal for each input. Their sums both equal one because no output variance arises from interaction.

When interactions are present:

$$\sum_i S_i\leq1, \qquad \sum_i S_{T_i}\geq1$$

The gap below one in the first-order sum reflects variance assigned to interactions. Total-effect indices double-count shared interaction variance across the participating inputs, so their sum can exceed one without indicating an error.

A worked additive example

Let $X_1$ and $X_2$ be independent Uniform$(0,1)$ inputs and define $Y=X_1+2X_2$. Each input has variance $1/12$, and the model has no interaction term.

$$V(Y) =V(X_1)+4V(X_2) =\frac{1}{12}+\frac{4}{12} =\frac{5}{12}$$

The first-order indices are:

$$S_1 =\frac{1/12}{5/12} =0.20$$ $$S_2 =\frac{4/12}{5/12} =0.80$$

Because the model is additive, $S_{T_1}=S_1$ and $S_{T_2}=S_2$. Under the specified equal-width input distributions, uncertainty in $X_2$ explains 80% of output variance because its coefficient is twice as large and its variance contribution is squared.

A worked interaction example

Now let $Y=X_1X_2$ with the same independent Uniform$(0,1)$ inputs. The output variance is $V(Y)=7/144$, and both inputs have symmetrical influence.

The first-order component for $X_1$ is:

$$V_{X_1}\left(E[Y\mid X_1]\right) =V\left(\frac{X_1}{2}\right) =\frac{1}{48}$$

Therefore:

$$S_1=S_2 =\frac{1/48}{7/144} =\frac{3}{7} \approx0.4286$$

The second-order interaction index is:

$$S_{12} =1-S_1-S_2 =\frac{1}{7} \approx0.1429$$

Each total-effect index is $S_{T_1}=S_{T_2}=4/7\approx0.5714$. The example shows why a first-order index alone understates an input's overall importance when it participates in interactions.

Sampling designs support Sobol estimation

Sobol indices are usually estimated from structured Monte Carlo or quasi-Monte Carlo samples rather than from an arbitrary set of model runs. Two base sample matrices are combined to isolate changes in selected input columns while preserving the required distribution.

For $k$ inputs and base sample size $N$, one common design requires approximately $N(k+2)$ model evaluations for first-order and total-effect indices. Designs that estimate all second-order indices commonly require approximately $N(2k+2)$ evaluations, although exact cost depends on the estimator and software implementation.

Quasi-random sequences can improve coverage

Low-discrepancy sequences such as Sobol sequences are designed to fill the unit hypercube more evenly than ordinary pseudo-random draws. They can improve estimation efficiency for smooth models when sample sizes and sequence construction are handled correctly.

A Sobol sampling sequence and a Sobol sensitivity index share a name but are different objects. The sequence generates points, while the index measures variance contribution; either can be used without assuming that the other has been implemented correctly.

Monte Carlo error must be quantified

Estimated sensitivity indices vary with finite sample size. Small samples can produce unstable rankings, confidence intervals that overlap widely, and even slightly negative first-order estimates because of estimator noise.

Convergence should be assessed by increasing the base sample size, examining repeated or nested estimates, and reporting uncertainty intervals from an appropriate bootstrap or resampling method. Truncating every negative estimate to zero before diagnosing convergence can hide inadequate sampling or model-evaluation problems.

Morris screening for models with many inputs

The Morris elementary-effects method is a screening approach that perturbs one input at a time at many locations in the uncertainty space. It is less computationally demanding than a full variance decomposition and can identify inputs that merit more detailed analysis.

For scaled input point $\mathbf{x}$ and step $\Delta$:

$$EE_i(\mathbf{x}) =\frac{f(\mathbf{x}+\Delta\mathbf{e}_i)-f(\mathbf{x})}{\Delta}$$

The mean absolute elementary effect $\mu_i^*$ measures overall influence, while the standard deviation $\sigma_i$ indicates nonlinearity or interaction. Morris results are qualitative screening measures and should not be reported as percentages of variance explained.

Standardised regression and rank methods

Regression-based sensitivity measures can be efficient when the model relationship is approximately linear. Partial rank correlation coefficients, or PRCCs, can summarise monotonic relationships after adjusting for other sampled inputs.

These methods can be misleading when effects are non-monotonic, strongly interactive, heteroscedastic, or discontinuous. Model fit and diagnostic plots should support their use, and a low correlation should not be interpreted as proof of negligible global influence.

Moment-independent methods examine more than variance

Variance-based indices focus on changes in output variance and can underrepresent changes confined to tails, skewness, multimodality, or threshold exceedance. Moment-independent methods compare conditional and unconditional output distributions rather than only their second moments.

They can be valuable when rare adverse outcomes, budget exceedance, or a decision boundary matters more than overall variance. Their estimands and scales differ from Sobol indices, so results should not be combined in a single ranking without explanation.

Correlated inputs need specialised treatment

Classical Sobol decomposition assumes independent inputs. Health-economic parameters are frequently correlated because they come from the same regression, evidence synthesis, patient sample, calibration, or accounting identity.

Applying an independence-based design after discarding correlation explores implausible combinations and changes the question. Dependence-aware Sobol variants, Shapley effects, grouped inputs, conditional sampling, posterior draws, or another method appropriate to the joint distribution should be selected and reported explicitly.

Shapley effects allocate shared influence

Shapley effects adapt ideas from cooperative game theory to allocate output variance among inputs, including under dependence. They average an input's marginal contribution across possible orders in which inputs can join a coalition.

The resulting effects provide a non-negative allocation that sums to one under standard variance-based definitions. They can be computationally expensive, and under dependence they may allocate influence arising from statistical association as well as direct model response, so causal interpretation is not automatic.

Grouped sensitivity can match evidence structure

Inputs may be more meaningful as a group than as isolated parameters. Examples include a vector of treatment-effect coefficients, correlated survival parameters, transition rows constrained to sum to one, or all parameters from one evidence source.

Group indices preserve the joint variation within each block and can answer which evidence module or model component drives output uncertainty. Group definitions should be prespecified because regrouping parameters after seeing results can manufacture a desired ranking.

Global sensitivity and probabilistic sensitivity analysis

Probabilistic sensitivity analysis, or PSA, propagates joint parameter uncertainty to distributions of costs, effects, and decision outcomes. GSA uses those or purpose-designed evaluations to attribute variation in selected outputs to inputs or groups.

PSA answers how uncertain the decision result is, while GSA asks which uncertainties drive a specified aspect of that result. A PSA sample is not automatically suitable for every GSA estimator, particularly when the sampling design, dependence structure, or required pick-and-freeze combinations differ.

Choosing a health-economic output

Sensitivity rankings depend on the output analysed. An input can strongly affect total costs but weakly affect incremental net benefit because its effect is similar in both strategies.

Useful outputs can include incremental cost, incremental QALYs, incremental net monetary benefit, budget impact, event counts, life-years, or the probability of crossing a clinically meaningful threshold. Separate GSA should be performed for outputs that answer materially different decision questions rather than assuming one ranking applies to the whole model.

Global sensitivity of net monetary benefit

Incremental net monetary benefit provides a continuous decision-scale output for cost-effectiveness GSA. It combines incremental health and cost on one scale while retaining the direction and magnitude of decision value. At willingness-to-pay threshold $\lambda$, define:

$$INMB(\lambda) =\lambda\Delta E-\Delta C$$

Sobol or other indices for $INMB$ can change with $\lambda$ because the relative influence of cost and effect parameters changes. Reporting results at one threshold should not imply that the same drivers dominate across every decision-maker's threshold range.

Decision indicators need careful interpretation

A binary indicator such as $I\{INMB>0\}$ directly represents whether an intervention is cost effective in each simulation. GSA of that indicator attributes variance in a discontinuous binary output, which answers a different question from GSA of continuous net benefit.

Inputs that move simulations near the decision boundary may dominate the binary indicator even if they explain modest overall variance in $INMB$. Both outputs can be useful, but their indices should be labelled and interpreted separately.

Global sensitivity is not value of information

A sensitivity index measures influence on a chosen model output under a specified uncertainty distribution. Value of information measures the expected decision benefit from reducing uncertainty, accounting for the consequences of choosing the wrong option.

A parameter can explain substantial output variance without having high value of information when it rarely changes the preferred decision. Conversely, uncertainty concentrated near a decision boundary can have decision value that is not obvious from a variance ranking alone.

Structural assumptions can be analysed explicitly

GSA need not be limited to continuous scalar parameters. Alternative extrapolation models, treatment-waning rules, state structures, data sources, or adherence assumptions can sometimes be encoded as categorical inputs or analysed as separate model structures.

Categorical structural choices require a sampling and sensitivity method that respects their meaning. Assigning arbitrary numeric codes and applying ordinary correlations can create an order that the alternatives do not possess.

Stochastic model noise can obscure sensitivity

Microsimulation, discrete-event simulation, and agent-based models may return different outputs for the same parameter vector because of first-order Monte Carlo noise. If that noise is large, sensitivity estimators can attribute unexplained simulation variability to residual output variance and weaken parameter rankings.

Common random numbers, larger simulated populations, replication at each design point, metamodels that represent stochastic variance, or analytic expectations can reduce the problem. The uncertainty due to model inputs should be separated from random simulation error and from real individual heterogeneity.

Emulators can reduce computational cost

An emulator or surrogate model approximates an expensive original model using a designed set of runs. Gaussian processes, polynomial chaos, regression models, and machine-learning methods can support GSA when direct evaluation is prohibitive.

Sensitivity indices from an emulator are trustworthy only within the validated input domain. Cross-validation, independent holdout runs, tail and boundary checks, uncertainty from the emulator itself, and reproduction of decision-relevant features are required before replacing the original model.

Implementing a teaching example in Excel

Excel can demonstrate global sampling and the additive worked example, although specialist software is preferable for formal Sobol designs and uncertainty intervals. A transparent workbook should separate the input distributions, sample matrices, model outputs, estimators, and convergence checks.

For $Y=X_1+2X_2$, columns can store reproducible draws for $X_1$ and $X_2$, the calculated output, and conditional or designed evaluations used by the chosen estimator. Volatile RAND() formulas should be frozen for audit or generated from a documented reproducible source; ordinary regression importance from random draws should not be relabelled as Sobol indices.

Software implementation safeguards

Sensitivity software usually couples a specific sampling design with a compatible estimator. Changing row order, omitting failed runs, or analysing outputs generated from a different design can invalidate results without producing a visible software error.

The workflow should preserve a unique run ID, all input values, model version, output status, random seed where relevant, and estimator settings. Failed model evaluations should be diagnosed rather than silently deleted, because failures can cluster in influential regions of the input space.

Validation and convergence checks

Validation should establish that the model runs, sampling design, estimator, and interpretation are internally consistent. A colourful sensitivity chart is not evidence that the indices are stable or answer the intended question.

• Distribution checks: Sampled marginal distributions, ranges, moments, bounds, and dependence match the declared uncertainty model.
• Design checks: The number and arrangement of runs match the selected estimator, with no reordered, duplicated, or missing rows.
• Analytic checks: Additive and interaction test functions reproduce known indices within Monte Carlo error.
• Convergence checks: Indices and their rankings stabilise as the base sample size increases.
• Interval checks: Bootstrap or estimator uncertainty is reported and is narrow enough for the claimed distinctions.
• Output checks: Each analysed output has the intended units, strategy contrast, time horizon, and willingness-to-pay threshold.
• Noise checks: Repeated runs quantify and control stochastic model noise at identical parameter values.
• Replication checks: An independent implementation reproduces selected indices from the same model evaluations.

Indices should also be checked for mathematical patterns such as $S_i\leq S_{T_i}$ under the independent-input formulation and approximate decomposition of variance. Small violations can arise from estimator noise, but large violations require investigation.

Common errors and misleading conclusions

GSA can create authoritative-looking rankings even when the input specification or estimator is inappropriate. The most serious mistakes arise from ignored dependence, insufficient sampling, and confusion between influence and decision value.

• Using arbitrary input ranges makes sensitivity rankings reflect convenience rather than evidence.
• Discarding correlation explores impossible parameter combinations and changes the underlying uncertainty question.
• Calling a tornado diagram global analysis ignores simultaneous variation and interactions.
• Reporting only first-order indices can miss inputs whose influence is mainly interactive.
• Treating Morris measures as variance shares gives screening statistics an interpretation they do not possess.
• Ignoring estimator uncertainty overstates small differences between parameter rankings.
• Equating influence with causality confuses mathematical model response with a real-world intervention effect.
• Equating influence with research value overlooks whether reducing uncertainty could change the decision.
• Analysing one output only can hide different drivers of cost, health outcomes, budget impact, and net benefit.

GSA explains variation generated by the model and uncertainty space supplied to it. It cannot reveal an omitted mechanism, biased evidence source, or structural assumption that was never represented.

A practical workflow

A defensible GSA starts with a decision-relevant output and a governed joint input specification. The workflow then matches the screening or quantitative method to the model, computational budget, dependence structure, and interpretation required.

1. Define the purpose: State whether the analysis will screen inputs, quantify variance contributions, investigate interactions, or prioritise model review.
2. Select the outputs: Identify the exact cost, effect, net-benefit, budget, clinical, or threshold outcome and its decision context.
3. Specify input uncertainty: Document ranges, distributions, dependence, groups, structural alternatives, and evidence sources.
4. Choose the method: Match Sobol, Morris, Shapley, regression, moment-independent, or another method to the question and assumptions.
5. Design the sample: Use an estimator-compatible random or quasi-random design with a justified base sample size.
6. Run reproducibly: Preserve run IDs, inputs, outputs, model version, seeds, failures, and stochastic-noise controls.
7. Estimate uncertainty: Calculate indices with confidence intervals or other estimator-precision measures.
8. Check convergence: Increase the sample size, replicate known test functions, and assess ranking stability.
9. Interpret for decisions: Compare drivers across outputs and thresholds and distinguish influence from causal effect and value of information.
10. Report transparently: Publish the uncertainty space, method, equations, run count, software, diagnostics, indices, intervals, and limitations.

What should be reported

A reproducible report identifies the model version, outputs, strategies, perspective, time horizon, willingness-to-pay thresholds, inputs, distributions, ranges, dependence, grouping, and structural alternatives. It also states the sensitivity method and estimator, sampling design, base sample size, total model evaluations, software, randomisation or sequence settings, treatment of failed runs, stochastic-noise control, convergence evidence, and interval method.

Health-economic interpretation should present first-order and total-effect results for each relevant output and explain why the dominant inputs differ. The report should distinguish model influence from evidence quality, controllability, causal importance, and expected value of further research.