Sobol Indices

Sobol Indices

Sobol indices quantify how much of a model output's variance is associated with uncertain inputs and their interactions over specified input distributions. They are a form of global, variance-based sensitivity analysis: the model is evaluated across the joint input space rather than changing one variable around a single base case. The results answer a conditional modelling question, because changing input ranges, distributions, dependence or the chosen output changes the indices.

Model and variance decomposition

Let $Y=f(X_1,\ldots,X_k)$ be a scalar model output with finite positive variance $V=\operatorname{Var}(Y)$. In the classical Sobol decomposition, input variables are mutually independent under their specified distributions. The output variance can then be uniquely partitioned into nonnegative contributions from individual inputs, pairs and higher-order groups:

$$V=\sum_i V_i+\sum_{i<j}V_{ij}+\cdots+V_{1,\ldots,k}.$$

The first-order contribution of input $i$ is $V_i=\operatorname{Var}_{X_i}\{\operatorname{E}(Y\mid X_i)\}$. It measures how much the conditional mean output varies when $X_i$ changes across its distribution, averaging over the other uncertain inputs. The first-order index is $S_i=V_i/V$. It is a fraction of output variance under the specified input distribution, not a derivative, causal effect, probability that an input is important, or fraction of the mean outcome.

For two inputs, the second-order contribution $V_{12}=V-V_1-V_2$ captures variance attributable to their interaction after removing individual main effects. In more than two inputs, a pairwise term is defined after subtracting all lower-order contributions from the variance of the conditional expectation given that pair. The independent-input decomposition ensures that the sum of all nonempty subset indices is one. First-order indices alone can sum to less than one when interactions matter.

Total-effect index

The total-effect index $S_{T_i}$ sums every variance component containing input $i$, including all interactions. For independent inputs it can also be calculated by the complementary expression

$$S_{T_i}=1-\frac{\operatorname{Var}_{X_{-i}}\{\operatorname{E}(Y\mid X_{-i})\}}{V}=\frac{\operatorname{E}_{X_{-i}}\{\operatorname{Var}(Y\mid X_{-i})\}}{V},$$

where $X_{-i}$ denotes all inputs except $i$. The gap $S_{T_i}-S_i$ signals interaction participation. In the classical independent-input setting, $0\leq S_i\leq S_{T_i}\leq1$. Total-effect indices generally sum to more than one when interactions are counted for multiple inputs. A small $S_{T_i}$ suggests that fixing input $i$ at an appropriate value may remove little variance, conditional on the chosen model and input distributions; it does not establish that the input is clinically or ethically irrelevant.

Exact two-input worked example

Consider an illustrative model $Y=X_1+X_2+X_1X_2$ with two independent inputs that each take 0 or 1 with equal probability. Think of the output as a dimensionless teaching score, not a calibrated health economic outcome. All four input combinations have probability $1/4$, so the calculation can be checked without simulation.

The mean is $\operatorname{E}(Y)=(0+1+1+3)/4=5/4=1.25$. The second moment is $\operatorname{E}(Y^2)=(0+1+1+9)/4=11/4=2.75$, giving $V=11/4-(5/4)^2=19/16=1.1875$. Conditional means are $\operatorname{E}(Y\mid X_1=0)=0.5$ and $\operatorname{E}(Y\mid X_1=1)=2$; the same values apply symmetrically to $X_2$.

Thus $V_1=V_2=\{(0.5-1.25)^2+(2-1.25)^2\}/2=9/16$. The remaining pairwise interaction contribution is $V_{12}=19/16-9/16-9/16=1/16$. Their normalized indices are:

The three distinct variance shares add to $9/19+9/19+1/19=1$. The two total-effect indices add to $20/19$, because the interaction is included in both. The total-effect gap is $S_{T_1}-S_1=1/19$, and likewise for input 2. These numbers depend on the binary input distributions; a different probability of 1 would change the indices even though the formula $f$ stays the same.

Estimating indices for a complex model

For a model too large to enumerate, draw input vectors from the stated independent distributions and evaluate the model repeatedly using a documented Monte Carlo or low-discrepancy design. Pick-freeze estimators combine two independent sampling matrices with columns swapped one at a time to estimate conditional variance contributions and total effects. The exact estimator and matrix construction matter; do not mix formulae from different designs or count an output generated for one input vector as if it belonged to another.

Specify the base sample size, number of inputs, actual number of model evaluations, random or quasi-random generator, scrambling or seed, estimator, convergence checks and uncertainty intervals for estimated indices. A common first/total-order design uses on the order of $N(k+2)$ model evaluations, but additional matrices or higher-order indices can require more; state the actual computation. Bootstrap intervals or replicated independent designs can show Monte Carlo uncertainty. Finite-sample estimates can fall slightly below zero or above one because of sampling noise; diagnose convergence and report uncertainty rather than asserting a negative population variance contribution.

If the model is stochastic, separate variation due to model randomness from variation due to uncertain inputs, for example by controlled random seeds or replication within each input vector. If output variance is zero, the indices with denominator $V$ are undefined. A surrogate model can reduce computation but introduces approximation error that needs validation on independent model runs, especially for interaction terms.

Assumptions and interpretation in health economics

Ordinary Sobol indices are built on the independent-input decomposition. Costs, treatment effects, baseline risks and utilities can be correlated; independently recombining them may create implausible clinical or economic scenarios. For dependent inputs, state a defensible joint distribution and choose a dependence-aware attribution method, conditional indices or grouped inputs with an explicit interpretation. Applying the independent-input formula to correlated inputs can make the familiar variance partition and sum-to-one claims invalid.

In a cost-effectiveness model, analyse clearly labelled outputs such as incremental net monetary benefit at a stated willingness-to-pay threshold, total cost, QALYs, budget impact or probability of a decision change. A large Sobol index for net benefit identifies a contributor to its variance under the assumed distribution; it need not identify the parameter with the highest value of additional research or the greatest probability of changing the preferred intervention. Inspect decision thresholds and expected value of information where the question is whether uncertainty is worth resolving.

Input distributions should express the relevant epistemic and population uncertainty without conflating variation between patients with uncertainty about model parameters. State how correlated parameters, structural alternatives and time dependence are handled. Sensitivity indices conditional on one model structure do not capture omitted structural uncertainty. Compare indices across alternative plausible structures and distributions when those choices affect the decision.

Reporting and source guidance

Report the model version, output definition and units, input distributions, ranges and dependence, estimator, number of evaluations, sampling uncertainty, convergence and higher-order interaction policy. Show first-order and total-effect estimates together, with a clear statement that total effects can overlap. Do not interpret the sum of total effects as a percentage of unique output variance.

Sobol's original global sensitivity indices paper develops variance-based indices and Monte Carlo estimation. Saltelli and colleagues' design and estimator paper addresses efficient estimation of total effects. The European Commission Joint Research Centre primer gives broader sensitivity-analysis context. The two-input example above is derived directly from its four stated input combinations.