Variance-Based Sensitivity

Global sensitivity analysis techniques, including Sobol indices, decomposing total output variance into components attributable to each parameter and their interactions.

Source: Sobol 2001

Variance-based sensitivity analysis is regarded as rigorous because it defines an input's importance by the exact share of the output's total variance it accounts for, a precise and additive measure rather than an informal impression. It captures effects across the whole range of every input at once, including interactions, and does not assume the model responds in any particular way, such as linearly. This generality and precision make it a benchmark, at the cost of needing many model runs. Sobol indices are its central tool. Saltelli and colleagues (2008) describe it.

Source: Saltelli et al. 2008

Variance-based sensitivity works by decomposing the output variance into contributions from each input and their combinations, estimated by Monte Carlo procedures using structured samples that vary subsets of inputs while averaging over the rest. First-order indices give the variance from each input alone, and total-effect indices give each input's full contribution including interactions. Comparing them reveals interactions. The result is a set of measures, each a share of output variance, quantifying how the inputs and their joint effects drive the uncertainty in the output.

Source: Saltelli et al. 2008

Variance-based sensitivity provides first-order sensitivity indices, giving the share of output variance from each input varying alone; total-effect indices, giving each input's full contribution including all interactions; and higher-order indices, giving the variance from specific groups of inputs acting jointly. The first-order index suits ranking inputs for prioritisation, and the total-effect index suits identifying inputs that can be fixed. Together these measures quantify individual and joint influences on output uncertainty, offering a complete decomposition of the variance among the inputs and their interactions.

Source: Sobol 2001

Variance-based sensitivity is used because it gives a rigorous, quantitative decomposition of output uncertainty among the inputs, valid for non-linear models and capturing interactions, unlike simpler local or one-way methods that can misjudge influence. It supports prioritising influential parameters, fixing negligible ones, and detecting interactions, informing where to focus data collection and how the model behaves. Where a precise understanding of the drivers of uncertainty is needed, variance-based sensitivity provides a thorough basis, justifying its greater computational cost over cruder sensitivity measures.

Source: Sobol 2001

Variance-based sensitivity can be computationally expensive, since estimating the indices accurately requires many model runs, which is demanding for complex or slow models, though screening can reduce the inputs first. It depends on the assumed input distributions, so the decomposition changes if these change, and correlated inputs complicate interpretation. It characterises uncertainty through variance, which may not capture all aspects of interest. These limitations mean variance-based sensitivity is applied where its precise decomposition justifies the cost, with attention to input assumptions and, often, prior screening.

Source: Saltelli et al. 2008