First-Order Index

First-Order Index: Concept Architecture

Orientation and learning roadmap

This page places the index within global variance-based sensitivity analysis and derives what it attributes to a single uncertain model input. It then calculates an illustrative two-input result, compares first-order and total-order measures, and explains the assumptions needed in a health-economic model. The closing checks address correlated inputs, simulation error, and decision relevance.

Define the output and input distributions

Let a model output $Y=f(X_1,\ldots,X_k)$ depend on uncertain inputs assigned a joint probability distribution. A first-order Sobol index describes the variation in the conditional mean output as one input varies across its distribution, expressed as a share of the unconditional output variance. Its numerical value depends on the selected output, input distributions, and model structure; it is not an intrinsic importance score for a parameter.

$$S_i=\frac{\operatorname{Var}_{X_i}\!\left[\operatorname{E}(Y\mid X_i)\right]}{\operatorname{Var}(Y)}, \qquad \operatorname{Var}(Y)>0.$$

The denominator must be nonzero. Under the classical independent-input variance decomposition, $0\leq S_i\leq1$, and the sum of all first-order indices is at most one. Dependence among inputs changes what can be attributed uniquely to a single input and requires a specified alternative interpretation or method.

Separate an input's main effect from interactions

The first-order numerator averages over uncertainty in all other inputs before measuring how the conditional expectation varies with $X_i$. It captures the main effect attributable to input $i$ under independent-input decomposition, excluding contributions that arise only through its interactions with other inputs. A small index can therefore coexist with a large effect through interactions.

For two independent inputs, the functional decomposition can be written

$$Y=f_0+f_1(X_1)+f_2(X_2)+f_{12}(X_1,X_2), \qquad \operatorname{Var}(Y)=V_1+V_2+V_{12}.$$

Here $V_1$ and $V_2$ are main-effect variance components and $V_{12}$ is the interaction component, provided the terms meet the usual orthogonality conditions. Then $S_1=V_1/\operatorname{Var}(Y)$ and $S_2=V_2/\operatorname{Var}(Y)$, leaving $V_{12}/\operatorname{Var}(Y)$ outside their sum. This residual is not necessarily model error; it can be genuine interaction between uncertain inputs.

Calculate an illustrative two-input result

Suppose $X_1$ and $X_2$ are independent, each equally likely to be $-1$ or $+1$, and a centred illustrative net-benefit deviation is $Y=2X_1+X_2+X_1X_2$. These four outcomes are arbitrary model units, not actual cost-effectiveness evidence. Because all three nonconstant terms have zero mean and are mutually uncorrelated, their variance contributions are 4, 1, and 1.

The output mean is zero and its variance is $(4+4+0+16)/4=6$. Since $\operatorname{E}(Y\mid X_1)=2X_1$ and $\operatorname{E}(Y\mid X_2)=X_2$, the indices are $S_1=4/6\approx0.667$ and $S_2=1/6\approx0.167$. Their sum is $5/6$, and the remaining $1/6$ is the interaction share.

Compare first-order and total-order indices

A total-order index includes both an input's main effect and every interaction involving that input. For independent inputs, one equivalent definition measures how much output variance remains when all inputs other than $X_i$ are fixed in the conditional expectation. Comparing total and first-order indices identifies the importance of interactions, while neither measure is a causal effect of changing an input in practice.

$$S_{T_i}=1-\frac{\operatorname{Var}\!\left[\operatorname{E}(Y\mid X_{\sim i})\right]}{\operatorname{Var}(Y)}.$$

In the example, $S_{T_1}=(4+1)/6=5/6$ and $S_{T_2}=(1+1)/6=2/6$. Thus input 2 has a modest first-order share but participates in an interaction contributing another sixth of total variance. The sum of total-order indices can exceed one because an interaction is included in each participating input's total effect.

Use the index in health-economic decisions

For a cost-effectiveness model, specify a decision-relevant scalar output, such as incremental net monetary benefit at a stated willingness-to-pay threshold, before computing indices. The first-order index then shows which uncertain parameters explain variation in that selected numerical output under the specified uncertainty distributions. A high share can help prioritise model investigation, but it is not the expected value of collecting more information or proof that changing that input would improve care.

An index for incremental cost may rank parameters differently from one for QALYs or net monetary benefit. An ICER may behave poorly when incremental effect approaches zero, so a net-benefit output may be more interpretable for global sensitivity in such cases. Parameter uncertainty must also be distinguished from first-order stochastic or patient-level variability, which is a different use of the words “first order” in health-economic modelling.

Estimate and audit the result

Pick-freeze Monte Carlo designs and suitable surrogate models can estimate first-order and total-order indices for costly models. Specify the sampling scheme, parameter ranges or distributions, random seed, model runs, estimator, and uncertainty in the estimated indices. Finite-sample estimates can fall slightly outside theoretical bounds because of numerical or Monte Carlo error, which should prompt diagnostic checks rather than an unexamined substantive interpretation.

• Check that the uncertain inputs and their joint distribution represent the intended decision uncertainty.
• Check that $\operatorname{Var}(Y)$ is materially above zero and that the output uses consistent units.
• Compare first-order with total-order indices to identify inputs involved in interactions.
• Quantify Monte Carlo or surrogate approximation error before ranking close index estimates.
• Use a dependency-aware method if inputs are correlated, rather than treating the classical independent-input decomposition as a unique attribution.
• Recalculate for alternative decision thresholds, structures, or plausible distributions when they could change the conclusion.

Sources and technical basis

The Sobol method establishes variance-based global sensitivity indices, and the SciPy and SALib documentation describe the first-order and total-order distinction and computational assumptions. ISPOR-SMDM guidance provides the health-economic context for reporting parameter uncertainty without confusing it with patient-level stochastic variability. The two-input arithmetic here is an original teaching example with fully enumerated outcomes.

• Sobol, Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates.
• SciPy, sobol_indices documentation.
• SALib, Basics of sensitivity analysis.
• ISPOR-SMDM, Model Parameter Estimation and Uncertainty Analysis.