Concept Architecture
Concept
Theoretically, First-Order Index is a variance-based global sensitivity measure that quantifies the proportion of the variance in a model output attributable solely to variation in a single input parameter. It forms part of Sobol sensitivity analysis and measures the main effect of an individual parameter independently of its interactions with other parameters. In health economics, the first-order index is used to identify the relative importance of uncertain model inputs influencing costs, health outcomes and cost-effectiveness results.
Mathematically, the first-order index is defined as the ratio of the variance explained by an individual input parameter to the total variance of the model output. The index ranges from 0 to 1, where a value of 0 indicates that the parameter has no direct effect on model output and a value of 1 indicates that it explains all output variability without interaction effects.
In practice, first-order indices are estimated using variance decomposition methods within global sensitivity analysis, most commonly through Monte Carlo or quasi-Monte Carlo simulation. They are applied to decision trees, Markov models and microsimulation models to rank influential parameters, guide model refinement and prioritise future data collection.
Purpose
Used to quantify the direct contribution of individual input parameters to overall model uncertainty and identify the most influential parameters in global sensitivity analysis.
Mathematical Formulae
Primary Formula
S? = Var(E(Y|X?)) / Var(Y)
where:
S? = first-order sensitivity index
Y = model output
X? = input parameter
Supporting Formulae
Total model variance:
Var(Y) = E(Var(Y|X?)) + Var(E(Y|X?))
Range of the first-order index:
0 � S? � 1
Related Mathematical Methods
- Sobol Sensitivity Analysis
- Global Sensitivity Analysis
- Variance Decomposition
- Monte Carlo Simulation
- Quasi-Monte Carlo Simulation
- Total-Order Index
Example
A probabilistic sensitivity analysis evaluates uncertainty in a Markov model. The first-order index for treatment efficacy is estimated as 0.46, indicating that 46% of the variance in incremental net monetary benefit is attributable directly to uncertainty in treatment efficacy alone. The first-order index for utility values is 0.12, demonstrating a substantially smaller independent contribution to overall model uncertainty.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| VAR.S | =VAR.S(B2:B1001) | Estimate the total variance of model outputs. |
| AVERAGE | =AVERAGE(B2:B1001) | Calculate conditional means required for variance decomposition. |
| SUMPRODUCT | =SUMPRODUCT(A2:A101,B2:B101) | Support variance decomposition calculations during sensitivity analysis. |
| RANK.EQ | =RANK.EQ(C2,C$2:C$20,0) | Rank parameters according to their first-order sensitivity indices. |
VBA (Optional)
VBA can automate Monte Carlo sampling and calculate first-order Sobol sensitivity indices for large health economic models.
Sources
- Sobol IM. Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Mathematics and Computers in Simulation. 2001;55(1?3):271?280.
- Saltelli A, Ratto M, Andres T, et al. Global Sensitivity Analysis: The Primer. John Wiley & Sons.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
- NICE. Health Technology Evaluation Manual.
Related Concepts (3)
Library
Publications
1
Statistical Analysis of Cost-Effectiveness Data — Willan & Briggs, 1st Edition ed., 2006 (John Wiley & Sons)
A synthesis of statistical methods for analysing cost-effectiveness data, including net-benefit regression, confidence intervals for the ICER, cost-effectiveness acceptability curves, and covariate adjustment. Part of the Wiley Statistics in Practice series.
BookView source →
Frequently Asked Questions (6)
What is a first-order index?
A variance-based sensitivity measure, also called a Sobol index, quantifying the share of a model's output variance from a single parameter varying alone.
Source: Sobol 2001
What does a first-order index leave out?
A first-order index measures the share of output variance a parameter accounts for through its own variation alone, averaged over all values of the others. What it leaves out is the variance that parameter contributes only in combination with others, through interactions, which is captured instead by the total-effect index. So a parameter with a small first-order index but a large total-effect matters mainly through interactions. Comparing the two reveals how much of an input's influence is interactive. Saltelli and colleagues (2008) describe these indices.
Source: Saltelli et al. 2008
How is a first-order index calculated?
A first-order index is calculated as the variance of the conditional expectation of the output given the input, divided by the total output variance, capturing how much of the output's variability is explained by that input alone. In practice it is estimated by Monte Carlo procedures using specially structured samples that vary the input of interest while averaging over the others. The result, between zero and one, gives the fraction of output variance due to that input's main effect, excluding interactions.
Source: Saltelli et al. 2008
What does a first-order index indicate?
A first-order index indicates what share of the output variance a single input explains on its own, so a large index marks an input whose uncertainty strongly drives the output's variability, and a small one an input with little individual influence. It shows how much the output variance would fall, on average, if that input were known exactly. First-order indices are used to rank inputs for prioritisation, identifying where reducing uncertainty would most reduce the uncertainty in the result.
Source: Sobol 2001
How does a first-order index differ from a total-effect index?
A first-order index measures the output variance explained by an input varying alone, excluding interactions, while a total-effect index measures the input's full contribution, including all its interactions with other inputs. So the total-effect index is at least as large as the first-order index, and a gap between them signals interaction effects. First-order indices suit factor prioritisation, identifying individual influence, whereas total-effect indices suit factor fixing, since an input is negligible only if its total effect, not just its first-order effect, is small.
Source: Saltelli et al. 2008
What are the limitations of first-order indices?
First-order indices capture only each input's individual contribution and not interactions, so they can understate an input's importance when it acts jointly with others, and their sum falls short of one when interactions are present. Estimating them accurately by Monte Carlo can require many model runs, which is demanding for complex models. They also depend on the assumed input distributions. These limitations mean first-order indices are interpreted alongside total-effect indices and with attention to the sampling effort and input assumptions.
Source: Sobol 2001
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Verified by Dr Darrin Baines
British health economist
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Verification date: 27 Oct 2025
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