Total Effect Index

Concept

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Theoretically, Total Effect Index is a variance-based global sensitivity measure that quantifies the total contribution of an individual input parameter to the variance of a model output, including both its direct effect and all interaction effects involving that parameter. It forms part of Sobol sensitivity analysis and exists to identify parameters whose uncertainty materially influences model results even when their independent contribution is small. In health economics, the total effect index is used to identify influential inputs in complex decision models and to support parameter prioritisation.

Mathematically, the total effect index is defined as the proportion of output variance attributable to an input parameter and every interaction containing that parameter. It may be represented through the expected conditional variance given all other parameters or as one minus the variance explained by all parameters except the parameter of interest. The difference between the total effect index and the corresponding first-order index represents the contribution of interactions involving that parameter.

In practice, total effect indices are estimated using Monte Carlo or quasi-Monte Carlo sampling within variance-based global sensitivity analysis. They are applied to health economic models to rank uncertain parameters, distinguish independent from interaction-driven effects and determine which parameters may be fixed without materially reducing the variance of costs, QALYs or net monetary benefit.

Purpose

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Used to quantify the complete contribution of an uncertain parameter, including all associated interaction effects, to variance in health economic model outputs.

Mathematical Formulae

Primary Formula

ST? = E[Var(Y|X??)] / Var(Y)

Supporting Formulae

Equivalent representation:

ST? = 1 ? Var(E(Y|X??)) / Var(Y)

First-order index:

S? = Var(E(Y|X?)) / Var(Y)

Interaction contribution involving parameter i:

ST? ? S?

Range:

0 � ST? � 1

Related Mathematical Methods

• Sobol Sensitivity Analysis
• Sobol Indices
• First-Order Index
• Global Sensitivity Analysis
• Variance Decomposition
• Monte Carlo Simulation
• Quasi-Monte Carlo Simulation

Example

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A global sensitivity analysis of a Markov model estimates a first-order index of 0.32 and a total effect index of 0.55 for treatment efficacy. Treatment efficacy therefore explains 32% of the variance in incremental net monetary benefit through its direct effect and a further 23% through interactions with other uncertain parameters.

Excel Implementation

Function	Example Formula	Health Economics Application
VAR.S	=VAR.S(OutputRange)	Estimate the total variance of simulated model outputs.
AVERAGE	=AVERAGE(ConditionalOutputRange)	Calculate conditional means used in variance decomposition.
SUMPRODUCT	=SUMPRODUCT(WeightRange,VarianceRange)	Combine weighted conditional variance estimates.
RANK.EQ	=RANK.EQ(B2,$B$2:$B$20,0)	Rank parameters according to their total effect indices.

VBA (Optional)

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VBA can automate quasi-Monte Carlo sampling, calculate total effect indices and rank uncertain parameters by their complete contribution to model output variance.

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.
• Saltelli A, Annoni P, Azzini I, Campolongo F, Ratto M, Tarantola S. Variance based sensitivity analysis of model output: design and estimator for the total sensitivity index. Computer Physics Communications. 2010;181(2):259?270.
• Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.