Concept Architecture
Concept
Theoretically, the Probability of Cost-Effectiveness is the probability that a healthcare intervention is cost-effective relative to one or more comparators at a specified cost-effectiveness threshold, conditional on uncertainty in model parameters. It is grounded in statistical decision theory and the net benefit framework and represents decision uncertainty rather than the probability that an intervention produces a clinical effect.
Mathematically, the Probability of Cost-Effectiveness is represented as the probability that an intervention has the greatest net monetary benefit among the alternatives being compared. In a two-intervention comparison, it is the probability that incremental net monetary benefit is positive. It is commonly estimated from probabilistic sensitivity analysis as the proportion of Monte Carlo simulations in which the intervention satisfies the relevant net benefit decision rule.
In practice, the Probability of Cost-Effectiveness is estimated by assigning probability distributions to uncertain model parameters, repeatedly sampling from those distributions, recalculating costs and outcomes, and recording the intervention with the highest net monetary benefit in each simulation. Estimates are calculated across a range of cost-effectiveness thresholds and commonly presented using a cost-effectiveness acceptability curve.
Purpose
Used to quantify decision uncertainty and show how the likelihood that an intervention is cost-effective changes across alternative cost-effectiveness thresholds.
Mathematical Formulae
Primary Formula
For two interventions:
P(cost-effective | ?) = P(??E ? ?C > 0)
where:
- ? = cost-effectiveness threshold per unit of health outcome
- ?E = incremental health effect
- ?C = incremental cost
- ??E ? ?C = incremental net monetary benefit
For multiple interventions:
P?(?) = P[NMB?(?) = max??? NMB?(?)]
where:
- P?(?) = probability that intervention i is cost-effective at threshold ?
- J = set of competing interventions
- NMB?(?) = net monetary benefit of intervention j
Supporting Formulae
Monte Carlo estimator for two interventions:
P?(cost-effective | ?) = (1/S) ????? I(??E? ? ?C? > 0)
where:
- S = number of probabilistic simulations
- I(�) = indicator function equal to 1 when the condition is satisfied and 0 otherwise
- ?E? = incremental effect in simulation s
- ?C? = incremental cost in simulation s
Net monetary benefit:
NMB?? = ?E?? ? C??
Related Mathematical Methods
- Probabilistic Sensitivity Analysis
- Monte Carlo Simulation
- Net Monetary Benefit
- Incremental Net Benefit
- Cost-Effectiveness Acceptability Curve
- Expected Value of Perfect Information
Example
A probabilistic sensitivity analysis compares a new heart failure treatment with standard care using 10,000 Monte Carlo simulations. At a threshold of �20,000 per QALY, the new treatment has positive incremental net monetary benefit in 7,350 simulations.
P?(cost-effective | �20,000) = 7,350 / 10,000 = 0.735
The probability that the new treatment is cost-effective at �20,000 per QALY is therefore 73.5%. This result indicates uncertainty about the preferred decision and does not imply a 73.5% probability that the treatment is clinically effective.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| IF | =IF(($B$1*C2)-B2>0,1,0) | Identifies whether incremental net monetary benefit is positive in each probabilistic simulation |
| COUNTIF | =COUNTIF(D2:D10001,1)/COUNT(D2:D10001) | Estimates the probability of cost-effectiveness as the proportion of simulations with positive incremental net benefit |
| SUMPRODUCT | =SUMPRODUCT(--(D2:D10001>0))/ROWS(D2:D10001) | Calculates the proportion of simulations in which an intervention is cost-effective |
| MAX | =MAX(E2:G2) | Identifies the highest net monetary benefit among multiple interventions in each simulation |
| MATCH | =MATCH(MAX(E2:G2),E2:G2,0) | Identifies which intervention has the greatest net monetary benefit in each simulation |
VBA (Optional)
Automate probabilistic simulations across multiple cost-effectiveness thresholds and calculate the probability that each intervention has the highest net monetary benefit.
Sources
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
- Fenwick E, O'Brien BJ, Briggs A. Cost-effectiveness acceptability curves: facts, fallacies and frequently asked questions. Health Economics. 2004;13(5):405?415.
- van Hout BA, Al MJ, Gordon GS, Rutten FFH. Costs, effects and C/E-ratios alongside a clinical trial. Health Economics. 1994;3(5):309?319.
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. Oxford University Press.
- NICE. Health Technology Evaluations: The Manual.
Related Concepts (3)
Library
Publications
2
Applied Methods of Cost-Effectiveness Analysis in Healthcare — Gray, Clarke, Wolstenholme & Wordsworth, 1st Edition ed., 2011 (Oxford University Press)
A practical, worked-example guide to conducting cost-effectiveness analysis, structured around outcomes, costs, modelling with decision trees and Markov models, and presenting results. Volume 3 in the Handbooks in Health Economic Evaluation series, developed from the University of Oxford course.
BookView source →Representing Uncertainty: The Role of Cost-Effectiveness Acceptability Curves — Elisabeth Fenwick, Karl Claxton and Mark Sculpher, 10(8):779–787 ed., 2001 (Health Economics)
Foundational explanation of cost-effectiveness acceptability curves and their proper role alongside expected net benefit.
Journal ArticleView source →
Frequently Asked Questions (6)
What is the probability of cost-effectiveness?
The likelihood, given uncertainty in the cost and effect estimates, that an intervention's ICER falls below a specified willingness-to-pay threshold.
Source: Fenwick, Claxton & Sculpher 2001
How is the probability of cost-effectiveness calculated?
Every uncertain parameter is assigned a distribution and the model is run many times drawing a value from each, producing a cost and an effect for each option in every simulation. At a given threshold, the proportion of simulations in which an option has the highest net benefit is its probability of being cost-effective. Repeating the calculation across a range of thresholds produces the acceptability curve, which is the standard presentation. The number of simulations should be large enough that the reported figure is stable, and the number used should be stated so a reader can judge that.
Source: Fenwick, Claxton & Sculpher 2001
What does the probability of cost-effectiveness measure?
The chance that an option is the best available choice given the uncertainty the model represented, at a stated willingness to pay. It measures the likelihood of being right rather than the magnitude of being wrong, so it conveys nothing about how much would be lost by choosing the alternative. Two options can carry similar probabilities while the consequences of an incorrect choice differ substantially between them. Reporting it alongside expected net benefit gives both the direction of the recommendation and the confidence attaching to it, which neither figure conveys alone.
Source: Briggs, Claxton & Sculpher 2006
Should the option with the highest probability of cost-effectiveness be chosen?
Not necessarily, and this is the most common misreading. The decision should follow expected net benefit, which is the average across simulations, rather than the frequency of being best. Where outcome distributions are skewed, an option can be optimal in most runs while having lower expected net benefit, because the runs in which it loses are the runs in which it loses heavily. Choosing on probability maximises the chance of being right rather than the expected health produced. The acceptability frontier addresses this by reporting the probability attaching to the option that expected net benefit actually selects rather than to whichever is most frequently best.
Source: Fenwick, O'Brien & Briggs 2004
How does the probability of cost-effectiveness relate to further research?
One minus the probability attaching to the chosen option is the chance the decision is wrong, which is the starting point for asking whether additional evidence would be worthwhile. Combining that chance with the health lost if the decision proves incorrect gives the expected value of removing the uncertainty. A decision with a substantial error probability and large consequences is where research is most likely to repay its cost. A high probability with small consequences attached rarely justifies delaying a decision to gather more evidence. Value of information analysis formalises this comparison rather than leaving it to judgement.
Source: Briggs, Claxton & Sculpher 2006
What limits the probability of cost-effectiveness?
It reflects only the uncertainty that was parameterised, so a model omitting a source produces a reassuring figure, and structural uncertainty about the model itself never appears. It depends entirely on the distributions assigned to inputs, which are frequently chosen for tractability rather than derived from evidence. And with several options the picture is harder to read, since the most probable option changes across the threshold range. Reporting which parameters were and were not made uncertain is therefore as important as reporting the probability itself. Structural uncertainty is normally addressed through scenario analysis instead, which sits outside the probabilistic calculation entirely.
Source: Fenwick, Claxton & Sculpher 2001
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 7 Aug 2025
Content version: 1.0.0
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