Probability of Cost-Effectiveness

Probability of Cost-Effectiveness: Concept Architecture

Orientation and learning roadmap

This page begins with a specified comparator and willingness-to-pay threshold, then counts the model draws in which an intervention's incremental net benefit is positive. It uses a small worked example to separate that probability from the strategy with the highest expected net benefit and closes with interpretation and reporting safeguards.

Define the decision comparison

For two strategies evaluated over the same population, horizon, and perspective, let $\Delta Q$ be incremental quality-adjusted life-years (QALYs) and $\Delta C$ incremental cost for the intervention minus comparator. At willingness to pay $\lambda$ per QALY, incremental net monetary benefit (INMB) is $\lambda\Delta Q-\Delta C$.

The probability in this setting is the proportion of the specified joint uncertainty distribution for which INMB is positive. Its meaning depends on the evidence, parameter distributions, correlations, structural assumptions, comparator, and threshold; it is not an unconditional probability that a technology is good. If values can tie exactly, the analysis must state whether it counts $\mathrm{INMB}=0$ as favourable, unfavourable, or a tie.

Calculate it from probabilistic draws

Probabilistic sensitivity analysis (PSA) draws uncertain parameters, calculates both strategies' costs and QALYs for each draw, and forms their paired difference. The estimated two-strategy probability at threshold $\lambda$ is the fraction of draws for which the incremental net benefit meets the stated criterion.

$$\widehat{P}_{\mathrm{CE}}(\lambda) =\frac{1}{N}\sum_{r=1}^{N} \mathbf 1\!\left(\lambda\Delta Q_r-\Delta C_r>0\right).$$

Here $N$ is the number of appropriately generated joint draws and $\mathbf 1(\cdot)$ equals one when the event is true and zero otherwise. Repeating the calculation across thresholds gives a cost-effectiveness acceptability curve (CEAC) for the specified comparison. An observed fraction also has finite-simulation Monte Carlo error, which should be small enough for the intended level of reported precision.

Worked example: probability versus expected value

Suppose a deliberately tiny teaching PSA compares an intervention with usual care at $\lambda=£20{,}000$ per QALY, and every draw yields $\Delta Q=0.05$ QALYs. Incremental costs vary across five equally weighted draws, so the health benefit is worth £1,000 in net-benefit terms for each draw.

Three of five draws favour the intervention, so the illustrative estimated probability is $3/5=0.60$. Yet mean INMB is $[3(100)+2(-1{,}500)]/5=-£540$, which favours usual care under a rule choosing the highest expected net benefit. Five draws are far too few for a stable policy estimate; their purpose is to show why the strategy most likely to win in individual draws need not maximise expected value.

More than two strategies

With several options, “cost effective” can mean that one option has the highest net benefit among all strategies in a draw. For strategy $j$, calculate $\Pr\{NB_j(\lambda)=\max_k NB_k(\lambda)\}$ using a clearly specified tie rule, rather than comparing it only with one convenient baseline.

Individual CEACs report how often each strategy ranks first over joint draws, while the cost-effectiveness acceptability frontier highlights the probability associated with the strategy that has the highest expected net benefit at each threshold. A strategy can have a high probability of being optimal without having the greatest expected net benefit, as the worked two-strategy example demonstrates. Neither graphic states the expected size of the gain or the expected loss if the wrong decision is made.

Interpretation and decision limitations

The probability is conditional on the model and evidence, including the way structural uncertainty and parameter correlation were handled. More simulation runs reduce numerical noise but cannot make an implausible survival extrapolation, utility mapping, or cost estimate reliable.

Decision-makers may consider expected net benefit, expected value of information, budget impact, equity, implementation, and risk of large losses in addition to the CEAC. A 60% probability is not the same as a 60% treatment success rate for an individual patient, nor does it mean that a threshold is itself uncertain. If the decision threshold changes, recompute the net-benefit comparison with that threshold rather than relabelling a result from another value.

Reporting for health technology assessment

Report the threshold range, strategies, comparator, perspective, horizon, PSA input distributions, joint sampling approach, and number of draws. Display or describe incremental costs and QALYs, expected net benefit, the acceptability curve, and the frontier when multiple choices make that comparison useful.

If uncertainty was omitted for important parameters or correlations, state that limitation alongside the probability. Outcomes such as treatment effectiveness can also be heterogeneous across patients; a population-average model probability does not substitute for subgroup-specific evaluation when decisions differ by subgroup. Rounding should reflect Monte Carlo precision rather than suggesting a level of certainty unsupported by the simulation.

Sources and further learning

The foundational CEAC paper explains the interpretation of acceptability curves, and the methodological guidance explains how probabilities fit into decisions made under parameter uncertainty. Institutional guidance adds reporting expectations for acceptability frontiers and incremental analysis.

• Fenwick, O'Brien, and Briggs (2004), “Cost-Effectiveness Acceptability Curves: Facts, Fallacies and Frequently Asked Questions”, Health Economics, 13:405–415. This addresses CEAC construction and common misinterpretations.
• ISPOR-SMDM Modeling Good Research Practices Task Force (2012), “Model Parameter Estimation and Uncertainty Analysis”, Value in Health. This places CEACs alongside other decision-uncertainty summaries.
• NICE, NICE technology appraisal and highly specialised technologies guidance: the manual, economic evaluation. This addresses CEAC presentation and the frontier.