Probabilistic Sensitivity Analysis

Probabilistic Sensitivity Analysis in Health Economic Models

Probabilistic sensitivity analysis is the standard way to show how uncertainty about the inputs of a cost-effectiveness model becomes uncertainty about which treatment to fund. Each uncertain input is given a probability distribution, the model is recalculated for a large number of joint draws, and the spread of results gives expected costs and QALYs, the chance that each option is best, and the expected cost of choosing wrongly. The 2026 HEMA report on uncertainty (Steuten and colleagues) recommends probabilistic analysis as the general default for characterising uncertainty in health technology assessment, and the NICE technology appraisal manual (PMG36) asks for the committee's preferred cost-effectiveness estimate to come from a probabilistic analysis when possible unless the model is linear. A probabilistic analysis is only as credible as the distributions and correlations put into it, and some important uncertainties usually sit outside it.

Why decision models are run probabilistically

The HEMA report gives two reasons for the default. Decision models are non-linear in their inputs, so running the model once at the mean of every input does not necessarily give the expected costs and health effects. Averaging the model outcomes over the joint distribution of the inputs does give them, even when decision uncertainty is not the main interest.

The second reason is that a probabilistic analysis brings all the characterised uncertainty together and shows its combined effect on a decision quantity such as incremental net health benefit. It can also estimate the probability that each option has the highest expected net health benefit and the expected consequences of that uncertainty in health or monetary terms. Doubilet and colleagues described the method in 1985 for decision trees, varying every probability and utility at once and recording how often each strategy came out best.

Deterministic sensitivity analysis keeps a narrower role. The HEMA report describes it as useful for understanding how the model behaves, exposing coding and conceptual errors and finding influential inputs, but notes that it cannot quantify decision uncertainty because it varies only selected inputs and misses their joint variation. Varying an input by a conventional amount such as 20% shows how sensitive the model is to that change, not how uncertain the input is, and the ISPOR-SMDM task force on parameter estimation (Briggs and colleagues, 2012) likewise says that arbitrary ranges of this kind should not be used to represent uncertainty.

Running a probabilistic analysis step by step

A probabilistic analysis starts from the deterministic version of the model. The analyst replaces each uncertain point estimate with a distribution that describes what is known about the true mean value, then recalculates the model many times. Each recalculation uses one complete set of draws, applied to every option being compared, so that incremental results keep their natural dependence.

1. List the uncertain parameters. Every input estimated from evidence belongs in the analysis, including those with little evidence behind them.

2. Assign distributions. Each distribution should match the parameter's range and the way it was estimated.

3. Represent correlation. Parameters estimated together are sampled together.

4. Draw one parameter set and run every option. Costs, QALYs and net benefits are stored for each option in that iteration.

5. Repeat until the results are stable. The number of simulations is reported with evidence of stability.

6. Summarise the decision. Expected values, the probability that each option is best and the expected loss are calculated from the stored iterations.

With $N$ iterations, the expected incremental net health benefit and the probability that the new option is cost-effective are estimated as

$$\overline{\Delta NHB}=\frac{1}{N}\sum_{i=1}^{N}\left(\Delta E_i-\frac{\Delta C_i}{\lambda}\right),\qquad \hat{p}=\frac{1}{N}\sum_{i=1}^{N}\mathbf{1}\!\left[\Delta E_i-\frac{\Delta C_i}{\lambda}>0\right]$$

where $\Delta E_i$ and $\Delta C_i$ are the incremental QALYs and incremental cost in iteration $i$, $\lambda$ is the cost-effectiveness threshold in money per QALY, and $\mathbf{1}[\cdot]$ equals 1 when the condition holds and 0 otherwise. The first guides the choice; the second shows how often the new option comes out ahead across the draws.

PMG36 (section 4.7.11) asks for enough simulations to minimise Monte Carlo error and suggests reviewing the variance of net benefits or ICERs as the number of simulations rises, and the HEMA report asks reports to give the number of simulations with evidence that the results are stable. The ISPOR-SMDM task force notes that in patient-level simulations stochastic (first-order) uncertainty between simulated patients, sometimes called Monte Carlo error, has to be eliminated (a different use of the term from PMG36's) before parameter uncertainty can be assessed, whereas cohort models can address parameter uncertainty without that concern.

Choosing distributions and correlations

The HEMA report notes that the choice of distribution is usually the more constrained of the two main specification tasks, because the form follows from the parameter's support and how it was estimated. The usual matches come from the HEMA report, the task force and, for multinomial probabilities, Briggs, Ades and Price (2003).

Correlation is where analyses more often fall short. The HEMA report observes that correlation is estimable when parameters are measured in the same dataset, that published evidence usually supplies only part of it, and that correlations supplied by assumption are elicited quantities that should be reported and justified. PMG36 (section 4.7.19) asks analysts to consider evidence on correlation between parameters and reflect it in the probabilistic analysis. For normally distributed parameters, correlated draws can be generated from the covariance matrix through a Cholesky decomposition.

Two further pitfalls recur in the HEMA report. Cutting the analysis back to a convenient handful of key parameters leaves it materially incomplete, and the standard error of a sample mean taken from one study reflects sampling in that study's setting, not the true uncertainty about the parameter in the population and period the decision covers. The task force adds that parameters should never be left out on the grounds that there is too little information to estimate their uncertainty, and PMG36 (section 4.7.12) states that uncertainty around individual parameters is not in general a reason to exclude them from probabilistic analyses.

Uncertainty that a probabilistic analysis misses

A standard probabilistic analysis covers parameter uncertainty. The HEMA report notes that concerns about bias, transferability, extrapolation, model structure and missing outcomes usually remain outside it, and are most often explored through scenario analysis, which it treats as a form of deterministic analysis because it compares discrete alternatives without weighting them.

The report recommends greater use of model averaging, switch parameters and structured expert elicitation to bring such uncertainty into the probabilistic analysis where feasible, and recommends that every scenario be accompanied by a probabilistic analysis of that scenario and at a minimum the expected value of perfect information. PMG36 (section 4.7.13) similarly expects scenario analyses to be probabilistic in general. The report calls the case where every characterised uncertainty sits in one probabilistic analysis Tier 1 and the common case of partial integration Tier 2, and asks that reports state which parameters, correlations or other sources were left out, because uncertainty about clinical benefit and uncertainty about costs point to different decision options.

Worked example with a skewed survival gain

This illustrative example shrinks a probabilistic analysis to five equally likely draws so that every step can be checked; the numbers do not describe a real technology. A new treatment costs a fixed GBP 15,000 more than usual care. Survival is simplified to a constant annual mortality rate $m$, so mean survival is 1 / $m$ years, and every life-year is valued at 0.5 QALYs, without discounting. Usual care has $m = 0.25$, giving 4 years. The new treatment's rate is uncertain, and the five draws are 0.10, 0.20, 0.20, 0.25 and 0.25, with mean 0.20. The threshold is GBP 25,000 per QALY, the lower value in PMG36.

1. Deterministic result. At the mean rate of 0.20, survival is 1 / 0.20 = 5 years, a gain of 1 year or 0.5 QALYs. The ICER is 15,000 / 0.5 = GBP 30,000 per QALY, above the threshold.

2. Draw by draw. Survival is 10, 5, 5, 4 and 4 years, so the gains are 6, 1, 1, 0 and 0 years, or 3, 0.5, 0.5, 0 and 0 QALYs. The mean gain is 4 / 5 = 0.8 QALYs, larger than the 0.5 at the mean input because survival is non-linear in $m$.

3. Net health benefit. The health forgone is 15,000 / 25,000 = 0.6 QALYs, so the incremental net health benefit in each draw is 2.4, -0.1, -0.1, -0.6 and -0.6 QALYs. Its mean is 1.0 / 5 = 0.2 QALYs, and the probabilistic ICER is 15,000 / 0.8 = GBP 18,750 per QALY.

4. Probability and loss. Only one draw in five favours the new treatment, so the probability that it is cost-effective is 0.2. The expected loss from adopting it is (0.1 + 0.1 + 0.6 + 0.6) / 5 = 0.28 QALYs, against 2.4 / 5 = 0.48 QALYs from keeping usual care, and the difference of 0.2 equals the expected net health benefit.

5. Interpretation. The deterministic analysis rejects the treatment while the probabilistic analysis favours it, because one large gain in a minority of draws raises the mean. The probability of 0.2 on its own would point to usual care, the option with the lower expected net benefit. The expected loss of 0.28 QALYs per patient is the per-patient EVPI and sets an upper limit on what research to resolve the mortality rate could be worth for each patient.

Reading and reporting the outputs

The decision rests on expected values. PMG36 (section 4.2.16) requires reported ICERs to be the ratio of expected additional cost to expected additional QALYs, and asks for expected net health benefits at GBP 25,000 and GBP 35,000 per QALY. The other outputs describe the uncertainty around that choice.

The cost-effectiveness plane shows the joint spread of incremental costs and effects for a pair of options. The cost-effectiveness acceptability curve gives the probability that each option is cost-effective across thresholds, and PMG36 (section 4.7.15) asks for that probability at GBP 25,000 to GBP 35,000 per QALY together with the error probability. The HEMA report notes that the curve does not show the consequences of a wrong choice, so expected loss should accompany it, and that the cost-effectiveness acceptability frontier is needed with several options because the most probable option is not necessarily the one with the highest expected net benefit. It also asks for the deterministic result with an explanation of any material difference.

For HTA bodies the analysis feeds the wider judgement on decision uncertainty. The HEMA report recommends that the population EVPI, a value of information measure, be reported at a minimum and that both the probability and the consequences of a wrong decision inform the decision options, including access as part of further research.

Common misreadings of probabilistic results

Most errors treat the analysis as more decisive than it is. The HEMA report warns that an analysis which integrates some sources of uncertainty while ignoring others can under- or overstate decision uncertainty.

• Reading the probability as the decision rule. The worked example shows a 0.2 probability alongside a positive expected net health benefit.
• Treating more iterations as a cure for poor inputs. The HEMA report says the remedy for weak inputs is better characterisation, not more computation.
• Using arbitrary ranges as distributions. A plus or minus 20% range shows how sensitive the model is to that change but says nothing about how uncertain the input is.
• Assuming the analysis covers structural uncertainty. Model structure, bias and transferability usually sit outside it unless deliberately brought in.

Sources

• Briggs AH, Ades AE, Price MJ. Probabilistic sensitivity analysis for decision trees with multiple branches: use of the Dirichlet distribution in a Bayesian framework. Medical Decision Making. 2003;23(4):341-350. https://doi.org/10.1177/0272989X03255922
• Briggs AH, Weinstein MC, Fenwick EAL, Karnon J, Sculpher MJ, Paltiel AD. Model parameter estimation and uncertainty analysis: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force Working Group-6. Medical Decision Making. 2012;32(5):722-732. https://doi.org/10.1177/0272989X12458348
• Doubilet P, Begg CB, Weinstein MC, Braun P, McNeil BJ. Probabilistic sensitivity analysis using Monte Carlo simulation: a practical approach. Medical Decision Making. 1985;5(2):157-177. https://doi.org/10.1177/0272989X8500500205
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). London: NICE; 2022, last updated 13 October 2026. Sections 4.2.16 and 4.7. https://www.nice.org.uk/process/pmg36
• Steuten L, Thavorn K, Cipriano LE, Guertin JR, McQueen B, Schmidler G, Sullivan S, Wailoo A, Sculpher M. Methods to Analyze and Present Uncertainty in Economic Evaluation to Support HTA Decisions. HEMA Report 2. Health Economics Methods Advisory; 2026. https://hemamethods.org/wp-content/uploads/2026/10/HEMA-Report-2_For-Publication_100726.pdf