Parameter Uncertainty

Parameter Uncertainty in Health Economic Models

Every probability, cost, utility and treatment effect in a decision model is an estimate, and parameter uncertainty is the doubt about their true values that those estimates carry into the model's results. It is the form of uncertainty that analysts most often quantify, by giving each input a probability distribution and sampling all of them together in a probabilistic sensitivity analysis. Because most decision models are nonlinear in their inputs, expected costs and QALYs generally need the inputs to be averaged over those distributions, and the chance of a wrong decision and the value of further research come from them too. The HEMA Report 2 by Steuten and colleagues warns that such an analysis is only as credible as the distributions and dependencies it encodes.

Where parameter uncertainty comes from

The ISPOR-SMDM task force on parameter estimation (Briggs and colleagues) explains that the probabilities governing outcomes are themselves uncertain because they are estimated. Sample size and variance in the data drive part of it, and the task force adds conflicting studies, weak internal or external validity, limited generalisability and a lack of empirical data.

The HEMA report separates sampling uncertainty, which affects precision, from bias, which affects validity, so an estimate can be precise but biased. Its table of evidence situations shows how far each is parameterised in current practice.

The boundary with structural uncertainty depends on how the model is specified. The HEMA report notes that waning of a treatment effect can be written as a waning-rate parameter and the choice between survival curves as weights on each. It asks analysts to state what is uncertain, whether it has been parameterised and, if not, how it has been characterised.

Parameter uncertainty, stochastic uncertainty and heterogeneity

The task force distinguishes parameter (second-order) uncertainty from stochastic (first-order) uncertainty, the random variation in outcomes between identical patients, and from heterogeneity, the variation explained by patient characteristics. It compares the first distinction to that between a standard error, which describes the precision of an estimate, and a standard deviation, which describes how individuals vary. More data shrink parameter uncertainty but leave stochastic variation in place. In patient-level simulations stochastic uncertainty must be eliminated before parameter uncertainty can be assessed, a problem cohort models avoid. The wider typology is set out under uncertainty.

Choosing a distribution for each input

The task force recommends that distributions follow standard statistical methods for the way the parameter was estimated, such as a beta distribution for binomial data. The usual match for each type of input is tabulated under probabilistic sensitivity analysis, including the Dirichlet distribution that Briggs, Ades and Price propose for several mutually exclusive events so that sampled probabilities still sum to one.

The task force favours continuous distributions over the parameter's theoretical range and questions convenient but implausible forms such as the triangular. The NICE manual (PMG36, section 4.7.10) states that distributions for probabilistic analysis should not be chosen arbitrarily but should represent the evidence and be justified. Where data are lacking it points to formal elicitation, and both sources reject leaving an input fixed because its uncertainty is hard to estimate.

Published evidence often reports a mean and standard error, which can be converted to distribution parameters by the method of moments. For a beta distribution the shape parameters are found as follows.

$$\alpha = \mu\left(\frac{\mu(1-\mu)}{s^{2}} - 1\right), \qquad \beta = (1-\mu)\left(\frac{\mu(1-\mu)}{s^{2}} - 1\right)$$

where $\mu$ is the reported mean, $s$ its standard error, and $\alpha$ and $\beta$ the two shape parameters. The formula only works when $s^{2}$ is smaller than $\mu(1-\mu)$. For a gamma distribution the matching values are shown next.

$$k = \frac{\mu^{2}}{s^{2}}, \qquad \theta = \frac{s^{2}}{\mu}$$

where $k$ is the shape parameter and $\theta$ the scale parameter. Some software uses the rate $\theta^{-1}$ instead, so the parameterisation should be recorded.

Correlation between inputs

Inputs estimated together, such as regression coefficients, have direct evidence on their correlation, which the task force says should be reflected through the variance-covariance matrix. Its example concerns progression probabilities with and without treatment, which share uncertainty about natural history and so cannot be sampled independently. The recommended fix is to give distributions to the baseline probability and the relative risk, which can plausibly be independent, and derive the on-treatment probability as their product.

The HEMA report finds that correlations between inputs are routinely ignored, for instance between progression-free and overall survival, and its advice on reporting assumed correlations is covered with PMG36 (section 4.7.19) under probabilistic sensitivity analysis.

How each type of evidence carries uncertainty

The HEMA report finds that sampling uncertainty from an indirect comparison can usually enter the probabilistic analysis, but uncertainty from assumptions about exchangeability or population adjustment may stay outside it. Baseline risks from large registries may have very small standard errors, which do not show that the estimates apply to the decision population.

Utilities estimated from patient or general-population samples carry sampling uncertainty. The value sets that turn health states into utilities are themselves estimated, and mapping between measures adds more. For costs, the report separates resource use, which can be given distributions and sampled jointly with the clinical events that generate it, from unit costs taken from fixed tariffs or schedules. Those have no sampling distribution and may be better examined through scenarios.

A recurring pitfall, according to the report, is to use the standard error of a mean taken from a single study. That figure reflects sampling in that study's setting, not the uncertainty about the mean for the population, place and period to which the decision applies. The task force likewise warns that relying on one study is likely to understate uncertainty.

Propagating parameter uncertainty through the model

Probabilistic analysis draws every input together many times, and PMG36 (section 4.7.12) expects the committee's preferred estimate to come from it when possible unless the model is linear. Its practice, and the narrower role of deterministic sensitivity analysis, are covered under probabilistic sensitivity analysis.

Worked example with a probability, a rate and net benefit

The numbers in this example are illustrative. The three steps are separate illustrations, and the threshold is GBP 25,000 per QALY.

1. Response probability. A trial reports 30 responders among 120 patients, so the input is given a Beta(30, 90) distribution. Its mean is 30 / 120 = 0.25 and its variance is (30 × 90) / (120² × 121) = 0.00155, a standard deviation of 0.039. Feeding this mean and standard deviation back into the moment formulas returns 30 and 90.

2. A nonlinear output. An annual mortality rate has mean 0.10 and standard error 0.02. The gamma parameters are k = 0.01 / 0.0004 = 25 and θ = 0.0004 / 0.10 = 0.004. With a constant rate, life expectancy is the reciprocal of the rate, which is 10 years at the mean rate. Averaged over the gamma distribution, the expected reciprocal is 1 / ((k − 1)θ) = 1 / (24 × 0.004) = 10.42 years, so a run at the mean input understates expected survival by about 0.42 years.

3. Net benefit and correlation. Suppose incremental QALYs have mean 0.50 with standard error 0.05 and incremental cost has mean GBP 10,000 with standard error GBP 2,000. Mean incremental net monetary benefit is 25,000 × 0.50 − 10,000 = GBP 2,500, and its variance follows from the formula below.

$$\operatorname{Var}(\mathrm{INMB}) = \lambda^{2}\operatorname{Var}(\Delta E) + \operatorname{Var}(\Delta C) - 2\lambda\operatorname{Cov}(\Delta E, \Delta C)$$

where $\lambda$ is the threshold, $\Delta E$ the incremental QALYs and $\Delta C$ the incremental cost. Treating the two as independent gives 1,562,500 + 4,000,000 = 5,562,500, a standard deviation of about GBP 2,358. Under a normal approximation the probability that INMB is positive is Φ(2,500 / 2,358) = Φ(1.06), about 0.855. With a correlation of 0.5 the covariance is 0.5 × 0.05 × 2,000 = 50, so the variance falls by 2 × 25,000 × 50 = 2,500,000 to 3,062,500. The standard deviation is then GBP 1,750 and the probability is Φ(1.43), about 0.923.

4. Interpretation. The beta and gamma forms keep each input within its natural range, and the second step shows why a run at mean values can misstate expected outcomes. The last step shows that ignoring a positive correlation between costs and effects here nearly doubles the apparent chance of a wrong decision, from about 7.7% to about 14.5%.

Using parameter uncertainty in HTA decisions

Probabilistic output supports the cost-effectiveness plane, cost-effectiveness acceptability curves and value of information analysis, including the EVPI and EVPPI. PMG36 (section 4.7.15) asks submissions to present the probability that a treatment is cost effective at GBP 25,000 to GBP 35,000 per QALY gained, and the error probability. These summaries describe decision uncertainty, which PMG36 (section 4.7.2) defines as the probability that a different decision would be reached if the true cost effectiveness of each technology were known.

PMG36 separates uncertainty in the choice of data sources (section 4.7.8), explored with alternative data sets, from uncertainty in parameter precision (section 4.7.10), represented by distributions. The HEMA report also asks assessment reports to name the parameters and correlations left out of the probabilistic analysis, since uncertainty about benefit and about cost point to different decision options.

Limitations and common misreadings

A standard error describes sampling uncertainty under the source study's assumptions. The HEMA report stresses that it does not show an estimate is unbiased, relevant to the decision population or transferable, so a precise input can still be highly uncertain for the decision. A narrow distribution elicited from experts shows that they were confident and in agreement, which is not the same as the quantity being known precisely.

Pruning a probabilistic analysis to a handful of convenient inputs leaves it materially incomplete. The HEMA report also warns that one evidence limitation can be counted several times, for instance in a cautious base case and again in a wider distribution. The task force notes that any representation of uncertainty in a model is conditional on its structural assumptions, so a probabilistic analysis of parameters says nothing about uncertainty in the structure itself.

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; ISPOR-SMDM Modeling Good Research Practices Task Force. 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
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36), section 4.7 Exploring uncertainty. London: NICE; 2022, last updated 13 October 2026. https://www.nice.org.uk/process/pmg36/chapter/economic-evaluation-2
• 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