Concept Architecture
Concept
Theoretically, Parameter Distribution is a probability distribution assigned to an uncertain model parameter to represent the range of plausible values that the parameter may assume and the probability associated with each value. It provides the mathematical framework for representing parameter uncertainty arising from sampling variation, observational evidence or expert judgement. In health economics, parameter distributions are fundamental to probabilistic sensitivity analysis because they enable uncertainty to be propagated through decision models.
Mathematically, a parameter distribution is defined by an appropriate probability distribution selected according to the characteristics of the parameter being modelled. Continuous parameters may follow normal, gamma or log-normal distributions, probabilities commonly follow beta or Dirichlet distributions, and count data may follow Poisson distributions. Distribution parameters are estimated from observed data, summary statistics or Bayesian posterior distributions.
In practice, parameter distributions are specified using estimates obtained from clinical trials, observational studies, meta-analyses, registries or expert elicitation. During probabilistic sensitivity analysis, random values are repeatedly sampled from these distributions and propagated through decision trees, Markov models and microsimulation models to quantify decision uncertainty.
Purpose
Used to represent uncertainty in model parameters by assigning appropriate probability distributions for probabilistic sensitivity analysis and simulation-based health economic modelling.
Mathematical Formulae
Primary Formula
There is no universally recognised canonical mathematical formula.
Supporting Formulae
Normal distribution:
X ~ N(?, ��)
Beta distribution:
X ~ Beta(�, ?)
Gamma distribution:
X ~ Gamma(�, ?)
Log-normal distribution:
X ~ LogNormal(?, ��)
Related Mathematical Methods
- Probability Distribution
- Beta Distribution
- Gamma Distribution
- Normal Distribution
- Dirichlet Distribution
- Monte Carlo Simulation
- Probabilistic Sensitivity Analysis
Example
A health economic model assigns a beta distribution to the annual probability of disease progression because the parameter is bounded between 0 and 1. The probability is estimated as 0.25 using Beta(30,90), while treatment costs are assigned a gamma distribution and treatment effects a normal distribution. During probabilistic sensitivity analysis, values are sampled from each parameter distribution for every simulation, allowing uncertainty to propagate through the model.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| NORM.INV | =NORM.INV(RAND(),Mean,SD) | Sample normally distributed treatment effects or regression coefficients. |
| BETA.INV | =BETA.INV(RAND(),Alpha,Beta) | Sample probabilities or utility values bounded between 0 and 1. |
| GAMMA.INV | =GAMMA.INV(RAND(),Alpha,Beta) | Sample positively skewed cost parameters. |
| LOGNORM.INV | =LOGNORM.INV(RAND(),Mean,SD) | Sample positively skewed continuous parameters such as relative risks or costs. |
VBA (Optional)
VBA can automate repeated sampling from multiple parameter distributions during probabilistic sensitivity analysis and export simulation results for decision uncertainty analyses.
Sources
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. Oxford University Press.
- 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. Medical Decision Making. 2012;32(5):722?732.
- NICE. NICE Health Technology Evaluations: The Manual.
Related Concepts (3)
Library
Tools & Resources
1
SAVI — Sheffield Accelerated Value of Information — Mark Strong, Jeremy Oakley & Penny Breeze (University of Sheffield), Web application ed., 2024 (University of Sheffield)
A free, open-access web calculator that computes value-of-information measures (EVPI, partial EVPI/EVPPI and EVSI) directly from a model’s probabilistic sensitivity analysis output — no need to re-run the model. Also reports payer strategy-specific and uncertainty burden.
Web Tool (R Shiny)View source →
Frequently Asked Questions (6)
What is a parameter distribution?
The specific probability distribution, such as a beta, gamma, or normal distribution, assigned to represent uncertainty in a given input parameter.
Source: Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press; 2006. doi:10.1093/oso/9780198526629.001.0001.
Why is the shape of a parameter's distribution matched to what it represents?
Each uncertain input is given a distribution whose shape and range suit the quantity it stands for, so the choice follows from the parameter's nature. A probability, bounded between zero and one, is given a beta distribution, a cost, positive and skewed, a gamma or log-normal, and a relative effect on the log scale, a normal. Matching the distribution to the parameter's permissible values ensures every sampled draw is possible and the uncertainty is represented faithfully. A mismatched distribution would sample impossible values. Briggs and colleagues (2006) set out these choices.
Source: Briggs et al. 2006
How is a parameter distribution chosen?
A parameter distribution is chosen to match the nature of the parameter and the evidence about it: a beta distribution for a probability bounded between zero and one, a gamma or log-normal for a cost that cannot be negative and may be skewed, a Dirichlet for a set of probabilities summing to one, and a normal distribution for an unbounded, symmetric quantity such as a log-scale estimate. The distribution's parameters are set from the data or estimates, so the choice reflects both the parameter's constraints and its uncertainty.
Source: Vose 2008
Why does the choice of parameter distribution matter?
The choice of parameter distribution matters because it determines the range and shape of sampled values and hence how uncertainty propagates to the result: an inappropriate choice can produce implausible values, such as negative costs or probabilities outside zero to one, or misrepresent skewness, distorting the output uncertainty. Matching the distribution to the parameter's bounds and shape keeps the sampled values plausible and the uncertainty realistic. So the parameter distribution is chosen carefully, since it directly affects the credibility of the probabilistic analysis.
Source: Briggs, Claxton & Sculpher 2006
What distributions are commonly used for parameters?
Distributions commonly used for parameters include the beta distribution for probabilities and utilities bounded between zero and one; the gamma and log-normal distributions for costs, which are non-negative and often right-skewed; the Dirichlet distribution for sets of transition probabilities summing to one; the normal distribution for unbounded symmetric quantities such as log hazard ratios; and the log-normal for relative risks. Each is chosen to respect the parameter's range and typical shape, so that the sampled values and the represented uncertainty are appropriate to the input.
Source: Briggs, Claxton & Sculpher 2006
How are the parameters of a parameter distribution set?
The parameters of a parameter distribution are set from the evidence about the input, so that the distribution reflects both the estimate and its uncertainty: for a probability, a beta distribution's parameters can come from the counts of events and non-events; for a cost, a gamma distribution's parameters from the estimated mean and standard error; and for a normal distribution, the mean and standard error of the estimate. Using the underlying data or estimates ensures the distribution captures the parameter's central value and the precision with which it is known.
Source: Vose 2008
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 29 Oct 2025
Content version: 1.0.0
Canonical Identity
- Term code
- HE-EM-UA-050
Stable URI · Machine-readable · Resolvable · CC BY 4.0