Concept Architecture
Concept
Theoretically, Distribution Fitting is the process of selecting and parameterising a probability distribution that best represents observed data. It provides a mathematical description of uncertainty and variability in model inputs or outcomes and is fundamental to statistical inference and stochastic modelling. In health economics, distribution fitting is used to characterise costs, utilities, survival times, treatment effects and epidemiological parameters for probabilistic sensitivity analysis and decision modelling.
Mathematically, Distribution Fitting estimates the parameters of candidate probability distributions using recognised statistical methods, most commonly maximum likelihood estimation or the method of moments. Competing distributions are compared using goodness-of-fit statistics and information criteria, such as the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC), to identify the distribution that most appropriately represents the observed data while balancing model fit and complexity.
In practice, Distribution Fitting is performed using patient-level trial data, observational datasets, disease registries or published summary statistics. Appropriate distributions are selected according to the characteristics of the variable being modelled, for example beta distributions for probabilities and utilities, gamma or log-normal distributions for costs, and Weibull or Gompertz distributions for survival data. The fitted distributions are subsequently used to generate parameter values during probabilistic sensitivity analysis and simulation modelling.
Purpose
Used to estimate probability distributions that accurately represent uncertain model parameters, enabling realistic stochastic simulation and probabilistic sensitivity analysis in health economic evaluation.
Mathematical Formulae
Primary Formula
Maximum likelihood estimation:
?? = arg max??? ????� f(x?; ?)
or equivalently,
?? = arg max??? ????� ln f(x?; ?)
where:
- ?? is the estimated parameter vector
- f(x?; ?) is the probability density or mass function
- n is the sample size.
Supporting Formulae
Akaike Information Criterion:
AIC = 2k ? 2 ln(L)
Bayesian Information Criterion:
BIC = k ln(n) ? 2 ln(L)
where:
- k is the number of estimated parameters
- L is the maximised likelihood
- n is the sample size.
Related Mathematical Methods
- Maximum likelihood estimation
- Method of moments
- Goodness-of-fit testing
- Akaike Information Criterion
- Bayesian Information Criterion
- Probability distribution modelling
- Probabilistic sensitivity analysis
Example
A health economic evaluation requires modelling annual inpatient costs for patients with chronic heart failure.
Observed cost data from 2,500 patients are fitted to several candidate distributions using maximum likelihood estimation.
Model comparison produces:
| Distribution | AIC |
|---|---|
| Gamma | 18,240 |
| Log-normal | 18,312 |
| Weibull | 18,486 |
The gamma distribution has the lowest AIC and is therefore selected for probabilistic sensitivity analysis. Random samples are subsequently drawn from the fitted gamma distribution during Monte Carlo simulation.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| AVERAGE | =AVERAGE(B2:B1001) | Estimate the sample mean for parameter estimation. |
| STDEV.S | =STDEV.S(B2:B1001) | Estimate sample variability during distribution fitting. |
| GAMMA.DIST | =GAMMA.DIST(B2,Shape,Scale,FALSE) | Evaluate the fitted gamma probability density. |
| NORM.DIST | =NORM.DIST(B2,Mean,SD,FALSE) | Evaluate the fitted normal distribution. |
| LOGNORM.DIST | =LOGNORM.DIST(B2,MeanLog,SDLog,FALSE) | Evaluate the fitted log-normal distribution. |
VBA (Optional)
Automate fitting of multiple candidate probability distributions, compare goodness-of-fit statistics and select the optimal distribution for probabilistic sensitivity analysis.
Sources
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press; 2006.
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. 4th ed. Oxford University Press; 2015.
- Briggs AH, Weinstein MC, Fenwick EAL, et al. Model parameter estimation and uncertainty analysis: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force Working Group-6. Value in Health. 2012;15(6):835?842.
- Burnham KP, Anderson DR. Model Selection and Multimodel Inference. 2nd ed. Springer; 2002.
- National Institute for Health and Care Excellence (NICE). Health Technology Evaluation Manual. Latest edition.
Related Concepts (4)
Library
Publications
1
Parameter Estimation and Uncertainty: A Report of the ISPOR-SMDM Modeling Good Research Practices Task Force-6 — Briggs, Weinstein, Fenwick, Karnon, Sculpher & Paltiel, Task Force Report 6 ed., 2012 (Value in Health / Medical Decision Making)
Best-practice guidance on parameter estimation and the characterisation of uncertainty in decision models, covering probabilistic sensitivity analysis, distributional choices, and correlation between parameters.
Journal ArticleView source →
Frequently Asked Questions (6)
What is distribution fitting?
The statistical process of identifying which probability distribution and parameters best represent the observed variation or uncertainty in a model input.
Source: Vose 2008
Why must a fitted distribution respect a parameter's natural range?
Many model inputs can only take certain values, and the distribution chosen to represent their uncertainty must respect that. A probability lies between zero and one, so a distribution bounded to that interval is used for it, while a cost cannot be negative, so a distribution confined to positive values is chosen. Fitting a distribution that allows impossible values, such as a negative cost or a probability above one, would let the model sample nonsense. Matching the distribution to the parameter's range is therefore a basic requirement. Briggs and colleagues (2006) set out these choices.
Source: Briggs et al. 2006
How is distribution fitting done?
Distribution fitting is done by selecting a candidate probability distribution suited to the nature of the parameter, estimating its parameters from data, often by methods such as maximum likelihood, and assessing how well it fits, using graphical checks and goodness-of-fit tests. Several distributions may be compared, and the one that best represents the data, while respecting the parameter's constraints, is chosen. The result is a fitted distribution describing the parameter's uncertainty, ready for use in sampling.
Source: Vose 2008
Why does the choice of distribution matter?
The choice of distribution matters because it determines how a parameter's uncertainty is represented, including its range, shape, and the plausibility of extreme values, which affects the propagated uncertainty in the results. A distribution must respect the parameter's constraints, for instance being bounded between zero and one for a probability or non-negative for a cost. An inappropriate distribution can misrepresent the uncertainty or allow impossible values, biasing the analysis, so the distribution is chosen to suit the parameter and its evidence.
Source: Briggs, Claxton & Sculpher 2006
How are distributions chosen for different parameter types?
Distributions are chosen to suit the type and constraints of each parameter: beta distributions for probabilities and utilities bounded between zero and one; gamma or log-normal distributions for costs, which are non-negative and often skewed; and normal or log-normal distributions for relative effects such as log hazard ratios. The distribution reflects the parameter's plausible range and shape. Choosing a distribution appropriate to the parameter ensures that sampled values are plausible and that the uncertainty is represented correctly for probabilistic analysis.
Source: Briggs, Claxton & Sculpher 2006
What are the challenges of distribution fitting?
Distribution fitting is challenging when data are limited, so the fitted distribution is itself uncertain and several distributions may fit similarly while implying different tails and extreme values. Choosing among them involves judgement, and a poorly chosen distribution can distort the represented uncertainty. Where data are sparse, distributions may rest on assumption. These challenges mean fitted distributions are examined for plausibility and their influence tested, since the representation of parameter uncertainty affects the results of probabilistic analysis.
Source: Vose 2008
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 9 Oct 2025
Content version: 1.0.0
Canonical Identity
- Term code
- HE-EM-MP-011
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