Concept Architecture
Concept
Theoretically, Monte Carlo Simulation is a stochastic computational method that estimates the distribution of model outcomes by repeatedly sampling uncertain input variables from specified probability distributions. It provides a numerical approach for evaluating systems that are analytically intractable due to uncertainty or complexity. In health economics, Monte Carlo simulation underpins probabilistic decision modelling by propagating parameter uncertainty, patient variability or stochastic processes through economic models.
Mathematically, Monte Carlo Simulation is represented by repeated random sampling from probability distributions, with each simulation producing one realisation of the model. Expected outcomes are estimated by averaging results across a large number of independent iterations, while uncertainty is characterised by the empirical distribution of the simulated outputs. There is no universally recognised canonical mathematical formula.
In practice, Monte Carlo Simulation is implemented by specifying probability distributions for uncertain model parameters or individual events, generating pseudo-random samples, evaluating the model for each iteration and summarising the resulting distribution of costs, health outcomes and cost-effectiveness measures. It is used in probabilistic sensitivity analysis, individual patient simulation, microsimulation and discrete event simulation to quantify decision uncertainty.
Purpose
Used to estimate expected outcomes and quantify uncertainty in health economic models by repeatedly evaluating the model using random samples drawn from specified probability distributions.
Mathematical Formulae
Primary Formula
Monte Carlo estimator of an expected value:
E(Y) � (1/N) ????? Y?
where:
- Y? is the model outcome from simulation i
- N is the number of simulation iterations.
Supporting Formulae
Monte Carlo variance estimator:
V?ar(Y) = (1/(N ? 1)) ????? (Y? ? ?)�
Monte Carlo standard error:
SE(?) = �(V?ar(Y) / N)
Related Mathematical Methods
- Monte Carlo integration
- Random sampling
- Pseudo-random number generation
- Probability distributions
- Probabilistic sensitivity analysis
- Microsimulation
- Individual patient simulation
- Discrete event simulation
Example
A probabilistic sensitivity analysis evaluates a new oncology treatment. The treatment effect is modelled using a log-normal distribution, utility values follow a beta distribution and costs follow a gamma distribution. The model is evaluated over 10,000 Monte Carlo iterations.
The simulated outcomes are:
- Mean incremental cost = �2,150
- Mean incremental QALYs = 0.18
The estimated ICER is:
ICER = 2150 / 0.18 = �11,944 per QALY
Across the 10,000 simulations, 94% of iterations fall below a willingness-to-pay threshold of �20,000 per QALY, indicating a high probability that the intervention is cost effective.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| RAND | =RAND() | Generate pseudo-random numbers for simulation. |
| NORM.INV | =NORM.INV(RAND(),Mean,SD) | Sample normally distributed parameters. |
| GAMMA.INV | =GAMMA.INV(RAND(),Shape,Scale) | Sample positively skewed cost distributions. |
| BETA.INV | =BETA.INV(RAND(),Alpha,Beta) | Sample utility values or probabilities. |
| AVERAGE | =AVERAGE(Results) | Estimate expected model outcomes. |
| STDEV.S | =STDEV.S(Results) | Estimate simulation variability. |
VBA (Optional)
Automate thousands of Monte Carlo iterations, recalculate the economic model and summarise the resulting distributions of costs, health outcomes and cost-effectiveness estimates.
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.
- Robert CP, Casella G. Monte Carlo Statistical Methods. 2nd ed. Springer; 2004.
- ISPOR-SMDM Modeling Good Research Practices Task Force. Modeling Good Research Practices. Value in Health. 2012.
- National Institute for Health and Care Excellence (NICE). Health Technology Evaluation Manual. Latest edition.
Related Concepts (4)
Library
Publications
3
Microsimulation Modeling for Health Decision Sciences Using R: A Tutorial — Krijkamp, Alarid-Escudero, Enns, Jalal, Hunink & Pechlivanoglou, Vol. 38, No. 3 ed., 2018 (Medical Decision Making)
The DARTH workgroup’s tutorial on implementing individual-level (microsimulation) models in R, including vectorisation techniques that dramatically reduce run time — the standard reference for patient-level simulation in R.
Journal ArticleView source →Markov Chain Monte Carlo in Practice — W. R. Gilks, S. Richardson & D. J. Spiegelhalter (editors), 1st Edition ed., 1996 (Chapman & Hall)
A foundational collection on Markov chain Monte Carlo methodology and practice, linking stochastic simulation with Bayesian inference and Gibbs sampling.
BookView source →The BUGS Book: A Practical Introduction to Bayesian Analysis — David Lunn, Chris Jackson, Nicky Best, Andrew Thomas & David Spiegelhalter, 1st Edition ed., 2012 (Chapman & Hall / CRC Press)
A practical guide to Bayesian modeling with BUGS, covering prior specification, posterior simulation, model checking and Markov chain Monte Carlo computation.
BookView source →
Frequently Asked Questions (6)
What is Monte Carlo simulation?
A technique estimating a model's output under uncertainty by repeatedly sampling random values for its inputs from specified distributions across many iterations.
Source: Metropolis & Ulam 1949
Where does the name Monte Carlo simulation come from?
The method is named after the casino district of Monaco, a nod to the games of chance whose randomness it imitates. It was developed by physicists in the 1940s, among them Stanislaw Ulam and John von Neumann, who used repeated random sampling to solve problems too complex for direct calculation. Drawing many random values and averaging the results lets the method estimate quantities that resist exact solution. The gambling allusion captures its reliance on chance. Metropolis and Ulam (1949) introduced the approach.
Source: Metropolis & Ulam 1949
How does Monte Carlo simulation work?
Monte Carlo simulation works by specifying probability distributions for the uncertain inputs, then repeatedly drawing a random value for each input from its distribution, running the model with those values, and recording the output, over many iterations. The collection of outputs forms a distribution reflecting the combined effect of the input uncertainty. Summarising this distribution, by its mean, spread, or probabilities, gives the estimate and its uncertainty. Many iterations are used so the sampled distribution stabilises and approximates the true output distribution.
Source: Metropolis & Ulam 1949
Why is Monte Carlo simulation used?
Monte Carlo simulation is used to propagate uncertainty through a model, showing how uncertainty in the inputs translates into uncertainty in the output, which is often impossible to derive analytically for complex models. It is central to probabilistic sensitivity analysis in health economics, where parameter distributions are sampled to estimate the uncertainty in cost-effectiveness results. It is also used to approximate integrals and expectations that cannot be computed directly. Its generality makes it a standard tool for handling uncertainty and complex calculations.
Source: Metropolis & Ulam 1949
How is Monte Carlo simulation used in health economics?
In health economics, Monte Carlo simulation underlies probabilistic sensitivity analysis: distributions are assigned to the uncertain parameters of a cost-effectiveness model, and the model is run many times with parameters sampled from these distributions, producing a distribution of costs and effects. This shows the uncertainty in the results, supporting outputs such as the cost-effectiveness plane and acceptability curve and the probability that an option is cost-effective. It is also used in individual simulation to sample patient events, so it is a core computational tool in the field.
Source: Metropolis & Ulam 1949
What determines the accuracy of Monte Carlo simulation?
The accuracy of Monte Carlo simulation depends chiefly on the number of iterations, since the estimates are based on sampled values and their precision improves as more samples are drawn, with the sampling error decreasing as the number of iterations rises. It also depends on the inputs being sampled from appropriate distributions. Enough iterations must be run for the results to stabilise, so that the Monte Carlo error is small relative to the quantities of interest, which is checked by whether results settle as iterations increase.
Source: Metropolis & Ulam 1949
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Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 8 Oct 2025
Content version: 1.0.0
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