Concept Architecture
Concept
Theoretically, Monte Carlo Integration is a numerical integration method that estimates the value of definite integrals using random sampling. Rather than evaluating an integral analytically or through deterministic numerical methods, it approximates the expected value of a function by averaging randomly generated observations. In health economics, Monte Carlo integration underpins probabilistic sensitivity analysis, Bayesian inference and simulation-based estimation of expected costs, health outcomes and cost-effectiveness.
Mathematically, Monte Carlo integration estimates an integral by generating independent random samples from a probability distribution and calculating the sample mean of the integrand. The estimator is unbiased under appropriate sampling assumptions and converges to the true integral according to the Law of Large Numbers, with estimation error decreasing at a rate proportional to the inverse square root of the number of simulations.
In practice, Monte Carlo integration is implemented by repeatedly sampling uncertain model parameters from their probability distributions, evaluating the model for each sample and averaging the resulting outcomes. It is routinely applied in decision trees, Markov models, microsimulation, Bayesian computation and probabilistic sensitivity analysis to estimate expected costs, QALYs, net monetary benefit and decision uncertainty.
Purpose
Used to numerically estimate expected values and definite integrals through random sampling, supporting probabilistic sensitivity analysis and simulation-based health economic modelling.
Mathematical Formulae
Primary Formula
I = ?f(x)p(x)dx � (1/n) ? ?f(x?)
where:
I = expected value or integral
x? ~ p(x)
n = number of simulation samples
Supporting Formulae
Monte Carlo standard error:
SE = s / �n
Monte Carlo convergence:
Error ? 1 / �n
Related Mathematical Methods
- Monte Carlo Simulation
- Importance Sampling
- Latin Hypercube Sampling
- Probabilistic Sensitivity Analysis
- Bayesian Analysis
- Numerical Integration
Example
A probabilistic sensitivity analysis estimates the expected lifetime cost of a treatment by repeatedly sampling uncertain costs, transition probabilities and utility values from their probability distributions. After 100,000 simulations, the average simulated lifetime cost is estimated as �24,860, representing the Monte Carlo approximation to the expected value of the underlying multidimensional integral.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| RAND | =RAND() | Generate random samples for Monte Carlo integration. |
| AVERAGE | =AVERAGE(B2:B100001) | Estimate the expected value from simulated model outputs. |
| STDEV.S | =STDEV.S(B2:B100001) | Estimate simulation variability and Monte Carlo error. |
| NORM.INV | =NORM.INV(RAND(),Mean,SD) | Generate random samples from normally distributed model parameters. |
VBA (Optional)
VBA can automate large-scale Monte Carlo integration by repeatedly sampling model parameters, evaluating model outputs and calculating expected costs and health outcomes.
Sources
- Robert CP, Casella G. Monte Carlo Statistical Methods. Springer.
- Glasserman P. Monte Carlo Methods in Financial Engineering. Springer.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
- ISPOR-SMDM Modeling Good Research Practices Task Force Reports.
- NICE. Health Technology Evaluation Manual.
Related Concepts (4)
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 Monte Carlo integration?
A numerical technique estimating a complex integral's value by sampling random points repeatedly and averaging them, useful when no closed-form solution exists.
Source: Metropolis & Ulam 1949
Why does Monte Carlo integration scale well with dimension?
Grid-based numerical integration needs points spread across every dimension, so the number required explodes as dimensions are added, quickly becoming impossible. Monte Carlo integration instead scatters random points through the space and averages the function's value over them, and the accuracy of this average depends on the number of points rather than the number of dimensions. This is why it stays feasible for the high-dimensional integrals that arise in probabilistic models, where grid methods fail. Its error falls with sample size regardless of dimension. Robert and Casella (2004) describe it.
Source: Robert & Casella 2004
How does Monte Carlo integration work?
Monte Carlo integration works by recognising that an integral can be written as an expected value under some distribution, then estimating that expectation by drawing random samples from the distribution, evaluating the integrand at each, and averaging. Multiplying by the appropriate factor gives the estimate of the integral. As the number of samples grows, the average converges to the true value by the law of large numbers, and the error decreases with the square root of the number of samples, independent of the dimension.
Source: Press et al. 2007
Why is Monte Carlo integration used for high-dimensional problems?
Monte Carlo integration is used for high-dimensional problems because its error decreases with the square root of the number of samples regardless of the number of dimensions, whereas classical numerical quadrature methods require a number of points that grows rapidly with dimension, becoming infeasible. So for integrals over many variables, such as expectations in complex probabilistic models, Monte Carlo integration remains practical where grid-based methods do not. This dimension-independence of its convergence rate is the key reason it suits high-dimensional integration.
Source: Metropolis & Ulam 1949
How accurate is Monte Carlo integration?
The accuracy of Monte Carlo integration improves with the number of samples, with the statistical error decreasing in proportion to the square root of the sample size, so obtaining an extra digit of accuracy requires about a hundred times as many samples. The error is random and can be estimated from the sample variance, giving a standard error for the result. Variance reduction techniques, such as importance sampling or control variates, can improve accuracy for a given number of samples where a suitable scheme is available.
Source: Press et al. 2007
What are the limitations of Monte Carlo integration?
Monte Carlo integration converges slowly, with error falling only as the square root of the sample size, so achieving high precision can require very many samples and be computationally expensive, particularly if each evaluation is costly. Its estimates are random, carrying sampling error that must be quantified. For low-dimensional smooth integrals, classical quadrature is more efficient. These limitations mean Monte Carlo integration is favoured for high-dimensional or awkward integrals, often with variance reduction techniques, rather than where deterministic methods would be faster and more accurate.
Source: Metropolis & Ulam 1949
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 28 Oct 2025
Content version: 1.0.0
Canonical Identity
- Term code
- HE-EM-UA-038
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