Concept Architecture
Concept
Theoretically, Multivariate Sampling is the process of simultaneously sampling multiple correlated random variables from a joint probability distribution while preserving their marginal distributions and dependence structure. It is founded on multivariate probability theory and enables realistic representation of parameter uncertainty where model inputs are statistically correlated. In health economics, multivariate sampling is used in probabilistic sensitivity analysis to generate coherent combinations of costs, utilities, treatment effects and transition probabilities.
Mathematically, multivariate sampling generates random vectors from a joint distribution defined by marginal probability distributions and a dependence structure, typically represented by a variance-covariance matrix or correlation matrix. Methods such as multivariate normal sampling and Cholesky decomposition preserve observed correlations so that sampled parameter values remain statistically consistent.
In practice, multivariate sampling is implemented by estimating the joint distribution of model parameters from regression analyses, Bayesian posterior distributions or empirical data. The resulting samples are incorporated into Monte Carlo simulation, probabilistic sensitivity analysis and Bayesian decision models to ensure that correlated parameters are propagated appropriately through the economic model.
Purpose
Used to generate correlated samples from multiple uncertain model parameters while preserving their joint probability structure, improving the realism and validity of probabilistic health economic analyses.
Mathematical Formulae
Primary Formula
X = ? + LZ
where:
X = sampled multivariate random vector
? = mean vector
L = Cholesky factor of the variance-covariance matrix
Z ~ N(0, I)
Supporting Formulae
Variance-covariance matrix:
� = LL?
Multivariate normal distribution:
X ~ N(?, �)
Related Mathematical Methods
- Multivariate Normal Distribution
- Cholesky Decomposition
- Variance-Covariance Matrix
- Correlation
- Monte Carlo Simulation
- Probabilistic Sensitivity Analysis
Example
A probabilistic sensitivity analysis models treatment cost, treatment effect and utility, recognising that these parameters are correlated because they were estimated from the same clinical study. A variance-covariance matrix is used to generate multivariate samples, ensuring that each simulation preserves the observed relationships between the parameters before estimating incremental costs and QALYs.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| MMULT | =MMULT(L_range,Z_range) | Generate correlated parameter samples using the Cholesky transformation. |
| TRANSPOSE | =TRANSPOSE(L_range) | Obtain the transpose of the Cholesky factor where required. |
| NORM.S.INV | =NORM.S.INV(RAND()) | Generate independent standard normal random variables prior to transformation. |
| COVARIANCE.S | =COVARIANCE.S(A2:A1001,B2:B1001) | Estimate covariance between model parameters when constructing the variance-covariance matrix. |
VBA (Optional)
VBA can automate multivariate sampling from correlated probability distributions for probabilistic sensitivity analyses and simulation-based health economic models.
Sources
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
- Johnson RA, Wichern DW. Applied Multivariate Statistical Analysis.
- Gelman A, Carlin JB, Stern HS, et al. Bayesian Data Analysis.
- NICE. Health Technology Evaluation Manual.
- ISPOR-SMDM Modeling Good Research Practices Task Force Reports.
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 multivariate sampling?
The simultaneous drawing of random values for multiple correlated input parameters from their joint probability distribution, rather than sampling each independently.
Source: Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press; 2006. doi:10.1093/oso/9780198526629.001.0001.
What does multivariate sampling preserve that independent sampling loses?
Sampling each parameter independently ignores any relationship between them, so it can pair values that never occur together in reality, such as a high event rate with a low cost when the two are actually linked. Multivariate sampling draws the parameters together from their joint distribution, preserving the correlations between them, so every sampled set is internally consistent. This keeps a probabilistic analysis realistic where inputs move together. Independent sampling would distort the resulting uncertainty. Briggs and colleagues (2006) describe this.
Source: Briggs et al. 2006
How is multivariate sampling carried out?
Multivariate sampling is carried out by drawing from the joint distribution of the correlated parameters, often by generating independent draws and transforming them to impose the required correlation structure, for example using the Cholesky factor of the covariance matrix. This produces sampled parameter sets with the specified variances and correlations. For sets of proportions summing to one, a Dirichlet distribution is used. The method ensures that each sampled combination respects the dependence among the parameters, so the joint uncertainty is represented faithfully.
Source: Press et al. 2007
Why is multivariate sampling needed?
Multivariate sampling is needed when input parameters are correlated, because sampling them independently would generate combinations that ignore the dependence and could misrepresent the output uncertainty, either overstating or understating it. Correlated parameters, such as regression coefficients estimated together, must be drawn jointly so their relationships are preserved. By sampling from the joint distribution, multivariate sampling propagates the correct joint uncertainty through the model, which matters where ignoring correlation would give a misleading picture of the uncertainty in the results.
Source: Briggs, Claxton & Sculpher 2006
Where is multivariate sampling used in modelling?
Multivariate sampling is used in probabilistic sensitivity analysis where several parameters are correlated, such as coefficients from a regression with a known covariance matrix, or a set of transition probabilities that must sum to one. In these cases the parameters are drawn jointly, using methods like the Cholesky decomposition or the Dirichlet distribution, so their correlations are respected. This ensures the model reflects the joint uncertainty of related inputs rather than treating them independently, which is important for a correct characterisation of output uncertainty.
Source: Briggs, Claxton & Sculpher 2006
What are the limitations of multivariate sampling?
Multivariate sampling requires knowledge of the joint distribution, including the correlations, which may be difficult to estimate, particularly when parameters come from different sources without a known covariance structure. Methods such as the Cholesky decomposition capture only linear correlations and need a valid, positive-definite covariance matrix. More complex dependence may not be represented. These limitations mean multivariate sampling is applied where the joint distribution or correlation structure can be specified adequately, with simpler assumptions used where the dependence cannot be characterised fully.
Source: Press et al. 2007
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British health economist
Professional identity: darrinbaines.org
Verification date: 28 Oct 2025
Content version: 1.0.0
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