Concept Architecture
Concept
Theoretically, Cholesky Decomposition is a matrix factorisation technique that decomposes a symmetric positive-definite matrix into the product of a lower triangular matrix and its transpose. It provides an efficient mathematical method for preserving correlation structures among multiple variables and is widely used in multivariate statistical modelling. In health economics, Cholesky decomposition is principally applied to generate correlated random samples during probabilistic sensitivity analysis.
Mathematically, Cholesky decomposition factorises a variance-covariance matrix into a unique lower triangular matrix. Multiplying independent standard normal random variables by the Cholesky factor produces correlated random variables with the desired covariance structure. This approach preserves observed parameter correlations when propagating uncertainty through simulation models.
In practice, Cholesky decomposition is performed using estimated variance-covariance matrices obtained from regression analyses, maximum likelihood estimation or Bayesian posterior distributions. The resulting matrix is incorporated into Monte Carlo simulation, probabilistic sensitivity analysis and Bayesian modelling to ensure correlated costs, utilities, treatment effects and model parameters are sampled appropriately.
Purpose
Used to preserve parameter correlations by generating correlated random variables from variance-covariance matrices during probabilistic sensitivity analysis, Monte Carlo simulation and Bayesian health economic modelling.
Mathematical Formulae
Primary Formula
A = LL?
where:
A = symmetric positive-definite matrix
L = lower triangular matrix
L? = transpose of L
Supporting Formulae
Generation of correlated random variables:
X = ? + LZ
where:
? = mean vector
L = Cholesky factor
Z ~ N(0, I)
Cov(X) = LL?
Related Mathematical Methods
- Variance-Covariance Matrix
- Multivariate Normal Distribution
- Monte Carlo Simulation
- Probabilistic Sensitivity Analysis
- Bayesian Analysis
- Matrix Factorisation
Example
A probabilistic sensitivity analysis estimates two correlated parameters with covariance matrix:
A = [[25, 10], [10, 16]]
The Cholesky decomposition produces:
L = [[5, 0], [2, 3.464]]
Independent standard normal samples are multiplied by L to generate correlated parameter values, preserving the observed covariance between treatment cost and treatment effectiveness throughout the simulation.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| MMULT | =MMULT(L_range,Z_range) | Multiply the Cholesky factor by independent random variables to generate correlated samples. |
| TRANSPOSE | =TRANSPOSE(L_range) | Obtain the transpose of the Cholesky factor where required. |
| RAND | =RAND() | Generate uniform random numbers for simulation. |
| NORM.S.INV | =NORM.S.INV(RAND()) | Generate independent standard normal random variables prior to applying the Cholesky transformation. |
VBA (Optional)
VBA can automate Cholesky decomposition of variance-covariance matrices and generate correlated parameter samples for large probabilistic sensitivity analyses.
Sources
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
- Drummond MF, et al. Methods for the Economic Evaluation of Health Care Programmes. Oxford University Press.
- ISPOR-SMDM Modeling Good Research Practices Task Force Reports.
- NICE. Health Technology Evaluation Manual.
Related Concepts (8)
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 Cholesky decomposition?
A matrix factorisation technique decomposing a covariance matrix to generate correlated random samples from otherwise independently sampled parameters.
Source: Press et al. 2007
How does Cholesky decomposition turn independent draws into correlated ones?
Sampling parameters independently ignores any correlation between them, but Cholesky decomposition provides a way to impose the correct correlation. It factorises the covariance matrix describing how the parameters vary together into a triangular matrix, which is then multiplied by a set of independently drawn values to transform them into draws carrying the specified correlations. The independently sampled numbers go in, and correlated ones come out. This lets a probabilistic analysis respect relationships between parameters. Briggs and colleagues (2006) describe this technique.
Source: Briggs et al. 2006
How is Cholesky decomposition used to generate correlated samples?
To generate correlated samples, the covariance matrix of the parameters is factorised by Cholesky decomposition into a lower triangular matrix, and a vector of independent standard random draws is multiplied by this factor. The result is a vector of draws with the specified variances and correlations. Adding the means gives correlated samples with the desired joint distribution. This lets a probabilistic analysis reflect known correlations among parameters, so joint uncertainty is represented properly instead of assuming the parameters vary independently.
Source: Briggs, Claxton & Sculpher 2006
Why does Cholesky decomposition require a positive-definite matrix?
Cholesky decomposition requires the matrix to be symmetric and positive-definite because only such a matrix can be written as a triangular factor times its transpose with real entries; a valid covariance matrix has this property. If a supplied correlation or covariance matrix is not positive-definite, perhaps from inconsistent estimated correlations, the decomposition fails, signalling that the specified structure is not a valid covariance matrix. This requirement thus also serves as a check that the assumed correlation structure among parameters is internally consistent.
Source: Press et al. 2007
Where is Cholesky decomposition applied in health economic modelling?
In health economic modelling, Cholesky decomposition is applied in probabilistic sensitivity analysis to sample correlated parameters, such as regression coefficients whose estimated covariance matrix links them, so that their joint uncertainty is propagated correctly. Rather than sampling each parameter independently, the method uses the covariance matrix to produce draws that preserve the correlations. This matters where parameters are estimated together and correlated, since ignoring the correlation would misstate the uncertainty in the model's results, so Cholesky decomposition supports correct joint sampling.
Source: Briggs, Claxton & Sculpher 2006
What are the limitations of using Cholesky decomposition?
Cholesky decomposition requires a valid, positive-definite covariance matrix, and obtaining a consistent estimate of all pairwise correlations can be difficult, particularly when correlations come from different sources; an invalid matrix cannot be decomposed. It captures only linear correlations described by the covariance matrix, not more complex dependence. It also assumes the sampling distribution is suited to this linear transformation, as with multivariate normal draws. These limitations mean the method is used where a valid covariance matrix and linear correlation adequately describe the parameter dependence.
Source: Press et al. 2007
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British health economist
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Verification date: 24 Oct 2025
Content version: 1.0.0
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