Correlation Matrix

Concept

Theoretically, Correlation Matrix is a symmetric matrix that summarises the pairwise correlation coefficients between multiple random variables. It quantifies the strength and direction of linear association among model parameters while preserving the dependence structure between them. In health economics, correlation matrices are used to represent relationships between uncertain parameters, ensuring that probabilistic analyses account for parameter dependence rather than assuming statistical independence.

Mathematically, Correlation Matrix is represented as a square matrix whose diagonal elements equal one and whose off-diagonal elements are correlation coefficients bounded between ?1 and +1. The matrix must be symmetric and positive semi-definite to constitute a valid correlation structure. Correlation matrices are commonly combined with multivariate probability distributions and matrix decomposition methods, such as the Cholesky decomposition, to generate correlated random samples for probabilistic sensitivity analysis.

In practice, Correlation Matrix is estimated from observed data, covariance matrices or published statistical analyses. Health economic models use correlation matrices when jointly sampling costs, utilities, treatment effects or regression coefficients during probabilistic sensitivity analysis. Incorporating parameter correlation improves the realism of uncertainty analyses and avoids biased estimates that may arise when correlated parameters are sampled independently.

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Purpose

Used to represent the dependence structure between uncertain model parameters, enabling correlated sampling and more realistic probabilistic sensitivity analyses in health economic models.

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Mathematical Formulae

Primary Formula

Correlation matrix:

?? =

? 1     ???   ?   ??? ?
????    1     ?   ??? ?
? ?     ?     ?    ?  ?
????   ???    ?    1  ?

where:

• ??? is the Pearson correlation coefficient between variables i and j.

Supporting Formulae

Pearson correlation coefficient:

??? = Cov(X,Y) / (�?�?)

Covariance relationship:

Cov(X,Y) = ???�?�?

Related Mathematical Methods

• Multivariate probability distributions
• Covariance matrix estimation
• Cholesky decomposition
• Monte Carlo simulation
• Probabilistic sensitivity analysis
• Multivariate normal sampling
• Regression modelling

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Example

A probabilistic sensitivity analysis jointly samples treatment cost and health utility because the two parameters are estimated from the same clinical trial.

The estimated correlation matrix is:

?? =

? 1.00   ?0.35 ?
??0.35    1.00 ?

The negative correlation indicates that higher treatment costs tend to be associated with lower health utility in the observed data. The matrix is used with Cholesky decomposition to generate correlated parameter values during 10,000 Monte Carlo simulations, preserving the observed dependence between parameters.

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Excel Implementation

Function	Example Formula	Health Economics Application
CORREL	=CORREL(B2:B101,C2:C101)	Estimate the correlation between two model parameters.
COVARIANCE.S	=COVARIANCE.S(B2:B101,C2:C101)	Estimate covariance prior to constructing the correlation matrix.
MMULT	=MMULT(Matrix1,Matrix2)	Perform matrix multiplication during correlated sampling procedures.
TRANSPOSE	=TRANSPOSE(MatrixRange)	Verify matrix symmetry or manipulate matrix orientation.
MINVERSE	=MINVERSE(MatrixRange)	Perform matrix calculations used in multivariate statistical methods.

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VBA (Optional)

Automate construction of correlation matrices and generation of correlated Monte Carlo samples using matrix decomposition methods for probabilistic sensitivity analysis.

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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.
• Gentle JE. Random Number Generation and Monte Carlo Methods. 2nd ed. Springer; 2003.
• National Institute for Health and Care Excellence (NICE). Health Technology Evaluation Manual. Latest edition.