Concept Architecture
Concept
Theoretically, Working Correlation is the assumed correlation structure specified within Generalised Estimating Equations (GEE) to represent the dependence between repeated observations from the same individual or cluster. It provides an approximation to the true within-subject correlation and enables efficient estimation of regression parameters without requiring the correlation structure to be correctly specified. The concept is founded on quasi-likelihood theory and estimating equations for correlated data.
Mathematically, the working correlation matrix forms part of the covariance matrix used in Generalised Estimating Equations. It specifies the assumed pattern of correlation among repeated observations, with common structures including independence, exchangeable, autoregressive and unstructured correlation. Although the consistency of GEE regression estimates does not depend on the correctness of the working correlation structure, an appropriate choice improves statistical efficiency. Robust sandwich variance estimators provide valid standard errors even if the working correlation is misspecified.
In practice, the working correlation structure is selected according to the expected dependence between repeated observations and may be compared using model selection criteria such as the Quasi-likelihood Information Criterion (QIC). In health economics, working correlation structures are routinely applied when analysing longitudinal healthcare costs, repeated utility measurements, patient-reported outcomes and healthcare utilisation using Generalised Estimating Equations.
Purpose
Used to represent the assumed correlation between repeated observations within Generalised Estimating Equations, improving the efficiency of parameter estimation while maintaining robust statistical inference.
Mathematical Formulae
Primary Formula
V? = �A???�R(�)A???�
where:
- V? = working covariance matrix for subject i
- � = dispersion parameter
- A? = diagonal variance matrix
- R(�) = working correlation matrix
- � = working correlation parameter(s)
Supporting Formulae
Exchangeable correlation:
R(�) =
[1?�?�??]
[�?1?�??]
[???????]
Autoregressive AR(1) correlation:
Corr(Y??, Y??) = �^|t?s|
Robust sandwich covariance estimator:
Var(??) = M??BM??
Related Mathematical Methods
- Generalised Estimating Equations
- Sandwich Estimator
- Exchangeable Correlation
- Autoregressive Correlation
- Unstructured Correlation
- Quasi-Likelihood
- Quasi-likelihood Information Criterion
Example
A health economist analyses quarterly healthcare expenditure for patients with chronic disease over two years using a Generalised Estimating Equations model. Because expenditure measurements collected close together are expected to be more strongly correlated than those collected further apart, an AR(1) working correlation structure is specified. Regression coefficients remain consistent regardless of whether the assumed correlation exactly matches the true correlation, while robust sandwich standard errors provide valid statistical inference.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| CORREL | =CORREL(B2:B101,C2:C101) | Explore pairwise correlations between repeated measurements before specifying a working correlation structure. |
| COVARIANCE.S | =COVARIANCE.S(B2:B101,C2:C101) | Estimate covariance between repeated observations. |
| MMULT | =MMULT(B2:E5,F2:I5) | Perform matrix multiplication for exploratory covariance calculations. |
| MINVERSE | =MINVERSE(B2:E5) | Invert covariance matrices for matrix-based statistical calculations. |
VBA (Optional)
Automate the construction and export of alternative working correlation matrices for repeated measures analyses performed in specialised GEE software.
Sources
- Liang KY, Zeger SL. Longitudinal data analysis using generalized linear models. Biometrika. 1986;73:13?22.
- Hardin JW, Hilbe JM. Generalized Estimating Equations.
- Diggle PJ, Heagerty P, Liang KY, Zeger SL. Analysis of Longitudinal Data.
- Fitzmaurice GM, Laird NM, Ware JH. Applied Longitudinal Analysis.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
- Drummond MF, et al. Methods for the Economic Evaluation of Health Care Programmes.
Related Concepts (3)
Library
Publications
1
Bayesian Methods in Health Economics — Gianluca Baio, 1st Edition ed., 2012 (Chapman & Hall / CRC Press)
An overview of Bayesian statistical methods for the analysis of health economic data, covering economic evaluation concepts, statistical cost-effectiveness analysis, Bayesian computation and MCMC, and applied health economic evaluation.
BookView source →
Frequently Asked Questions (6)
What is a working correlation?
In generalised estimating equations, an assumed correlation structure among repeated observations, improving statistical efficiency even if not perfectly correct.
Source: Liang & Zeger 1986
What role does a working correlation play in generalised estimating equations?
A working correlation is an assumed pattern of correlation among the repeated observations on the same unit, supplied to generalised estimating equations to improve their efficiency. It is called working because it need not be exactly right: the method still gives valid estimates of the average effect even if the assumed structure is imperfect, though a closer guess yields tighter estimates. Common choices include treating all pairs as equally correlated or letting correlation fade with time. Guiding efficiency without needing to be exact is its role. Kirkwood and Sterne (2003) describe this.
Source: Kirkwood & Sterne 2003
Why is it called a working correlation?
It is called a working correlation because it is a working assumption about the correlation structure used to fit the model, not necessarily the true correlation, and generalised estimating equations give valid estimates of the average effects and, through robust standard errors, valid inference even if the working correlation is misspecified. So the term working reflects that the assumed structure is provisional and need not be correct, since the method is robust to its misspecification, which is why a working correlation is chosen to be a plausible approximation to improve efficiency, while the validity of the estimates does not depend on it being exactly right.
Source: Liang & Zeger 1986
What are common working correlation structures?
Common working correlation structures include independence, assuming no within-cluster correlation; exchangeable, or compound symmetry, assuming all pairs within a cluster are equally correlated; autoregressive, assuming correlation declines with the time between observations; and unstructured, estimating every pairwise correlation freely. So working correlation structures range from simple to flexible, with the choice reflecting assumptions about how observations within clusters are correlated, and a structure that better approximates the true correlation improves efficiency, though generalised estimating equations remain valid under misspecification, which is why the choice balances plausibility and simplicity against the number of correlation parameters to estimate.
Source: Liang & Zeger 1986
How does the choice of working correlation affect results?
The choice of working correlation affects the efficiency of the estimates, with a structure closer to the true correlation giving more precise estimates, but it does not affect the validity of the estimates of the average effects or, when robust standard errors are used, the validity of the inference, since generalised estimating equations are robust to misspecification of the working correlation. So the choice of working correlation influences efficiency but not validity, which is why a reasonable structure is chosen to improve precision while the robust standard errors ensure valid inference even if the structure is wrong, meaning the method tolerates an imperfect working correlation without biasing the estimated effects.
Source: Liang & Zeger 1986
How is the working correlation used in generalised estimating equations?
The working correlation is used in generalised estimating equations to weight the observations in estimating the population-average effects, incorporating the assumed within-cluster correlation into the estimating equations, while robust, or sandwich, standard errors provide valid inference regardless of whether the structure is correct. So the working correlation is used to guide estimation and improve efficiency, entering the equations that estimate the average effects, and its role is practical rather than requiring correctness, since the accompanying robust standard errors make the inference valid under misspecification, which is why generalised estimating equations can use a simple working correlation while still yielding trustworthy estimates and standard errors.
Source: Liang & Zeger 1986
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 26 Dec 2025
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/working-correlation
- Term code
- HE-ES-SA-238
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