Exchangeable Correlation

Exchangeable Correlation in GEE and Cluster Randomised Trials

Exchangeable correlation, also called compound symmetry, treats patients in the same general practice, hospital or trial centre as equally alike, with no pair closer than any other. Health economists meet it as a working correlation in generalised estimating equations fitted to cluster randomised trial data, and as the pattern implied by a random intercept in a mixed effects model. Under a random intercept its single parameter equals the intraclass correlation, and the same parameter drives the design effect that inflates sample sizes and standard errors in cluster trials. A wrong choice of structure leaves GEE estimates of incremental costs or QALYs consistent when the mean model is right and data are missing completely at random, but ignoring clustering makes a cost-effectiveness result look more certain than it is.

The exchangeable matrix and its single parameter

The structure describes the correlation among the $m$ observations in one cluster with a single number. Every off-diagonal entry is the same, so the order of observations within the cluster carries no information.

$$R(\rho) = \begin{pmatrix} 1 & \rho & \cdots & \rho \\ \rho & 1 & \cdots & \rho \\ \vdots & \vdots & \ddots & \vdots \\ \rho & \rho & \cdots & 1 \end{pmatrix}$$

where $R(\rho)$ is the $m \times m$ working correlation matrix for a cluster of $m$ observations, the diagonal entries of 1 are each observation's correlation with itself, and $\rho$ is the common correlation between any two different observations in the cluster. Liang and Zeger set this out as Example 3 of their 1986 paper introducing GEE, with the parameter called $\alpha$, and noted that any number of observations per subject is possible under it, so practices recruiting different numbers of patients pose no problem.

Not every value of $\rho$ gives a valid matrix. The matrix has one eigenvalue equal to $(m-1)\rho + 1$ and $m-1$ eigenvalues equal to one minus $\rho$, so it is positive definite only when $\rho$ lies strictly between $-1/(m-1)$ and 1. For clusters of 20 the lower limit is minus 1/19, about minus 0.053.

In GEE the working correlation is combined with the variance of each observation to give a working covariance matrix, in the form set out by Liang and Zeger.

$$V_i = A_i^{1/2} \, R(\rho) \, A_i^{1/2} / \phi$$

where $V_i$ is the working covariance matrix for cluster $i$, $A_i$ is the diagonal matrix of variance-function values for the observations in that cluster, and $\phi$ is the scale parameter of the generalised linear model. Mixed model software labels the same pattern compound symmetry, and the SAS documentation describes TYPE=CS as having two unknown parameters, a common covariance and a residual variance.

Random intercepts and the intraclass correlation

The exchangeable pattern is what a random intercept produces. Liang and Zeger describe their Example 3 as the exchangeable correlation structure obtained from a random effects model with a random level for each subject. Writing that model out shows why the parameter is the intraclass correlation.

$$y_{ij} = x_{ij}^{\top}\beta + b_i + e_{ij}$$

where $y_{ij}$ is the outcome of patient $j$ in cluster $i$, $x_{ij}$ is the vector of covariates with coefficients $\beta$, $b_i$ is the cluster's random intercept with variance $\sigma_b^2$, and $e_{ij}$ is the patient-level error with variance $\sigma_e^2$, independent of $b_i$. Two patients in the same cluster share $b_i$, so their covariance is $\sigma_b^2$, each has variance $\sigma_b^2 + \sigma_e^2$, and their correlation is as follows.

$$\rho = \frac{\sigma_b^2}{\sigma_b^2 + \sigma_e^2}$$

where $\rho$ is the same for every pair in every cluster, which is exactly the exchangeable form. The CONSORT extension for cluster randomised trials defines this intracluster correlation coefficient as the proportion of the total variance of the outcome that can be explained by the variation between clusters. A random intercept has a variance that cannot be negative, so it only produces $\rho \ge 0$, while a marginal exchangeable working correlation can take the small negative values allowed by the eigenvalue limit.

The two routes estimate different things. Rutterford, Copas and Eldridge describe mixed models as cluster-specific and GEE as population-averaged, differing in the interpretation of the treatment effect. For mean costs and QALYs on the identity scale the targets coincide, but odds ratios from a logistic model differ, as the longitudinal texts of Diggle and colleagues and of Fitzmaurice and colleagues explain.

How GEE uses the common correlation

GEE estimates $\rho$ by moments from Pearson residuals pooled across clusters, alternating with updates of the regression coefficients. The variance of those coefficients comes from the sandwich estimator, which puts observed cross-products of residuals within each cluster in place of the unknown true covariance.

$$\widehat{\mathrm{Var}}(\hat{\beta}) = M^{-1} \Big( \sum_{i=1}^{K} D_i^{\top} V_i^{-1} S_i S_i^{\top} V_i^{-1} D_i \Big) M^{-1}, \qquad M = \sum_{i=1}^{K} D_i^{\top} V_i^{-1} D_i$$

where $K$ is the number of clusters, $D_i$ is the matrix of derivatives of cluster $i$'s means with respect to the coefficients $\beta$, $V_i$ is the working covariance above, $S_i$ is the vector of residuals for cluster $i$, and $M$ is the model-based term that forms the bread of the sandwich. Liang and Zeger showed that the consistency of the estimates and of this variance depends only on the correct specification of the mean, not on the correct choice of working correlation, and their companion paper in Biometrics gives the result for discrete and continuous outcomes. The protection rests on independence between clusters.

The choice still affects precision. In one of Liang and Zeger's efficiency calculations, independence working equations kept 0.99 of the efficiency of the correctly specified exchangeable analysis when every subject had 10 observations. When the number of observations varied from 1 to 8, the relative efficiency fell to 0.82. Exchangeable weighting pays off when sizes differ.

Exchangeable, independence, AR(1) and unstructured compared

Working structures differ in how correlation changes with distance between observations and in how many parameters they need. For a cluster or patient with $n$ observations they compare as follows.

Rutterford and colleagues call the exchangeable assumption common and reasonable for cluster randomised trials, since patients in a practice have no natural order. They add that working independence is simple, can aid convergence, and with equal cluster sizes loses no efficiency against a true exchangeable correlation. For repeated health state utility values or costs from one patient, an exchangeable structure assumes baseline and 12-month values are as closely related as two adjacent visits, which often fails over long follow-up, where correlation that fades with time is more plausible.

The design effect implied by exchangeable correlation

The design effect for a cluster trial follows directly from the exchangeable matrix. The variance of a cluster mean is the sum of all entries of the covariance matrix divided by $m^2$, and the exchangeable matrix has $m$ diagonal entries of 1 and $m(m-1)$ off-diagonal entries of $\rho$.

$$\mathrm{Var}(\bar{y}_i) = \frac{\sigma^2}{m^2}\big[m + m(m-1)\rho\big] = \frac{\sigma^2}{m}\big[1 + (m-1)\rho\big]$$

where $\bar{y}_i$ is the mean outcome in a cluster of $m$ patients, $\sigma^2$ is the total variance of one patient's outcome, and the bracketed term is the factor by which clustering inflates the variance compared with $m$ independent patients. The CONSORT extension gives this inflation factor for clusters of equal size, and notes that $\rho$ is typically small (often below 0.05) but its impact can be considerable if clusters are large.

For unequal cluster sizes, Rutterford and colleagues give design effects for binary outcomes under GEE with exchangeable correlation and under working-independence GEE when the true correlation is exchangeable. Both reduce to this factor when all clusters are the same size.

Worked example with practice-level clustering of costs

All figures in this example are illustrative. A cluster randomised trial of a practice-based service measures 12-month NHS cost per patient. Practices are randomised, 20 per arm with 20 patients each, so 400 patients per arm, and the incremental cost is GBP 200.

Step 1 sets out the variance components. The between-practice variance is 72,000 and the within-practice variance is 1,368,000 (both in GBP squared). The total variance is 1,440,000, a standard deviation of GBP 1,200, and the common correlation is 72,000 / 1,440,000 = 0.05.

Step 2 gives the design effect. With 20 patients per practice it is 1 + 19 × 0.05 = 1.95, and its square root, the factor on standard errors, is 1.396. As a check, the variance of a practice mean is 72,000 + 1,368,000 / 20 = 140,400, the same as 1,440,000 / 20 × 1.95.

Step 3 compares standard errors for the difference in mean cost. Treating the 800 patients as independent gives a variance of 2 × 1,440,000 / 400 = 7,200 and a standard error of GBP 84.85. Allowing for clustering multiplies the variance by 1.95, to 14,040, and the standard error becomes GBP 118.49. The 95% interval for the incremental cost is GBP 33.69 to GBP 366.31 when clustering is ignored, and minus GBP 32.24 to GBP 432.24 when it is allowed for. The effective sample size is 400 / 1.95 = 205.1 per arm.

A correlation of only 0.05 nearly doubles the variance of the incremental cost, and a cost difference that looks clearly positive when clustering is ignored becomes uncertain once it is allowed for. The same inflation passes into incremental net benefit and acceptability curves.

Uses in trial-based economic evaluation

Gomes and colleagues, comparing methods for cost-effectiveness analysis of such trials, found that most studies use methods that ignore clustering. In their case study, mean incremental net benefits were similar across methods, but ignoring clustering underestimated statistical uncertainty and the value of information from further research.

Multi-centre and multinational trials raise the same issue with centres as clusters. Manca and colleagues note that centres and countries often differ in clinical practice, patient case-mix and unit costs, so costs and outcomes can be correlated among patients recruited in the same location, and they use multilevel models to estimate population-average and location-specific cost-effectiveness. A random centre intercept in such a model implies exchangeable correlation within each centre.

Small samples, missing data and other limits

Most problems come from the sandwich variance or from assumptions outside the correlation structure. Four recur in trial-based work.

1. Too few clusters. Mancl and DeRouen report that tests using the sandwich covariance estimator can have inflated size with few clusters, up to 2 to 3 times the nominal level in simulations with 10 to 40 subjects, and propose a bias-corrected estimator. Rutterford and colleagues note that with few clusters the mixed model is less biased than GEE, and Gomes and colleagues found GEE interval coverage of 0.841 with 3 clusters per arm and suggest 8 to 15 clusters per arm are needed.
2. Missing data. Liang and Zeger require data to be missing completely at random for consistency when the working correlation is wrong, a strong assumption for trial dropout.
3. Small correlations dismissed. Gomes and colleagues advise that studies should not justify methods that ignore clustering on the basis of low estimated intracluster correlations.
4. Costs and effects treated separately. An exchangeable structure describes dependence within one outcome, while a cost-effectiveness analysis also needs the correlation between costs and QALYs. For cluster trials Gomes and colleagues recommend multilevel models and the two-stage bootstrap.

Sources

• Campbell MK, Piaggio G, Elbourne DR, Altman DG. Consort 2010 statement: extension to cluster randomised trials. BMJ. 2012;345:e5661. https://doi.org/10.1136/bmj.e5661
• Diggle PJ, Heagerty P, Liang KY, Zeger SL. Analysis of Longitudinal Data. 2nd ed. Oxford University Press; 2002.
• Fitzmaurice GM, Laird NM, Ware JH. Applied Longitudinal Analysis. 2nd ed. Wiley; 2011.
• Gomes M, Ng ES, Grieve R, Nixon R, Carpenter J, Thompson SG. Developing appropriate methods for cost-effectiveness analysis of cluster randomized trials. Medical Decision Making. 2012;32(2):350-361. https://doi.org/10.1177/0272989X11418372
• Liang KY, Zeger SL. Longitudinal data analysis using generalized linear models. Biometrika. 1986;73(1):13-22. https://doi.org/10.1093/biomet/73.1.13
• Manca A, Rice N, Sculpher MJ, Briggs AH. Assessing generalisability by location in trial-based cost-effectiveness analysis: the use of multilevel models. Health Economics. 2005;14(5):471-485. https://doi.org/10.1002/hec.914
• Mancl LA, DeRouen TA. A covariance estimator for GEE with improved small-sample properties. Biometrics. 2001;57(1):126-134. https://doi.org/10.1111/j.0006-341x.2001.00126.x
• Rutterford C, Copas A, Eldridge S. Methods for sample size determination in cluster randomized trials. International Journal of Epidemiology. 2015;44(3):1051-1067. https://doi.org/10.1093/ije/dyv113
• SAS Institute. SAS/STAT User's Guide, The MIXED Procedure, Mixed Models Theory. Accessed 18 October 2026. https://go.documentation.sas.com/api/docsets/statug/v_023/content/statug_mixed_details01.htm
• Zeger SL, Liang KY. Longitudinal data analysis for discrete and continuous outcomes. Biometrics. 1986;42(1):121-130. https://doi.org/10.2307/2531248