Exchangeable GEE Contributions for Repeated Measures

Why Exchangeable Weighting Changes a Cluster Contribution

In a generalized estimating equation, a cluster contributes a weighted combination of its observed-minus-mean residuals. Under an independence working correlation, each repeated-measure residual is weighted separately. An exchangeable working correlation instead assigns the same working correlation to every pair of distinct observations in a cluster. Its inverse therefore makes each residual’s contribution depend on the other residuals in that cluster.

The correlation assumption concerns the weighting used in the estimating equation. It does not, by itself, change the specified marginal means. The derivation below makes the coupling explicit and shows how to calculate it.

Exchangeable Working Correlation and Covariance

Let cluster \(i\) contain \(n_i\) responses, collected in the vector \(Y_i\), with marginal mean vector \(\mu_i=E[Y_i]\). Let \(A_i\) be the diagonal matrix of marginal variance functions, with diagonal entries \(v(\mu_{ij})\). With dispersion parameter \(\phi>0\), an exchangeable working covariance is

Here \(I_{n_i}\) is the \(n_i\)-dimensional identity matrix, \(\mathbf{1}_{n_i}\) is the vector of \(n_i\) ones, and \(\rho\) is the common working correlation for distinct responses within the cluster. The diagonal entries of \(R_i(\rho)\) equal 1 and its off-diagonal entries equal \(\rho\). Thus, the working covariance between responses \(j\) and \(k\) is \(\phi\rho\sqrt{v(\mu_{ij})v(\mu_{ik})}\) when \(j\ne k\).

When Is the Matrix Positive Definite?

The form involving \(\mathbf{1}_{n_i}\mathbf{1}_{n_i}^{T}\) makes the eigenvalues easy to identify. The vector of ones is an eigenvector: multiplying \(R_i(\rho)\) by it gives \(\{1+(n_i-1)\rho\}\mathbf{1}_{n_i}\). Any vector orthogonal to \(\mathbf{1}_{n_i}\) has entries summing to zero, so multiplication by \(R_i(\rho)\) gives \(1-\rho\) times that vector. This orthogonal subspace has dimension \(n_i-1\).

Proof. The two subspaces just described are orthogonal and together span \(\mathbb{R}^{n_i}\). A real symmetric matrix is positive definite exactly when its eigenvalues are all positive. Therefore we require \(1-\rho>0\) and \(1+(n_i-1)\rho>0\). Rearranging gives \(\rho<1\) and \(\rho>-1/(n_i-1)\), respectively. For clusters with different sizes, the common parameter must lie in the intersection of their allowable intervals. A cluster of size 1 has \(R_i=[1]\), so it places no restriction on \(\rho\). \(\square\)

This condition is about the working matrix being invertible and positive definite, not about whether a negative correlation is plausible in a particular application. For example, with clusters of sizes 3 and 5, both matrices are positive definite only if \(-1/4<\rho<1\): the larger cluster gives the stricter lower bound.

Inverse Matrix and the Cluster-Level GEE

Write \(J_i=\mathbf{1}_{n_i}\mathbf{1}_{n_i}^{T}\). When \(R_i(\rho)\) is positive definite, its inverse is

To verify the expression, set \(a=1-\rho\) and \(b=\rho\), so \(R_i=aI_{n_i}+bJ_i\). Since \(J_i^2=n_iJ_i\), multiply \(aI_{n_i}+bJ_i\) by \(a^{-1}(I_{n_i}-cJ_i)\), where \(c=b/(a+bn_i)\). The product is

because \(c=b/(a+bn_i)\). Substituting \(a=1-\rho\) and \(a+bn_i=1+(n_i-1)\rho\) gives the stated inverse. The positivity conditions ensure its denominators are nonzero.

Let \(\beta\) denote the mean-model parameter, and let \(D_i=\partial\mu_i/\partial\beta^T\) be the \(n_i\)-by-\(p\) derivative matrix. The cluster contribution to the GEE is

This is the usual multivariate quasi-score form, with the working covariance used to weight the residual vector. Define the standardized residual vector \(e_i=A_i^{-1/2}(Y_i-\mu_i)\), whose \(j\)th entry is \(e_{ij}=(Y_{ij}-\mu_{ij})/\sqrt{v(\mu_{ij})}\). Substituting the covariance and inverse above yields

Derivation. Since \(V_i^{-1}=\phi^{-1}A_i^{-1/2}R_i(\rho)^{-1}A_i^{-1/2}\), the definition of \(e_i\) gives the first equality. The inverse formula gives \(R_i^{-1}e_i=(1-\rho)^{-1}\{e_i-\rho\mathbf{1}_{n_i}(\mathbf{1}_{n_i}^Te_i)/[1+(n_i-1)\rho]\}\). Finally, \(\mathbf{1}_{n_i}^Te_i=\sum_k e_{ik}\), which proves the displayed expression. \(\square\)

The shared residual sum is the source of the coupling: every standardized residual enters the adjustment for every response in that cluster. The adjustment is then combined with the response-specific derivatives and variance functions. Thus, it is not generally correct to replace a whole cluster by one unweighted residual sum.

Worked Calculations

Interpretation and a Common Pitfall

The exchangeable assumption is a parsimonious way to represent a common within-cluster association in the working covariance. It can be useful when repeated measurements are similarly related across their occasions, while the marginal mean model still describes the expected response at each occasion or covariate pattern. Its effect on the GEE is algebraic: the inverse working correlation alters how residual patterns contribute to the parameter estimate.

Common mistake: treating correlation as a common multiplier. It is tempting to say that positive correlation simply multiplies every independent-working contribution by the same factor. The inverse formula shows why this is wrong: the adjustment subtracts a multiple of the entire cluster’s standardized residual sum from each component, and the result is then combined with response-specific derivative and variance terms. A common multiplier is obtained only in special directions or models, not for an arbitrary cluster contribution.

Also note the limiting case \(\rho=0\). The inverse becomes \(I_{n_i}\), and the contribution reduces to the independence-working form \(\phi^{-1}D_i^TA_i^{-1}(Y_i-\mu_i)\). This check is useful when deriving or computing an exchangeable contribution.