Mean Models And Dependence Models In GEE

Two Specifications, Two Jobs

In clustered or repeated-measures data, observations from the same cluster are generally dependent. Generalized estimating equations (GEE) address this setting by separating two modeling choices: the marginal mean model, which describes the expected outcome as a function of predictors, and the working correlation model, which represents dependence among observations within a cluster for purposes of estimation. They are related in the estimating procedure, but they are not interchangeable.

This separation matters because the mean model defines the regression relationship of interest. The working correlation helps determine how residual information from a cluster contributes to estimating that relationship and its uncertainty. The working correlation is called “working” because it need not be the true correlation structure for GEE to estimate a correctly specified marginal mean consistently under suitable conditions.

Mean, Variance, and Working Correlation

The mean model states how the expected outcome changes with predictors on the scale determined by the link. For example, with a binary outcome, a logit link can specify the log odds of the marginal probability; with a count outcome, a log link can specify the log marginal mean. The coefficients describe these marginal relationships, as discussed in the earlier material on population-averaged mean models in GEE.

To form an estimating equation, GEE also represents the outcome variability within each cluster. Let \(A_i\) be a diagonal matrix whose \(j\)th diagonal entry is the variance-function value for observation \(j\), typically \(\phi v(\mu_{ij})\), where \(v\) is the chosen variance function and \(\phi\) is a scale parameter when appropriate. Let \(R_i(\alpha)\) be a working correlation matrix, potentially depending on nuisance parameter(s) \(\alpha\). The working covariance matrix is

Thus, the marginal mean determines the \(\mu_{ij}\) values and hence the variance-function terms in \(A_i\). The working correlation contributes the off-diagonal dependence representation in \(R_i(\alpha)\). Together they define \(V_i\), which weights the residual vector in the GEE. This connection does not make the two specifications identical: changing the mean formula changes the proposed expected outcomes; changing \(R_i(\alpha)\) changes how residuals are weighted and how uncertainty is calculated.

Write \(\mu_i=(\mu_{i1},\ldots,\mu_{in_i})^\mathsf{T}\), and let \(D_i=\partial\mu_i/\partial\beta^\mathsf{T}\) be the matrix of derivatives of the mean vector with respect to the regression parameters. The GEE for \(\beta\) is

The cluster residual vector \(Y_i-\mu_i\) is transformed by the inverse working covariance and then related to changes in the fitted means through \(D_i\). This is a weighted estimating equation, not a statement that the full joint distribution of \(Y_i\) has been specified.

Why the Working Covariance Changes the Calculation

A short derivation shows the role of the working covariance. In the special case of an identity link and a linear mean \(\mu_i=X_i\beta\), \(D_i=X_i\). If the working covariance is treated as fixed while solving for \(\beta\), the estimating equation becomes

Rearranging gives

The matrix \(V_i^{-1}\) determines the weight assigned to different residual directions within a cluster. If it changes, the solution can change when the data and design make those directions contribute differently. In some special designs the estimate stays the same, even though the weighting has changed; one should not assume that a different working correlation must change every finite-sample estimate.

What the Working Model Does—and Does Not—Claim

The working correlation is a device for representing within-cluster dependence in the estimating equation. It is not an additional predictor effect, and it does not define a separate mean for each pair of observations. Nor does a chosen working correlation alone assert that the joint distribution of outcomes is exactly correct. GEE targets the specified marginal mean while using the working covariance to organize the information in correlated observations.

Under standard regularity conditions, the central consistency argument is that the mean residual has expectation zero when the marginal mean is correctly specified. In particular, if clusters are independent and \(E(Y_i-\mu_i\mid X_i)=0\), then for a nonrandom working covariance conditional on the predictors,

Consequently, a working covariance that is not the true covariance need not introduce bias into the estimating equation’s population target, provided the mean is correct and the weighting is well behaved. Consistency and valid large-sample inference also require suitable conditions such as independent clusters, an identifiable and sufficiently informative design, finite moments, and enough independent clusters. A poor working model can reduce efficiency, and incorrect assumptions about the mean are not repaired by choosing a more elaborate working correlation.

Assumptions, Diagnostics, and Alternatives

The key assumption for the mean target is that the chosen predictors and link correctly describe the marginal means. The estimating procedure also relies on clusters being independent sampling units; dependence within a cluster is the feature being represented. The working covariance matrices must be valid for the calculations, in particular invertible (or handled with an appropriate method), and the predictor design must contain enough information to identify the parameters. The independent-cluster asymptotic argument is not a guarantee of good small-sample performance.

Diagnostics should therefore examine both specifications separately. Check whether fitted means and residual patterns suggest omitted predictors, a poor link, or systematic changes over time or across predictor values. Separately assess whether the working dependence representation appears plausible and whether the algorithm converges to stable estimates. Nonconvergence, extreme parameter estimates, or an invalid working correlation matrix can indicate a computational or specification problem. Residual dependence does not by itself prove that the mean is wrong, just as a plausible correlation pattern does not validate the mean.

A common mistake is to reason, “The working correlation is wrong, so the marginal regression coefficients must be biased.” That conclusion is too strong. With a correctly specified marginal mean and the required regularity conditions, a misspecified working correlation can leave the estimating equation centered at the right mean parameter, although it may affect efficiency and the accuracy of model-based uncertainty calculations. The reverse mistake is equally serious: a well-chosen working correlation cannot rescue a misspecified mean. Appropriate variance estimation is therefore important; the sandwich variance and its small-sample concerns are treated separately in later material.

The principal alternative in this course is the generalized linear mixed model (GLMM), whose mean is specified conditionally on random effects rather than directly as a population-averaged mean. GEE directly specifies the marginal mean and uses a working covariance for estimation. A GLMM instead models latent cluster-level variation as part of its conditional structure. These approaches answer different questions and rely on different modeling assumptions; a working correlation in GEE should not be interpreted as though it were a random effect.