Alternating Mean and Correlation Estimation

Why Alternate the Updates?

In generalized estimating equations (GEE), the regression mean and the within-cluster dependence are connected through the covariance matrix used to weight the estimating equation. The coefficient update needs a working correlation; the correlation update, in turn, uses residuals calculated from the current fitted means. Alternating updates let each part be estimated using the current values of the others.

This procedure is an iteration among estimating equations, not generally a likelihood-maximization algorithm. An update can change the next update, and convergence should be checked rather than presumed. In particular, one should track the regression coefficients, any estimated marginal variance or dispersion parameters, and the working-correlation parameters.

Setup and Update Equations

Suppose there are \(m\) independent clusters. Cluster \(i\) has \(n_i\) observed responses collected in \(Y_i\), and a mean model with regression parameter \(\beta\) gives fitted mean vector \(\mu_i(\beta)\). Let \(D_i(\beta)=\partial\mu_i(\beta)/\partial\beta^\mathsf{T}\). If the marginal variance has a dispersion parameter \(\phi\), write \(A_i(\beta,\phi)=\operatorname{diag}\{\phi v(\mu_{ij}(\beta))\}\), where \(v\) is the variance function. For a family with fixed dispersion, \(\phi\) is fixed rather than updated. The working covariance is \(V_i=A_i^{1/2}R_i(\alpha)A_i^{1/2}\), with working-correlation parameter \(\alpha\).

The coefficient step solves

For a nonlinear mean model, this equation is usually solved iteratively, for example by a Fisher-scoring or Newton-type step, while using the current working covariance. A typical dispersion update for a model with a common scale is the Pearson estimate

where \(p\) is the dimension of \(\beta\), provided \(N>p\). This is a common moment-based scale estimate, not a universal choice for every response family or variance model. A correlation update uses standardized residuals such as \(r_{ij}=(Y_{ij}-\hat\mu_{ij})/\sqrt{v(\hat\mu_{ij})}\), and applies the chosen correlation estimator to their within-cluster products. The exact moment formula depends on the working-correlation structure; for example, exchangeable correlation pools pairwise products, while an unstructured estimate uses products for each visit pair with observations available at both visits.

What Is Held Fixed?

A transparent implementation specifies the order of updates. One common cycle is: update \(\beta\) with \(\phi\) and \(\alpha\) fixed; update \(\phi\) with the new \(\beta\) fixed; then update \(\alpha\) using residuals from that new mean fit. When estimating correlation, the mean and marginal variance model are treated as fixed for that step. The updated parameters are then used in the next cycle.

Two Useful Properties of the Iteration

The first property explains why a common dispersion scale often need not be known to update the regression coefficients.

Proof. Since \((\phi W_i)^{-1}=\phi^{-1}W_i^{-1}\), the estimating equation becomes

Because \(\phi^{-1}\) is a nonzero scalar common to the whole sum, this equation equals zero exactly when \(\sum_iD_i^\mathsf{T}W_i^{-1}(Y_i-\mu_i)=0\). Its root is therefore unchanged by the value of \(\phi\). This cancellation does not apply if variance parameters alter the relative weights across observations or clusters. \(\square\)

The second property identifies what it means for an alternating procedure to have settled. Write the coefficient, dispersion, and correlation update rules as \(B(\phi,\alpha)\), \(S(\beta)\), and \(C(\beta,\phi)\), respectively, with each rule assumed to be defined for the inputs in question.

Proof. At a fixed point, one complete cycle leaves every component unchanged. The first step of that cycle replaces \(\beta^\ast\) by \(B(\phi^\ast,\alpha^\ast)\); since the cycle leaves \(\beta\) unchanged, this replacement must equal \(\beta^\ast\). The same reasoning for the second and third steps gives \(\phi^\ast=S(\beta^\ast)\) and \(\alpha^\ast=C(\beta^\ast,\phi^\ast)\). By definition, \(B\) returns a solution of the GEE with its supplied \(\phi\) and \(\alpha\), so the fixed-point coefficient solves that equation. This argument characterizes a fixed point; it does not establish that iteration reaches one. \(\square\)

Worked Updates

Practical Convergence Checks and a Common Pitfall

A useful stopping rule checks more than the regression coefficients. For example, require the maximum scaled change across \(\beta\), \(\phi\), and \(\alpha\) to be small, and also require the GEE residual to be small. One possible scaled change for a parameter vector \(\theta\) is \(\|\theta^{(k+1)}-\theta^{(k)}\|_\infty/\{1+\|\theta^{(k)}\|_\infty\}\). A score check can use the norm of \(U(\beta^{(k+1)};\phi^{(k+1)},\alpha^{(k+1)})\), scaled to suit the problem. The tolerance should reflect numerical precision and the scale of the parameters. Correlation and variance updates must also remain in the parameter domain required by the chosen model.

Common mistake: Treating one coefficient update as the final fit because it solves the GEE for the starting correlation. That equation is conditional on the working covariance used in the update. Once the correlation is re-estimated from the new residuals, the coefficient equation has changed and may have a different root. The correction is to repeat the full cycle and assess changes in every updated parameter, not just whether the first coefficient equation was solved.

Alternation makes the dependence between mean, variance, and working correlation explicit. It does not, by itself, guarantee a unique fixed point, convergence from every starting value, or a monotone improvement criterion. Those properties should not be inferred from small changes in one component alone.