Informative Cluster Size and GEE Targets

Why Cluster Size Can Change the Question Being Answered

In clustered data, a cluster may be a clinic, household, school, or other sampling unit, and its size is the number of observed responses it contributes. If cluster size is associated with outcomes, a cluster with many responses can represent a different outcome distribution from a cluster with few responses. Consequently, an analysis that sums response contributions across clusters need not answer the same question as one that gives every cluster equal importance.

This distinction is about the estimand, not just the standard error. A sandwich covariance estimator can account for dependence when estimating uncertainty around a GEE solution, but it does not by itself change which clusters or individuals receive more weight in the estimating equation.

Let clusters be indexed by \(i=1,\ldots,n\), and let \(M_i\geq 1\) be the number of observed individuals in cluster \(i\). Write \(Y_{ij}\) for the response of individual \(j\) in cluster \(i\), and define the cluster mean \( \bar Y_i=M_i^{-1}\sum_{j=1}^{M_i}Y_{ij} \). There are two natural ways to average these responses over a population of clusters.

The two means coincide if cluster size is unrelated to cluster mean outcome, but need not otherwise. The individual-average mean corresponds to choosing an individual from the population of individuals represented by the clusters. The cluster-average mean corresponds to choosing a cluster first and then summarizing it. Neither is universally preferable: the scientific question determines which population average is relevant.

What an Independence-Working GEE Targets

Consider the intercept-only mean model \(E[Y_{ij}]=\beta\), with an identity link and a common variance used in the working covariance. Under an independence working correlation, the GEE estimating equation is proportional to the sum of all response residuals:

Even though the equation is written as a sum over clusters, it gives each response one contribution. Thus a cluster with more responses can have more influence than a smaller cluster.

Proof. Rearranging the estimating equation gives \( \hat\beta=\sum_i\sum_jY_{ij}/\sum_iM_i \). Since \(\sum_jY_{ij}=M_i\bar Y_i\), divide numerator and denominator by \(n\):

The stated integrability assumptions give finite expectations for the numerator and denominator terms. The law of large numbers makes the numerator converge in probability to \(E[M_i\bar Y_i]\) and the denominator converge in probability to \(E[M_i]\). Since \(M_i\geq1\), \(E[M_i]\geq1>0\). The ratio therefore converges in probability to \(E[M_i\bar Y_i]/E[M_i]\). \(\square\)

The difference between these two means has a useful exact expression. Because \(E[M_i\bar Y_i]=E[M_i]E[\bar Y_i]+\operatorname{Cov}(M_i,\bar Y_i)\), we have

This identity applies when the expectations and covariance exist. Positive association between size and cluster mean shifts the individual-average mean upward relative to the cluster-average mean; negative association shifts it downward. Independence of size and cluster mean is sufficient for equality, though not necessary: equality also holds whenever their covariance is zero.

Which Treatment Effect Is Being Averaged?

The same distinction applies to treatment effects. Suppose treatment \(a\in\{0,1\}\) is assigned at the cluster level, and let \(\bar Y_i(a)\) denote cluster \(i\)’s mean outcome under treatment \(a\). Define the cluster-average treatment effect as \( E[\bar Y_i(1)-\bar Y_i(0)] \). The individual-average treatment effect is \( E[M_i\{\bar Y_i(1)-\bar Y_i(0)\}]/E[M_i] \), when cluster size is defined for the target population and does not change between treatment conditions. These effects can differ if treatment effects vary with cluster size.

Randomization can support causal comparisons, but it does not make these two averages identical. When effects vary across clusters, specifying “the treatment effect” without specifying the averaging population leaves the target ambiguous.

Working Correlation Can Change the Weighting Again

The independence-working intercept example has a particularly direct interpretation: its estimating equation counts responses, producing the individual-average target. With a different working covariance, the cluster-level contribution to the same intercept equation can instead give each cluster a size-dependent weight that is neither equal across clusters nor proportional to size.

Derivation. For a cluster of size \(m\), the working covariance is \(V=\phi\{(1-\rho)I_m+\rho J_m\}\), where \(I_m\) is the identity matrix and \(J_m\) is the matrix of ones. Direct multiplication verifies that

Indeed, multiplying the expression in braces by \((1-\rho)I_m+\rho J_m\) gives \(I_m\), using \(J_m^2=mJ_m\). With the intercept derivative vector \(\mathbf 1_m\), the cluster score is \(\mathbf 1_m^\mathsf{T}V^{-1}(Y_i-\beta\mathbf 1_m)\). Since \(\mathbf 1_m^\mathsf{T}(Y_i-\beta\mathbf 1_m)=m(\bar Y_i-\beta)\), substitution gives

Thus the cluster contributes with the stated weight. For \(0\leq\rho<1\), the matrix is positive definite, and the denominator defining the weight is positive. \(\square\)

This derivation is deliberately limited to an intercept-only model with the stated working covariance. In a general GEE, the target is determined by the root of the expectation of the full cluster estimating function. Covariates, mean-model specification, and working covariance all matter; it is not generally correct to label every standard GEE coefficient simply “the individual-average effect” or “the cluster-average effect.”

Common Pitfall and Practical Interpretation

Wrong reasoning: “The sandwich estimator is robust to within-cluster dependence, so it also corrects the effect of informative cluster size.” This confuses uncertainty with the target. A sandwich estimator changes the estimated covariance of the GEE solution; it does not alter the estimating equation whose solution defines the coefficient.

Correction: First identify the scientific target: an average over clusters or an average over individuals, and, for effects, the population over which treatment effects are averaged. Then inspect how the chosen GEE’s cluster contributions weight the data. If size is associated with outcomes, the resulting coefficient may target a different average from the one of interest, even when standard errors are valid for that coefficient.

In applied work, report the unit that receives equal weight and state how cluster size enters the target. Also distinguish outcome-related cluster size from ordinary dependence within clusters: dependence concerns the joint distribution of responses, whereas informative size concerns the relationship between the number of responses in a cluster and the outcomes or effects being averaged.