Why Cluster Size Can Change the Question
Unequal cluster sizes are common: clinics serve different numbers of patients, and participants contribute different numbers of repeated measurements. Unequal sizes alone do not establish a problem. The concern is whether size is related to the outcomes. If small clinics tend to have higher outcome rates than large clinics, an average that gives each clinic equal influence can differ substantially from one that gives each patient equal influence.
That difference is not automatically a flaw. It may reflect two legitimate but distinct questions: “What is the average outcome for a typical cluster?” and “What is the average outcome for a typical observed individual?” The analysis needs reconsideration when the chosen model or estimating equation silently answers a different question from the scientific target, or relies on an independence assumption between size and outcome-related cluster features that is not credible.
The definition applies to outcome distributions, not only to averages. For binary outcomes, for example, clusters of different sizes may have different success probabilities or different distributions of cluster-specific risks. In a GLMM, size may be associated with the latent random effects; in GEE, it may be associated with the observed outcome patterns. In either case, the dependence can matter even when the recorded covariates are the same.
The definition also requires care about the sampling design. If researchers deliberately choose fixed numbers of observations per cluster, the realized sizes are not necessarily evidence that size arose from the outcome process. Nevertheless, size can still affect the weighting of clusters in an analysis. The substantive question is whether outcomes or outcome-related cluster features differ systematically with size, and what population the analysis is intended to describe.
Two Averages and a Short Derivation
For clarity, suppose clusters are sampled from a population of clusters, each with size \(n_i\) and cluster mean \(\bar Y_i\). The cluster-average mean gives each cluster equal weight. The observation-average mean gives each observed individual equal weight, so a cluster with more observations contributes more.
The second expression follows because a randomly selected observation is more likely to come from a large cluster. Its cluster’s size determines how often that cluster can supply the selected observation. Expanding the numerator gives an exact relationship between the two averages:
Thus the two targets agree when \(\operatorname{Cov}(n_i,\bar Y_i)=0\), but can differ when cluster size and cluster outcome mean are associated. This derivation concerns these two population averages; it does not claim that every GEE or GLMM coefficient equals either average. Regression coefficients depend on the mean model, covariates, link function, and estimating or likelihood procedure.
Worked Example: Detecting a Size–Outcome Pattern
Consider four hypothetical clinics. Two have 3 patients each and record 2 successes among those 3 patients; two have 9 patients each and record 2 successes among those 9. Compare the average clinic success rates across the two size groups.
For each small clinic, the observed success rate is \(2/3=0.6667\). For each large clinic, it is \(2/9=0.2222\). The small clinics have higher observed rates, while also having fewer patients. The difference in group-average rates is \(0.6667-0.2222=0.4445\), or about 44.4 percentage points.
This pattern is a warning that size may be informative for the outcome. It does not by itself establish that the association would persist in a larger population: there are only four clinics, and their rates are estimated with considerable uncertainty. It does show why equal cluster sizes cannot be assumed and why the target of an analysis should be stated.
What Ordinary GEE Weighting Can Imply
GEE specifies a marginal mean for the outcomes and uses a working covariance structure to form estimating equations. With independent working correlation and an intercept-only mean model, the estimating equation gives each observed outcome one contribution. For a binary outcome, the fitted intercept corresponds to the observed success proportion. That is an observation-weighted summary, not an equal-weight average of cluster proportions.
A sandwich variance can provide robust standard errors under its usual conditions, including independent clusters and a sufficiently large number of clusters. But robust variance estimation concerns uncertainty around the estimating-equation solution; it does not generally change the solution’s weighting or turn an observation-average target into a cluster-average target. Nor does it make an unsuitable mean estimating equation unbiased for every desired estimand.
Worked Example: Observation-Weighted and Cluster-Weighted Rates
In a hypothetical set of three clusters, the first has 2 successes among 2 observations, the second has 2 among 4, and the third has 2 among 8. The corresponding cluster rates are \(1\), \(0.5\), and \(0.25\).
The equal-cluster average is
There are \(2+4+8=14\) observations and \(2+2+2=6\) successes. The observation-weighted rate is
The estimates differ by \(0.5833-0.4286=0.1547\), about 15.5 percentage points. Under the intercept-only, independent-working-correlation GEE described above, the fitted mean is \(6/14=0.4286\). It estimates the success rate for an observed individual in this sample, not the average success rate of a cluster chosen with equal probability. Neither rate is inherently the correct answer without a specified target.
For covariate models, the weighting is more complicated than this intercept-only example: the working covariance and design matrices influence how cluster residuals contribute. Still, the example isolates the basic issue. If size is associated with outcome patterns, standard GEE procedures should not be assumed to answer a cluster-average question merely because they account for within-cluster dependence.
Implications for GLMMs
A GLMM describes outcomes conditional on covariates and cluster-specific random effects, then specifies a distribution for those random effects. A common formulation treats the random effects as drawn from the same distribution across clusters, independently of cluster size and covariates unless the model states otherwise. Informative cluster size can contradict that assumption: for instance, clinics with higher underlying outcome risk may tend to be smaller.
If the random-effects distribution actually depends on \(n_i\), integrating the likelihood over a size-independent distribution may not represent the data-generating process. The fitted model’s conditional coefficients and its population-level summaries can then be difficult to interpret for the intended population. This does not mean that every GLMM fails whenever cluster sizes differ. The concern is dependence or a mismatch between the model’s assumptions and the size–outcome process.
Worked Example: Size-Related Random Intercepts
Suppose a hypothetical Gaussian random-intercept model is \(Y_{ij}=\beta_0+b_i+\epsilon_{ij}\), where \(b_i\) is a cluster effect and the errors have mean zero. Imagine four clusters with random-intercept values \(+1,+1,-0.5,-0.5\). The first two clusters each have size 2, and the last two each have size 8.
The equal-cluster average random intercept is
The observation-weighted average is instead
In this constructed example, smaller clusters have higher random intercepts, so size and cluster effect are associated. A model that assumes a size-independent random-effects distribution would not express that relationship. The calculation illustrates a possible mismatch; it is not a claim that a particular software fit must return either average as its intercept estimate. The interpretation depends on the likelihood, covariates, and target population.
Reconsideration, Diagnostics, and Alternatives
Start by asking what is sampled and what is to be averaged. If the target is a randomly selected person, observation weighting may be natural. If it is a randomly selected cluster, each cluster may need equal influence for a summary. For regression, state the target in terms of the population and covariate distribution rather than relying on labels such as “marginal” or “conditional” alone.
Useful exploratory checks include plotting cluster-level outcome summaries against \(n_i\), tabulating outcomes across size groups, and examining whether estimated random effects or residual summaries vary with size. These checks can reveal a strong pattern, but cluster summaries are noisy, particularly for small clusters. A lack of visible association is not proof that size is noninformative. When available, subject-matter knowledge about how cluster size was determined can be more informative than a small-sample diagnostic.
If size may be informative, possible responses depend on the target and design. Options include defining a cluster-weighted estimand and using a suitable cluster-weighted estimating approach; conditioning on or stratifying by cluster size when scientifically justified; or modeling the relationship between size and the outcome process, including the size dependence of random effects when appropriate. Inverse-size weighting can reduce the influence of large clusters for a cluster-average target, but it changes the estimand and may reduce precision. Adding \(n_i\) as a predictor is not a universal remedy: it does not automatically capture dependence between size and latent cluster characteristics or correct an inappropriate weighting scheme.
| Approach | What to check | Important limitation |
|---|---|---|
| Standard GEE | Whether its estimating equation and weighting match the target, and whether the mean specification is credible | A robust sandwich variance does not repair a changed or unintended estimand |
| Standard GLMM | Whether the random-effects distribution is plausible across cluster sizes | A size-independent random-effects assumption may be inappropriate when size and latent effects are associated |
| Cluster weighting or size adjustment | Whether the weighting or adjustment targets the desired population | Can change precision and does not automatically solve all forms of size–outcome dependence |
| Joint modeling of size and outcomes | Whether the joint model represents the design and substantive size process | Requires additional assumptions and may be difficult to identify from limited data |
No diagnostic or alternative method is automatic. A sound analysis explains how cluster size arose, identifies the target explicitly, and considers whether conclusions change under reasonable approaches to weighting or modeling. The central pitfall is to treat unequal size as only a standard-error issue: when size is informative, the estimated quantity itself can change.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- In one sentence, distinguish unequal cluster sizes from informative cluster size.
- Suppose the average cluster size is 5 and the covariance between cluster size and cluster mean outcome is \(-1\). If the equal-cluster mean is 0.6, what is the observation-average mean?
- Why does a robust sandwich variance not, by itself, change an observation-weighted GEE estimate into a cluster-weighted estimate?
- What assumption about random effects may be questionable if larger clinics tend to have lower underlying outcome risk?
- Name one diagnostic check and one limitation of that check.
Solutions. (1) Unequal sizes means \(n_i\) varies; informative size means the outcome distribution depends on \(n_i\). (2) By \(\mu_{\text{observation}}=\mu_{\text{cluster}}+\operatorname{Cov}(n_i,\bar Y_i)/E(n_i)\), the result is \(0.6+(-1)/5=0.4\). (3) The sandwich estimates uncertainty around the estimating-equation solution; it does not generally change the equation’s contributions or target. (4) A model assuming the random-effects distribution is independent of cluster size may be questionable. (5) One can plot cluster outcome summaries against size; such summaries can be noisy, so no visible pattern is not proof of independence.