Diagnosing Informative Cluster Size

Why Diagnose Cluster Size?

In clustered data, clusters can have different numbers of observed units. A clinic may contribute many patients while another contributes only a few; a study may record different numbers of repeated visits for different people. When cluster size is related to responses or covariates, larger clusters can contribute systematically different information from smaller ones. A standard GEE may then describe an individual-weighted population in which large clusters count more, rather than an average cluster.

The diagnostic goal is not to establish a causal explanation for why sizes differ. It is to look for evidence that cluster size predicts outcome levels, outcome distributions, or covariate composition, and then assess whether substantive conclusions are sensitive to how clusters are weighted or modeled.

Definition: Let cluster $i$ have size $N_i$, covariates $X_{ij}$, and responses $Y_{ij}$, for $j=1,\ldots,N_i$. Cluster size is outcome-informative when the distribution of the response vector depends on $N_i$, possibly conditional on measured covariates. In practice, dependence of cluster-level outcome summaries on $N_i$ is a useful diagnostic signal. A relationship between $N_i$ and the covariate distribution is also important: it can change which covariate profiles are prominent in an individual-weighted analysis, even if outcome levels show no clear size pattern.

A diagnostic should specify what is being compared. For a continuous response, one might examine each cluster’s mean. For a binary response, one might examine each cluster’s proportion of successes. Covariate summaries can include cluster-level means, proportions, or category counts. These summaries do not replace the outcome model; they make patterns associated with cluster size visible.

A Weighting Identity

The difference between an average cluster and an average individual can be written exactly. Suppose all responses are observed, and define the cluster mean $\bar Y_i=N_i^{-1}\sum_{j=1}^{N_i}Y_{ij}$. Let $m$ be the number of clusters. The cluster-weighted mean averages the $m$ cluster means equally, whereas the pooled individual mean gives cluster $i$ weight proportional to $N_i$.

Result: With empirical averages taken over the $m$ clusters, $$ \bar Y_{\mathrm{ind}}-\bar Y_{\mathrm{cl}} =\frac{\operatorname{Cov}_m(N_i,\bar Y_i)}{\bar N}, \qquad \bar N=\frac{1}{m}\sum_{i=1}^m N_i. $$ Thus, the two means agree when the empirical covariance between cluster size and cluster mean is zero; they differ when that covariance is nonzero.

Proof. By definition, $$ \bar Y_{\mathrm{ind}} =\frac{\sum_{i=1}^m N_i\bar Y_i}{\sum_{i=1}^m N_i} =\frac{m^{-1}\sum_{i=1}^m N_i\bar Y_i}{\bar N}. $$ The empirical covariance is $$ \operatorname{Cov}_m(N_i,\bar Y_i) =m^{-1}\sum_{i=1}^m N_i\bar Y_i-\bar N\bar Y_{\mathrm{cl}}, $$ where $\bar Y_{\mathrm{cl}}=m^{-1}\sum_i\bar Y_i$. Dividing this equality by $\bar N$ and rearranging gives the stated identity. $\square$

This identity is a diagnostic aid, not a universal formula for every GEE coefficient. It shows directly how size-related outcome levels can change a marginal mean. Regression coefficients also depend on covariates, the mean model, and the estimating equations; a difference between two fitted coefficients should therefore be interpreted in that fuller context.

Worked Example: Binary Outcomes and Unequal Cluster Weights

Four clinics have the following numbers of patients and observed successes:

Clinic	Size $N_i$	Successes	Proportion $\bar Y_i$
1	2	0	0
2	2	1	1/2
3	4	2	1/2
4	8	6	3/4

The cluster-weighted mean is $$ \bar Y_{\mathrm{cl}} =\frac{0+1/2+1/2+3/4}{4} =\frac{7}{16} =0.4375. $$ The pooled individual mean is $$ \bar Y_{\mathrm{ind}} =\frac{0+1+2+6}{2+2+4+8} =\frac{9}{16} =0.5625. $$ For a check using the identity, $\bar N=16/4=4$, and $$ \operatorname{Cov}_m(N_i,\bar Y_i) =\frac{0+1+2+6}{4}-(4)(7/16) =\frac{9}{4}-\frac{7}{4} =\frac{1}{2}. $$ Therefore $\bar Y_{\mathrm{ind}}-\bar Y_{\mathrm{cl}}=(1/2)/4=1/8$, which agrees with $9/16-7/16=1/8$. The larger clinics have higher proportions in this small example. This is a visible size pattern, not evidence by itself that clinic size caused higher success rates.

Outcome and Covariate Diagnostics

A practical assessment begins with a plot or table of cluster size against an appropriate cluster-level summary. Useful displays include a scatterplot of $\bar Y_i$ against $N_i$, summaries within prespecified size groups, and plots of cluster-level covariate means or proportions against size. For binary outcomes, plotting each cluster’s success proportion is often more interpretable than plotting its raw success count: counts tend to grow mechanically with cluster size.

The precision of a cluster summary may itself depend on size. For example, under independent Bernoulli responses with success probability $p$, a cluster proportion based on $N_i$ observations has variance $p(1-p)/N_i$. Thus, small clusters tend to have noisier proportions. A descriptive trend should be assessed with that varying precision in mind; treating all plotted proportions as equally precise can obscure the pattern or overemphasize small clusters.

Worked Example: Detecting Size-Related Covariate Composition

Suppose four clinics have sizes $2,2,6,6$. Their proportions of patients with a baseline risk factor are $0,0,1/2,1/2$, respectively. The cluster-weighted average risk-factor proportion is $$ \frac{0+0+1/2+1/2}{4}=\frac{1}{4}. $$ Across all patients, the number with the risk factor is $0+0+3+3=6$, out of $2+2+6+6=16$. The individual-weighted proportion is therefore $$ \frac{6}{16}=\frac{3}{8}. $$ The larger clinics have a higher risk-factor proportion, so the individual-weighted covariate distribution gives that profile more weight. This is a size-related difference in covariate composition. It can affect the interpretation of an outcome comparison even if the outcome summaries themselves do not show an obvious trend with size.

The conclusion is deliberately limited: these summaries identify an association in the observed clusters. They do not tell us whether the risk factor influenced clinic size, clinic size influenced which patients were observed, or some other process influenced both.

A second useful diagnostic is to summarize outcomes within size strata. For size groups $g$, let $M_g$ be the number of clusters in group $g$, and let $\bar Y_g$ be the average of their cluster means. The cluster-weighted mean can be expressed as $\sum_g(M_g/m)\bar Y_g$. The pooled individual mean instead weights each group according to the total number of individuals it contributes, $\sum_{i\in g}N_i$. Comparing outcome summaries across strata helps reveal whether size groups differ in outcome level; comparing their covariate summaries reveals whether they differ in composition.

Worked Example: A Cluster-Level Size Trend

Consider four clusters with sizes $2,2,4,4$, and continuous-outcome means $1,3,5,7$. To summarize the linear association descriptively, regress the four cluster means on cluster size using ordinary least squares, with equal weight for each cluster.

The mean size is $\bar N=3$, and the mean of the four outcome means is $\bar Y_{\mathrm{cl}}=4$. The slope is $$ \hat b =\frac{\sum_i(N_i-\bar N)(\bar Y_i-\bar Y_{\mathrm{cl}})} {\sum_i(N_i-\bar N)^2} =\frac{(-1)(-3)+(-1)(-1)+(1)(1)+(1)(3)} {(-1)^2+(-1)^2+1^2+1^2} =\frac{8}{4}=2. $$ This calculation reveals a positive linear size trend in these summaries. Indeed, the two clusters of size 2 average $(1+3)/2=2$, and the two clusters of size 4 average $(5+7)/2=6$.

For comparison, the pooled individual mean is $$ \frac{2(1)+2(3)+4(5)+4(7)}{2+2+4+4} =\frac{56}{12} =\frac{14}{3}. $$ The cluster mean is $4$. Here, the difference arises because the two larger clusters have higher means overall, consistent with the positive linear slope above.

Sensitivity Analyses Using GEE

Plots and summaries are descriptive. A complementary sensitivity analysis asks whether the fitted GEE conclusions change under reasonable ways of representing size dependence. One approach is to augment the mean model with $N_i$, or a prespecified transformation or size-group indicator, and examine whether it predicts outcome levels after the included covariates are accounted for. If effect modification is scientifically plausible, an interaction between size and a covariate of interest can be examined as a separate sensitivity model.

Because $N_i$ is constant for all observations in cluster $i$, it is a cluster-level predictor. Its association with responses should be assessed with inference that respects clustering, and with due caution when the number of clusters is small. Adding size terms is a diagnostic sensitivity analysis, not an automatic correction: it changes the mean model and may change the estimand. Compare estimates and uncertainty across the original model, the size-augmented model, and—when relevant—the cluster-weighted GEE introduced in earlier material. Report which conclusions are stable and which depend on the weighting or size specification.

A size term that is not statistically significant does not establish that cluster size is uninformative. The test may have low power, the relationship may be nonlinear, or relevant size dependence may concern covariate composition or response variability rather than the chosen mean summary. Conversely, a statistically detectable association does not identify its mechanism or prove that a particular GEE interpretation is valid.

Common Pitfalls and Practical Interpretation

Looking only at total counts. More successes in a larger cluster may simply reflect more observations. Compare rates or means as well as counts.
Equating a lack of a linear trend with independence. Size patterns can be nonlinear or concentrated in a few size groups. Use plots and stratified summaries alongside a fitted slope.
Ignoring covariate composition. Similar outcome averages can conceal size-related differences in the covariates represented by the clusters.
Treating an association as a causal explanation. Diagnostics describe observed dependence; they do not establish why cluster sizes differ.
Calling a sensitivity model a universal fix. Including size changes the model and may change the target. Explain the specification and compare conclusions rather than presenting one adjusted fit as definitive.

Key takeaway: Diagnose informative cluster size by examining both outcome summaries and covariate distributions across cluster sizes. Use size-stratified displays and GEE sensitivity analyses to see whether conclusions are robust, while distinguishing observed association from its cause and from the choice of population weighting.

Check Your Understanding

Answer each question using the distinction between cluster-level and individual-level weighting.

What does it mean for cluster size to be informative for the outcome, and how is that different from size-related covariate composition?
In the weighting identity, what feature of the clusters determines whether the cluster-weighted and individual-weighted means differ?
Why can a plot of raw success counts against cluster size be misleading?
Give one reason a nonsignificant linear size term does not establish that cluster size is uninformative.
Why should adding cluster size to a GEE be described as a sensitivity analysis rather than an automatic correction?