Individual-Weighted And Cluster-Weighted Marginal Means

Why the Unit Being Averaged Matters

A marginal mean summarizes expected outcomes across a target population. But a population can be sampled or described at more than one level. In clustered data, one might choose a cluster—such as a clinic or school—uniformly, or choose an individual uniformly from all individuals in the clusters. These choices generally do not select the same population when cluster sizes differ.

The difference is about the target of the average, not automatically about whether a model is correct. Giving each clinic equal weight answers a question about the average clinic. Giving each person equal weight answers a question about the average person in the pooled population. If large and small clusters have different expected outcomes, these averages can differ.

Here, \(\mu_j\) must refer to the same outcome and scenario for every cluster. For example, it might be the cluster-specific predicted probability of an outcome under a specified treatment condition, after applying a chosen covariate standardization. The earlier material on standardizing marginal effects establishes that the covariate distribution is part of an estimand. The weights across clusters are a separate choice. Define the cluster means using a consistent prediction target first; then specify whether the target averages clusters or individuals.

The counts \(n_j\) must also match the population the question names. For a person-level outcome, they might count eligible people, not visits or repeated records. If the target is an average visit, a different set of weights may be appropriate. Unequal counts alone do not make either estimand wrong: the right choice depends on the intended population and sampling unit.

Deriving the Difference

Let \(\bar n=N/J\) be the average cluster size and \(\bar\mu=J^{-1}\sum_{j=1}^{J}\mu_j=\mu_C\) the cluster-weighted mean. Subtracting the two estimands gives

To verify the last equality, note that \(N=J\bar n\), so the difference before the last equality is \(N^{-1}\sum_j(n_j-\bar n)\mu_j\). Since \(\sum_j(n_j-\bar n)=0\), subtracting \(\bar\mu\) from each \(\mu_j\) does not change that sum. This identity shows why the two means differ: their difference is the size–mean association across clusters, scaled by the total number of individuals.

If larger clusters tend to have larger means, the right-hand sum tends to be positive and individual weighting tends to give a larger mean. If larger clusters tend to have smaller means, it tends to be negative. The exact condition for equality is that the centered size–mean products sum to zero. Equal cluster sizes guarantee equality, as do equal cluster means, but neither is required: positive and negative products can also cancel.

What Each Average Represents

The cluster-weighted mean corresponds to selecting a cluster uniformly from the \(J\) target clusters and then using that cluster’s defined mean. It is useful when the scientific question concerns the average cluster, or when clusters are the units to which equal representation should be given. For example, it might be appropriate to describe the typical clinic when the goal is to compare clinics as organizations, regardless of patient volume.

The individual-weighted mean corresponds to selecting one individual uniformly from all eligible individuals in the target. A cluster containing 100 eligible individuals then contributes ten times as much as a cluster containing 10. This is often the natural target for a question about the expected outcome of a randomly selected patient from the pooled patient population.

These interpretations require a clearly defined target set. If the observed cluster sizes are merely a sample and the intended target is a broader population of clusters, plugging observed sizes directly into the formula may not represent the intended population average. Likewise, if a cluster’s size is related to who is observed or to the outcome process, the size-weighted average remains a well-defined descriptive target, but additional assumptions may be needed to interpret it as a broader population quantity.

Weighting Marginal Contrasts

The weighting choice also applies to contrasts, such as a difference in predicted probabilities between two treatment conditions. To target an average contrast across clusters, calculate each cluster’s contrast and average those contrasts equally. To target an average contrast across individuals, weight the cluster-specific contrasts by the same eligible individual counts. When comparing two conditions for the same target population, keep the target clusters, sizes, covariate definition, and weights consistent across conditions.

Assumptions, Diagnostics, and Alternatives

The arithmetic definitions themselves require only a specified set of clusters, positive eligible counts, and one comparable cluster-specific mean per cluster. Interpreting the results as model-based population means also requires that the cluster means be defined and estimated for the stated outcome, scenario, and covariate target. If predictions are intended to describe a wider population, the model and sampling assumptions supporting that interpretation must be appropriate; the formulas do not establish those assumptions.

Before reporting either average, check the following:

• Confirm the unit counted. Make sure \(n_j\) counts eligible members of the intended target, rather than visits, measurements, or duplicated records.
• Inspect the cluster sizes and means together. A table or plot of \(n_j\) against \(\mu_j\) can reveal whether a few large clusters dominate the individual-weighted result or whether size and mean tend to move together.
• Verify common definitions. The cluster means must refer to the same outcome scale, condition, and standardization target. Otherwise, the weighted average mixes unlike quantities.
• Check the weights sum to one. Cluster weights are \(1/J\); individual weights are \(n_j/N\). In either case their sum must equal 1.
• Separate point estimation from uncertainty. Individuals within a cluster are not automatically independent observations. Uncertainty calculations must reflect the clustered design and the way the means were estimated; the weighted point estimate alone does not supply a valid standard error.

A common alternative is to report both estimands, with their target populations stated explicitly. Another is to use design or survey weights when the target population’s sampling design requires them. Such weights are not interchangeable with \(n_j/N\): they answer a different population-representation problem and must be justified by the design. The choice is not “weighted is better” versus “unweighted is safer.” It is a choice between well-defined targets.

A discrepancy between the two averages is therefore informative, not necessarily evidence of a defective GLMM. It can indicate that cluster size and expected outcome are associated, or simply that the estimands represent different sampling schemes. Reporting the weighting rule and the target unit makes the result interpretable and helps readers see why another analysis might produce a different marginal mean.