Unequal Cluster Sizes in Nested Models

Unequal Sizes Do Not Alter the Hierarchy

Nested data are often unbalanced: schools may contain different numbers of classes, and classes may contain different numbers of students. This variation changes how much information is observed in each group, but it does not by itself change which groups contain which observations. The key modeling task is to represent the membership structure correctly and then consider whether the observed cluster sizes are compatible with the assumptions and the intended interpretation.

Let \(a=1,\ldots,A\) index the highest-level groups, \(b=1,\ldots,B_a\) index the lower-level groups within group \(a\), and \(i=1,\ldots,n_{ab}\) index observations within lower-level group \(b\) of group \(a\). Both \(B_a\), the number of lower-level groups in group \(a\), and \(n_{ab}\), the number of observations in lower-level group \(ab\), may vary. The lower-level group is nested in the higher-level group when each lower-level group belongs to exactly one higher-level group.

A useful distinction is between the membership map and the counts. The membership map says which observation belongs to which lower-level group and which lower-level group belongs to which higher-level group. Counts summarize how many observations or subgroups appear under each parent. A data set can have highly unequal counts and still have a simple, strictly nested hierarchy.

A Nested Random-Intercept GLMM

For a response \(Y_{abi}\) from observation \(i\) in lower-level group \(ab\), consider the following random-intercept GLMM:

Here \(g\) is the link function, \(\mathbf{x}_{abi}\) is the covariate vector, \(\boldsymbol{\beta}\) is the vector of fixed-effect coefficients, \(U_a\) is the random intercept for higher-level group \(a\), and \(V_{ab}\) is the random intercept for lower-level group \(ab\). A standard specification assumes \(U_a\sim N(0,\tau_A^2)\) and \(V_{ab}\sim N(0,\tau_B^2)\), mutually independent across groups and levels. Conditional on the random effects and covariates, the responses are independent and follow the chosen response distribution, with mean related to the linear predictor through \(g\).

The index ranges allow unequal sizes directly: the model has a random effect \(U_a\) for each observed higher-level group and a random effect \(V_{ab}\) for each observed lower-level group. There is no requirement that all \(B_a\) values be the same, or that all \(n_{ab}\) values be the same. Unequal counts do not require a different nesting formula.

These assumptions are substantive. The response distribution and link must suit the outcome; the random-effects distribution and covariance structure must be reasonable; and the membership identifiers must be correct. Conditional independence means that, after conditioning on the modeled random effects and covariates, there is no additional unmodeled dependence among responses. If a cluster-size process is related to latent group characteristics in a way relevant to the response, treating the observed sizes as innocuous can also be inadequate.

What Unequal Sizes Change: Information and Precision

Although unequal sizes do not change the hierarchy, they do affect how much information is observed about each group. In a random-intercept model, a group with many observations generally provides more information about its own conditional mean than a group with few observations. It does not follow that every group should receive equal influence in every summary or that the largest groups are automatically representative of the intended population.

The covariance structure makes the role of membership explicit. For illustration, take a Gaussian identity-link model with independent residuals: \(Y_{abi}=\beta_0+U_a+V_{ab}+\varepsilon_{abi}\), where \(U_a\sim N(0,\tau_A^2)\), \(V_{ab}\sim N(0,\tau_B^2)\), and \(\varepsilon_{abi}\sim N(0,\sigma^2)\), with all components independent. The variances are assumed constant across groups. For two distinct observations, covariance comes from the random effects they share.

The first case shares both random intercepts; the second shares only the higher-level intercept; the third shares neither. The group-size counts do not appear in these pairwise covariance expressions. They determine how many observations of each pair type are present, and therefore affect the amount and balance of information available for estimating model parameters.

Conditional Predictions and Cluster-Size Interpretation

For a nonlinear link, the cluster sizes do not enter the conditional mean formula directly. They affect which observations are available and how strongly data inform the fitted effects, while a particular conditional probability is calculated from that observation's covariates and its group's random effects. This distinction is important: changing the number of students in a class does not by itself change the probability assigned to a particular student when the covariates and random effects are held fixed.

A separate issue is whether the desired summary gives equal weight to groups or equal weight to individuals. Those are different targets when group sizes vary. For example, the unweighted average of group-specific outcome rates gives each group the same contribution, whereas pooling all individual outcomes gives more influence to groups with more observations. Neither summary is universally correct; the choice should match the scientific question and the sampling design.

Justification: Why the Membership Pattern Determines Covariance

The covariance formula follows directly from the model and shows why changing counts is not the same as changing the hierarchy. Consider two distinct responses in the same class and write each as the sum of its fixed mean, the shared school effect, the shared class effect, and its own residual. Fixed means do not contribute to covariance, and independent residuals for distinct observations have covariance zero. Independence of the random effects then gives \(\operatorname{Cov}(Y_{abi},Y_{abi'})=\operatorname{Var}(U_a)+\operatorname{Var}(V_{ab})=\tau_A^2+\tau_B^2\).

For observations in different classes of the same school, \(U_a\) is shared but the class effects and residuals are independent, so the covariance is \(\operatorname{Var}(U_a)=\tau_A^2\). For observations from different schools, no random effect is shared, so the covariance is zero under this model. The derivation depends on who shares which random effects, not on how many observations happen to be in each group. In a GLMM with a nonlinear link, these expressions describe the latent linear predictor's random-effect covariance structure; they are not generally the response-scale covariances.

Diagnostics, Alternatives, and Limitations

Unequal sizes can make estimation less balanced and can expose model weaknesses. Check the group counts and the membership identifiers before fitting: duplicate labels, groups assigned to multiple parents, or a mistaken hierarchy can create a different dependence model. Examine whether a small number of very large clusters dominate the information, whether some levels have too few observations to estimate their variability reliably, and whether model convergence or random-effect estimates are sensitive to a few groups. Residual and fitted-value checks should respect the response distribution and the conditional structure; apparent extra dependence may indicate omitted structure rather than an unequal-size problem itself.

A particularly important failure mode is informative cluster size. Cluster size is informative when it is associated with outcomes or latent group characteristics in a way relevant to the analysis. For instance, if clinics with higher underlying event risk systematically contribute more patients, a model that conditions on observed size without accounting for this relationship may not answer the intended population-level question. The assumption that size is noninformative is not guaranteed merely because the random effects are normally distributed. Consider the data-generation and sampling processes, and decide whether the target is an average group, an average individual, or another population quantity. If needed, model the size process jointly, use a design-appropriate weighting or sampling approach, or conduct sensitivity analyses.

Common alternatives have distinct consequences. Aggregating each group to one summary loses individual-level information and can obscure within-group covariate effects; it is not a neutral way to handle unequal sizes. Treating groups as fixed effects may be useful for specific within-group comparisons, but it changes the target and does not estimate a population distribution of group effects in the same way as random effects. A model with only one grouping level is simpler, but it is inappropriate when dependence genuinely occurs at multiple nested levels. The appropriate choice follows the design and estimand, not the convenience of a balanced data table.