Intraclass Correlation in Nested Hierarchies

Why a Nested Hierarchy Has More Than One ICC

In a nested design, two observations may share some units but not others. Two students in the same class share both their class and school; students in different classes at the same school share the school but not the class. Their outcomes therefore share different amounts of modeled variation. A single “the ICC” can conceal this structure: the relevant intraclass correlation depends on which units the pair shares.

The derivation below uses a nested random-intercept model. Its central idea is direct: independent random effects contribute covariance to a pair of outcomes only when both outcomes contain the same random effect. Dividing that shared covariance by the marginal variance gives the correlation.

Three Nested Levels: Schools, Classes, and Students

Let student \(i\) be in class \(j\) at school \(k\). Consider the Gaussian random-intercept model

where \(a_k\) is a school random intercept, \(b_{jk}\) is a class-within-school random intercept, and \(\varepsilon_{ijk}\) is the student-level residual. Assume

with all random effects and residuals mutually independent. The random effects are also assumed independent of the predictors in the fixed part of the model. These assumptions imply a common variance at each level and no additional dependence beyond that induced by shared random effects. The correlations below are marginal over the random effects, conditional on the fixed predictors; the fixed part changes the expected outcomes but does not add covariance under this model.

Deriving the Correlations From Shared Effects

For any one student, the random part of the outcome is \(a_k+b_{jk}+\varepsilon_{ijk}\). Independence makes its variance the sum of the component variances:

For two distinct students in the same class and school, the two outcomes contain the same \(a_k\) and \(b_{jk}\), but have different independent residuals. Their covariance is therefore \(\tau_S^2+\tau_C^2\). For two students in different classes at the same school, only \(a_k\) is shared, so their covariance is \(\tau_S^2\). Students in different schools share neither random intercept, so their covariance is zero.

The correlation is covariance divided by the product of the standard deviations. Since all outcomes have the same marginal variance \(V=\tau_S^2+\tau_C^2+\sigma^2\), each pair’s correlation is its covariance divided by \(V\):

The first ICC is at least as large as the second, because the same-class pair shares every effect shared by the same-school pair, plus the class effect. Both are nonnegative under the independent random-intercept assumptions. The zero correlation for different schools is a model implication, not a universal claim about people at different schools.

Extending the Rule to a Deeper Hierarchy

The same reasoning applies when there are more than two nested grouping levels. Suppose level 1 is the highest level, level \(L\) is the lowest grouping level, and observations have a random intercept at each level as well as an observation-level residual:

Assume the random effects at all levels and the residual are mutually independent, with a common variance within each level. For a pair of distinct observations that share the same unit at levels 1 through \(q\), but diverge below level \(q\), their shared random effects are exactly \(U^{(1)},\ldots,U^{(q)}\). Thus the covariance is \(\sum_{\ell=1}^{q}\tau_\ell^2\), while each observation’s marginal variance is \(\sum_{\ell=1}^{L}\tau_\ell^2+\sigma^2\). Their ICC is

This is a cumulative quantity: moving down the hierarchy to a more closely shared unit adds variance components to the numerator. The formula is not a claim that variance components themselves are correlations. A component must be divided by total variance, and a pair sharing multiple levels must include all the components shared at those levels.

What Changes in a Generalized Linear Mixed Model?

For a generalized linear mixed model, a response-scale variance decomposition is not always available in the same form as for Gaussian outcomes. A common convention for binary GLMMs is to define an ICC on an underlying latent-variable scale. For a logistic random-intercept model, the level-one logistic distribution has variance \(\pi^2/3\). Replacing the Gaussian residual variance with this latent residual variance gives a useful, explicitly scale-dependent ICC. This is not generally the correlation of the observed zero-one outcomes.

Assumptions, Diagnostics, and Alternatives

The formulas are exact for the stated random-intercept model. Before using them, check that the hierarchy reflects the sampling structure, that levels are genuinely nested, and that the model includes the intended random intercept at each level. Independence of random effects, constant variances within levels, and absence of additional shared effects are essential to the simple sums. In Gaussian models, the residual assumptions should also be plausible after accounting for fixed predictors.

Useful checks include examining whether residual spread changes across groups, whether fitted group-level patterns suggest a missing random slope, and whether estimated variance components are near zero or on a boundary. A near-zero estimate can make the corresponding ICC nearly zero; it does not prove that the population variance is exactly zero. ICC estimates also have sampling uncertainty, so avoid treating rounded point estimates as precise constants. For non-Gaussian outcomes, state clearly whether the reported ICC is latent-scale or response-scale.

A random-intercept model is appropriate when grouping shifts the outcome level similarly across predictor values. If group differences in a predictor’s effect are plausible, random slopes can change covariance with predictor values, so one constant ICC may no longer summarize all pairs. Crossed classifications can also make pairs share a unit that is not part of a single nested chain; the nested cumulative formula then does not apply unchanged. A marginal model or a robust covariance approach answers a different modeling question and does not automatically provide the same variance-partition interpretation.

A frequent mistake is to call the class variance \(0.50\) an ICC of 0.50, or to use it as the correlation for every pair in the same school. The correction is to identify the pair’s shared effects, add their variances, and divide by total marginal variance. Another mistake is to interpret a logistic latent-scale ICC as a directly observed binary correlation. Always identify the model scale and the pair relationship in the interpretation.