Deriving the Gaussian Intraclass Correlation

Why a Shared Random Intercept Creates Similarity

In clustered data, two observations from the same clinic, classroom, or other group may resemble one another because they share a cluster-level condition. A Gaussian random-intercept model represents one source of that resemblance with a shared cluster-specific deviation. The intraclass correlation coefficient (ICC) summarizes the marginal correlation between two distinct outcomes in the same cluster under this model.

The ICC is not the correlation between a response and its fitted value, nor is it the correlation of residuals after conditioning on the random intercept. It is a property of the model’s marginal outcome distribution: it compares the covariance induced by the shared random intercept with the total outcome variability.

Model and Assumptions

Let \(Y_{ij}\) be the continuous outcome for observation \(i\) in cluster \(j\), and let \(\mathbf{x}_{ij}\) be its fixed-effect covariate vector. The Gaussian random-intercept model is

Here \(\boldsymbol{\beta}\) is the vector of fixed-effect coefficients, \(b_j\) is the random intercept for cluster \(j\), and \(\varepsilon_{ij}\) is the observation-level residual. For the derivation below, assume

• \(b_j\sim N(0,\tau^2)\), independently across clusters;
• \(\varepsilon_{ij}\sim N(0,\sigma^2)\), independently across observations;
• all residuals are independent of all random intercepts; and
• \(\tau^2\geq 0\), \(\sigma^2\geq 0\), with fixed covariates treated as nonrandom for this calculation.

The fixed part \(\mathbf{x}_{ij}^{\mathsf T}\boldsymbol{\beta}\) can differ across observations. The random intercept is shared by every observation in a cluster, while each observation has its own residual. The model also assumes that, after accounting for the fixed effects and random intercept, there is no additional residual correlation within a cluster.

Deriving the ICC

Write \(\mu_{ij}=\mathbf{x}_{ij}^{\mathsf T}\boldsymbol{\beta}\). Conditional on \(b_j\), the outcome has mean \(\mu_{ij}+b_j\) and variance \(\sigma^2\). For two distinct observations \(i\) and \(k\) in cluster \(j\), their residuals are conditionally independent under the assumptions. The marginal covariance follows from the law of total covariance:

The fixed means are constants in this calculation, and the two outcomes share the same \(b_j\). For either observation, the law of total variance gives

The same marginal variance applies to \(Y_{kj}\). Therefore, if \(\tau^2+\sigma^2>0\), their correlation is their covariance divided by the product of their standard deviations:

This derivation also explains why the ICC is sometimes described as the proportion of marginal variance attributable to between-cluster variation. The numerator is the variance of the shared random intercept; the denominator is the sum of that variance and the residual variance. This description is specific to the stated model and its variance decomposition.

Boundary Cases and Conditional Residuals

Variance components can lie on the boundary of their allowed range, so the formula’s denominator and the interpretation of correlation need checking. If \(\sigma^2>0\), the residuals are conditionally independent and have correlation zero. If \(\sigma^2=0\), each residual is degenerate at zero, and its correlation is undefined, not zero: a constant random variable has zero variance, so the correlation denominator is zero.

For the marginal outcomes, if \(\tau^2=0\) and \(\sigma^2>0\), the ICC is \(0\): there is no shared random-intercept variation. If \(\sigma^2=0\) and \(\tau^2>0\), the ICC is \(1\): after subtracting their fixed means, outcomes in the same cluster are identical because they share \(b_j\). If both variance components are zero, the marginal outcome variance is zero, so the outcome correlation and ICC are undefined.

Interpretation, Alternatives, and Model Checks

The ICC is a concise summary of within-cluster similarity under the random-intercept model. It is useful when the question concerns how strongly outcomes from the same cluster are associated relative to the model’s total outcome variation. A high ICC indicates that the shared cluster component accounts for a substantial fraction of that variation; a low ICC indicates that the residual component is larger relative to the shared component.

The ICC is not the only way to describe clustering. The pairwise covariance \(\tau^2\) is in squared outcome units and gives the absolute shared variation; the ICC is dimensionless and expresses that covariance relative to total variance. Reporting both can be informative because the same ICC can arise from very different total variances. The variance components themselves also retain their distinct roles: \(\tau^2\) describes between-cluster intercept variation, while \(\sigma^2\) describes observation-level residual variation.

Before interpreting an ICC, check whether the model’s assumptions are plausible. In particular, look for remaining patterns in residuals within clusters, unequal residual spread, and clusters whose outcomes show structure not captured by a single shared intercept. If repeated observations have additional time-related dependence, for example, the independent-residual assumption may fail. A random-intercept ICC then describes only the dependence represented by that model and may not capture the full correlation pattern.

A common mistake is to say, “The conditional residual correlation is zero, so the ICC is zero.” These are different quantities. When \(\sigma^2>0\), conditional residual correlation is zero under the model, but the marginal outcome ICC can be positive because two outcomes share \(b_j\). When \(\sigma^2=0\), the conditional residual correlation is undefined rather than zero. Another mistake is to treat the ICC as a causal effect or as a universal measure of similarity independent of model assumptions. It is a model-based correlation, not evidence that being in a particular cluster causes an outcome.