IntraclusterKorrelation

IntraclusterKorrelation, commonly referred to as the intracluster correlation coefficient (ICC), is a statistic that describes the degree of similarity among observations that share the same cluster or group. It is widely used in cluster sampling, multilevel modeling, and any study design in which data are collected in naturally occurring groups such as classrooms, clinics, or households.

Mathematically, ICC reflects the proportion of total variance that is attributable to differences between clusters. In a simple random-effects model y_ij = μ + u_j + ε_ij, where u_j is the between-cluster effect with variance τ^2 and ε_ij is the within-cluster error with variance σ^2, ICC = τ^2 / (τ^2 + σ^2). This means that a higher ICC indicates more similarity among observations within the same cluster. In practice, ICC is often estimated from data using an ANOVA-based formula, such as ICC = (MS_between − MS_within) / (MS_between + (k − 1)MS_within) for k observations per cluster, or by fitting linear mixed-effects models.

Interpretation and implications: ICC values range from 0 to 1, with values near 0 suggesting little clustering and values near 1 indicating strong within-cluster similarity. The ICC influences the design effect and effective sample size: DEFF = 1 + (m − 1) × ICC, where m is average cluster size. Consequently, higher ICCs reduce the information gained from each additional observation within a cluster and require larger overall samples.

Estimation can be frequentist (REML/ML) or Bayesian, and for binary outcomes, ICC concepts extend via latent-variable formulations. IntraclusterKorrelation provides a concise summary of clustering structure essential for analysis planning and interpretation in hierarchical data settings.