Intraclass Correlation

Study guide: Exam MAS-II

The intraclass correlation (ICC) is the share of total variance in a Linear Mixed Model that sits between groups — equivalently, the correlation between two observations drawn from the same group.

\mathrm{ICC} = \rho = \frac{\sigma_b^{2}}{\sigma_b^{2} + \sigma^{2}}

• \sigma_b^2 is the random-intercept variance and \sigma^2 the residual variance — the two variance components
• \rho = 0: groups are indistinguishable and ordinary regression would do. \rho \to 1: observations within a group are near-copies, so a group of n carries little more information than a group of 1
• It is the correlation the random intercept induces — the off-diagonal of a compound-symmetric covariance structure
• The design effect 1 + (n-1)\rho says how much a clustered sample inflates the variance of a mean; effective sample size is n / [1 + (n-1)\rho]
• Ignoring a non-zero ICC is the classic mistake — standard errors come out too small and t-statistics too large

Effective Sample Size Under Clustering Claim severities are grouped by claims office, with \sigma_b^2 = 0.4 and \sigma^2 = 1.6. Each of 25 offices contributes 20 claims. What is the ICC, and how many independent claims is the 500-claim sample worth?

Answer \rho = \frac{0.4}{0.4 + 1.6} = 0.20 Design effect = 1 + (20 - 1)(0.20) = 4.8, so n_{\text{eff}} = \frac{500}{4.8} \approx 104 500 clustered claims carry about as much information as 104 independent ones. Treating them as independent would understate standard errors by a factor of \sqrt{4.8} \approx 2.2.