46  ICC Derivation

Often, the ICC is described as the correlation between observations that share the same group membership. While you can look at the visualizer to get some intuition on what this means, here is a short proof adapted from S52 materials.

Consider the variance components model (this is the random intercept model with no covariates):

\[ y_{ij} = \beta_0 + \zeta_j + \varepsilon_{ij} \]

The correlation between an observation \(y\) and an observation from the same group \(y'\) is the standardized covariance:

\[ \rho(y, y') = \frac{cov(y,y')}{\sqrt{var(y)var(y')}} \]

We can expand the numerator, the covariance between \(y\) and \(y'\) and substitute in the definition of \(y\) from our model:

\[ cov(y,y') = cov(\beta_0 + \zeta_j + \varepsilon_{ij}, \beta_0 + \zeta_j + \varepsilon'_{ij}) \]

By definition, \(\beta_0\) is the same for everyone (i.e., the “constant” term), and \(\zeta_j\) will be the same for both observations because we are looking within a single cluster. The only difference between the two groups are the individual level error terms, \(\varepsilon_{ij}\). The rules of covariance tell us that the constant drops out and the \(\varepsilon\) too because it is independent of \(\zeta_j\), we can simplify our equation:

\[ cov(y,y') = cov(\zeta_j, \zeta_j) \]

The covariance of a variable with itself is the variance:

\[ cov(y,y') = cov(\zeta_j, \zeta_j) = var(\zeta_j) = \sigma^2_\zeta \]

Conceptually, the \(\zeta_j\) represents the shared influences on \(y\) that would cause the similarity between observations in the same group.

We know from our variance decomposition in the ICC formula that \(var(y)\) is the sum of the between-group and within-group variance components (note the independence of random effects assumption is key here):

\[ var(y) = var(y') = \sigma^2_\zeta + \sigma^2_\varepsilon \]

We can substitute these quantities into the original formula:

\[ \rho(y, y') = \frac{cov(y,y')}{\sqrt{var(y)var(y')}} = \frac{\sigma^2_\zeta}{\sigma^2_\zeta + \sigma^2_\varepsilon} = ICC \] Thus, the ICC is both the proportion of total variance accounted for by group membership and the correlation between pairs of observations drawn from the same group. QED!