# Understanding the intra-class correlation coefficient

Suppose we fit a simple mixed / multilevel model with no predictors, which some call a variance components model:

$$y_{ij} = \beta_0 + u_j + e_{ij}$$

where $u_j$ are the level-2 residuals and $e_{ij}$ are the level-1 residuals, and we obtain estimates, $\hat{\sigma_u^2}$ and $\hat{\sigma_e^2}$ for the variance of $u_j$ and $e_{ij}$ respectively. I get it that

$$\nu = \frac{\hat{\sigma_u^2}}{\hat{\sigma_u^2} +\hat{\sigma_e^2}}$$ is the proportion of total variance in the response due to between-level-2 units, and so it makes sense that this is called the variance partition coefficient.

What I don't get is why $\nu$ can also be interpreted as the correlation between response scores for 2 randomly selected observations from within the same level-2 unit (ie the intra-class correlation). It makes sense that $\nu$ should be related to this correlation, but why should be be exactly equal ?

Common assumptions are that $$\textrm{Cov}(\mathbf{u}, \mathbf{e}) = \mathbf{0}$$ $$\textrm{Cov}(\mathbf{e}) = \sigma^2_e \mathbf{I}.$$

Let $i \neq i'$.

On the one hand, we have \begin{align*} \textrm{Var}(y_{ij}) & = \textrm{Var}(\beta_0 + u_j + e_{ij}) \\ & = \textrm{Var}(u_j + e_{ij}) \\ & = \textrm{Var}(u_j) + \textrm{Var}(e_{ij}) + 2 \textrm{Cov}(u_j, e_{ij})\\ & = \sigma^2_u + \sigma^2_e. \end{align*}

On the other hand, we have \begin{align*} \textrm{Cov}(y_{ij}, y_{i'j}) & = \textrm{Cov}(\beta_0 + u_j + e_{ij}, \beta_0 + u_j + e_{i'j}) \\ & = \textrm{Cov}(u_j + e_{ij}, u_j + e_{i'j}) \\ & = \textrm{Cov}(u_j, u_j) + \textrm{Cov}(u_j, e_{i'j}) + \textrm{Cov}(e_{ij}, u_j) + \textrm{Cov}(e_{ij}, e_{i'j}) \\ & = \sigma^2_u. \end{align*}

Hence \begin{align*} \textrm{Cor}(y_{ij}, y_{i'j}) & = \frac{\textrm{Cov}(y_{ij}, y_{i'j})}{\sqrt{\textrm{Var}(y_{ij})}\sqrt{\textrm{Var}(y_{i'j})}} \\ & = \frac{\sigma^2_u}{\sqrt{\sigma^2_u + \sigma^2_e} \sqrt{\sigma^2_u + \sigma^2_e}} \\ & = \frac{\sigma^2_u}{\sigma^2_u + \sigma^2_e}. \end{align*}

The latter is the correlation between measurement $y_{ij}$ and measurement $y_{i'j}$ ($i \neq i'$), i.e., the correlation between "any two responses having the same $j$".

• Thank you. This is a really nice answer ! But $\textrm{Cov}(\mathbf{e}) = \sigma^2_e \mathbf{I}$ seems like a strange assumption....isn't that more of a definition ? Commented Dec 5, 2012 at 16:09
• It just says that the error terms have constant variance and are uncorrelated. This might be true and this might be wrong... but that's the default option. Commented Dec 5, 2012 at 16:57
• Does this also hold for the other ICCs (ICC2,1; ICC2,k; ICC3,1; ICC3,k)?
– バシル
Commented Jul 4, 2023 at 9:38