A random effects model in the context of ANOVA looks like:
$$ Y_{ij} = \mu + \alpha_j + \varepsilon_{ij}, $$
where $j \in 1, \ldots, a$ are the $a$ levels in the population. Both $\alpha_j$ and $\varepsilon_{ij}$ are Normal random variables with mean 0 and variance $\sigma_{\alpha}^2$, $\sigma_{\epsilon}^2$, respectively. Then, it is commonly stated that the intraclass correlation is:
$$ \frac{\sigma_\alpha^2}{\sigma_\alpha^2+\sigma_\varepsilon^2}. $$
I am wondering how this is computed?
The intra-class covariance is, $$ Cov(Y_{1j},Y_{2j}) = E((\alpha_j+\epsilon_{1j})(\alpha_j+\epsilon_{2j})) = E(\alpha_j^2) = \sigma_\alpha^2$$ and the variance is $$ Var(Y_{1j}) = E((\alpha_j+\epsilon_{1j})^2) = E(\alpha_j^2)+E(\epsilon_{1j}^2)=\sigma_\alpha^2+\sigma_\epsilon^2$$ so the correlation is $$ Cor(Y_{1j},Y_{2j}) = \frac{Cov(Y_{1j},Y_{2j})}{\sqrt{Var(Y_{1j})Var(Y_{2j})}} = \frac{\sigma^2_\alpha}{\sigma^2_\alpha+\sigma^2_\epsilon}.$$
Going between equalities, I used are definitions, multiplied stuff out, and used the fact that the $\sigma_j$ $\epsilon_{1j}$ and $\epsilon_{2j}$ are mean zero and uncorrelated.