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That's fine, thanks for clarifying. Can I ask, if the effect levels are independent, how did you arrive at the covariance for individuals in different groups being $\sigma^2_{\alpha}+\sigma^2_{\epsilon}$? because for $i \neq i^{\prime}$, $Cov[y_{ij},y_{i^{\prime}k}]=0$ because $\alpha_{i}$, $\alpha_{i^{\prime}}$, $\epsilon_{ij}$ and $\epsilon_{i^{\prime}k}$ are independent, right?
The reason I'm confused, is that I interpret that sentence as $\alpha_i$ and $\alpha_j$ are independent given $\alpha$ - which doesn't really make a ton of sense to me!
Hi Dan, cheers for the answer. I'm a little confused. You refer to $\alpha$ as a parameter, but it is a random variable, right? And please may you expand on what "The effects for different levels are independent, given the underlying parameter $\alpha$" means? Thanks.
