Assumptions of mixed effects model I am reading a book which has a chapter on mixed effects models, but I am a little confused by both the model and the underlying assumptions.
Suppose we have the following basic mixed effects model:
$$y_{ij}=\mu+\alpha_{i}+\epsilon_{ij},$$
for levels $i=1,...,a$, observations $j=1,...,n$, and where the $\alpha_{i} \sim N(0,\sigma^2_{\alpha})$ and $\epsilon_{ij}\sim N(0,\sigma^2_{\epsilon})$ are independent and identically distributed random effects, and the $\mu$ is a fixed effect.
My questions are as follows...

*

*The random effects for different levels, $\alpha_i$ and $\alpha_j$, $i \neq j$, are modelled as identically distributed i.e. they share a common mean and variance. The goal of inference is to estimate the common variance parameter $\sigma_{a}^2$. Does this mean that we are not interested in the effect of any in particular level, but only how the effects of the levels vary?

*In the book they talk about correlation between levels, and claim that "when there is no variation between the levels, $\sigma_{a}^2=0$, and when the variation between the levels is much larger than within the levels, $\sigma_{a}^2$ is large". I don't think I understand how there can be any variation between the levels, when both the effects $\alpha_i$ and the errors $\epsilon_{ij}$ are modelled as independent?

I don't think I understand the underlying assumptions of the model. Namely, are the distributions for the different random effects identically distributed? Are the effects for different levels independent? If I were to sample many observations from the same level, would they only vary through the $\epsilon_{ij}$ term, because $\alpha_i$ isn't indexed by $j$?
Thanks. If anyone can recommend a thorough introduction to mixed effects modelling I would be grateful.

Random effects model in my opinion is easiest to understand when you write down the probabilistic model behind it:
$$\begin{align}
a_i &\sim \mathcal{N}(0, \, \sigma^2_a) \\
y_{ij} &\sim \mathcal{N}(\mu + a_i, \, \sigma^2_\epsilon)
\end{align}$$
As you immediately see, $a_i$ and $y_{ij}$ are both random variables. In fixed effects model $a_i$ would be just "fixed" parameters like $\mu$ in the above example.
Your first question relates more to the nuances of the frequentist setting, because in Bayesian setting all the parameters are considered as random variables, so there is no distinction between fixed and random effects. Hopefully, we already have many good answers that explain it in detail, see What is the mathematical difference between random- and fixed-effects? or What is a difference between random effects-, fixed effects- and marginal model?. The biggest difference is that in fixed effects model $a_i$ are independent, "fixed" parameters, while in random effects model they are assumed to be random variables with the same distribution. In this sense we care about the variance of the distribution of those parameters. When fitting random effects models we use special methods that help us to precisely estimate the variance. You may also want to read What is the upside of treating a factor as random in a mixed model? or Fixed effect vs random effect when all possibilities are included in a mixed effects model for deeper insight on the differences between fixed and random effects, because it is quite subtle.
Answering your second question, what is meant is that $a_i$ are independent and identically distributed random variables, with variance $\sigma^2_a$. If the variance $\sigma^2_a=0$ this means that the distribution is degenerate, because it is constant, so all the $a_i$ are the same. With large $\sigma^2_a$ variance, the $a_i$ would vary a lot. It has nothing to do with $\epsilon_{ij}$ errors.
