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.

*

*As far as I understand, we may be interested in the effect of any particular level, but we may not. I.e. it is possible that we care about the $\alpha_i$s, but it is also possible the $\alpha_i$s are nuisance parameters and we only care about $\sigma_{\alpha}^2$. This depends on the situation.


*As you say, the distributions for different random effects are indeed (at least in the basic model) identically distributed. The effects for different levels are independent.
And as you further note, if you sample different observations from the same level, they would only vary through $\epsilon_{ij}$. The point the book is making, is about how much observations in different groups will vary.
Say we have two observations, $y_{k}$ and $y_{l}$. If they are in the same group (say $i$), then $$y_k - y_l = \alpha_i + \epsilon_k - \alpha_i - \epsilon_l = \epsilon_k - \epsilon_l \sim N(0, 2\sigma^2_{\epsilon})$$
If they are in different groups (say $i$ and $j$ respectively), then $$y_k - y_l = \alpha_i + \epsilon_k - \alpha_j - \epsilon_l \sim N(0,  2(\sigma^2_{\alpha} + \sigma^2_{\epsilon}))$$
So individuals in different groups vary more than individuals in the same group.
I find it helpful to think of in terms of how $\sigma^2_{\alpha}$ affects the variation between individuals in different groups. First let's rewrite the model in an equivalent form.
Note you could equally define the model as
\begin{equation}
y_{ij} = \alpha_j + \epsilon_{ij}
\end{equation}
where $\alpha_j \sim N(\mu,\sigma_{\alpha}^2)$ (note these $\alpha$ are different to yours as they have mean $\mu$ rather than 0).
You can show that the estimate for the mean $\hat{\alpha_i}$ is given by a weighted sum of the mean of the $i$th group $\bar{y_i}$ and the population mean for the $\alpha$, which is $\mu$. So

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*As $\sigma^2_{\alpha} \to 0$, the $\alpha_i \to \mu$ are all the same and individuals in different groups differ only by their error $\epsilon$, so there is no between-group variation, as if we had not included the $\alpha_i$s but set all equal to $\mu$.

*As $\sigma^2_{\alpha} \to \infty$, $\alpha_i \to \bar{y_i}$, the mean of the $i$th group. This is as if we had not modelled the $\alpha_i$s by assuming them to be normally distributed, but treated them as separate parameters in a fixed effects model.

I would recommend reading "Data Analysis Using Regression and Multilevel/Hierarchical Models" by Gelman and Hill.
