Assumptions of the random-effects ANOVA Wilcox (1994) writes on p.289 that

The usual random effects model is that for $J$ randomly sampled
groups,
$X_{ij} = \mu + a_j + \epsilon_{ij},$
where $X_{ij}$ is the $i$-th observation randomly sampled from the
$j$-th group ($i$ = $1$, ... , $n_j$; $j = 1$, ..., $J$), $\mu$ is the
grand mean, $a_j = \mu_j - \mu$, $\mu_j$ is the mean associated with
the $j$-th group, $a_j$ is normally distributed with mean 0 and
variance $\sigma^2_a$, and $\epsilon_{ij}$ is normally distributed
with a mean of $0$ and variance $\sigma^2$ with $a_j$ and
$\epsilon_{ij}$ independent.

My questions are:

*

*Why does the random-effects ANOVA assume that $a_j$ and $\epsilon_{ij}$ will be independent? How would we test this assumption? Does this assumption have a "counterpart" in fixed-effects ANOVA?

*Is it the case, that as per this question with no answer, independent observations are not an assumption of the random-effects ANOVA? Why is that?

*Are there any other assumptions made by the random-effects ANOVA but not mentioned by Wilcox in the text above? I'm aware that other conditions may be helpful, e.g. that inferences might be suspect if we don't have equal variances. But I wonder if there are any other strict assumptions that underpin the model.

Wilcox, R. R. (1994). A one-way random effects model for trimmed means. Psychometrika, 59(3), 289-306.
 A: 
Why does the random-effects ANOVA assume that $a_j$ and $\epsilon_{ij}$ will be independent? How would we test this assumption? Does this assumption have a "counterpart" in fixed-effects ANOVA?

This assumption is there to handle omitted variable bias, that is, to exclude the possibility of other variable(s) not in the model being associated with the outcome, so from that point of view there is a parallel with regular ANOVA (and any other regression model for that matter). As far as testing it goes, I suppose we could just plot them and hope to see no pattern, as we would for a plot of residuals vs fitted values. We could maybe go a bit further and collapse the $\epsilon_{ij}$ to, say $\gamma_{j}$ in order that it can have the same dimension as $a_j$ and simply compute their correlation, although that would not rule out some nonlinear dependence.

Is it the case,independent observations are not an assumption of the random-effects ANOVA? Why is that?

That is because with repeated measures we expect observations within one group to be more similar to observations in the same group, than to observations in other groups and that is what $a_j$ is there to capture. The more similar they are within each group, that is, the more they are correlated, the larger $\sigma^2_a$ will be.

Are there any other assumptions made by the random-effects ANOVA but not mentioned by Wilcox in the text above? I'm aware that other conditions may be helpful, e.g. that inferences might be suspect if we don't have equal variances. But I wonder if there are any other strict assumptions that underpin the model.

Not that I am aware of.
