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.