# Random effects ANOVA: what happened to the assumption of independence?

It's often stated that regular (fixed effects) ANOVA assumes independence of observations, but that in random effects ANOVA there is no such assumption.

I think the following represents a standard way to represent the two ANOVAs with formulae:

• $Y_{ij} = \beta_{0} + \beta_{j} + \epsilon_{ij}$ (fixed effects ANOVA)

• $Y_{ij} = \gamma_{00} + u_{j} + \epsilon_{ij}$ (random effects ANOVA)

I think I understand the basics of the difference between the two, inasmuch as in the fixed effects ANOVA the null hypothesis is that the $\beta_{j}$s is equal to zero, whereas in random effects ANOVA the null hypothesis is that $u_{j}$ has zero variance.

However, those two null hypotheses seem fundamentally similar to me, and I don't see how they license the relaxation of the assumption of independence.

In the random effects ANOVA the observations $$Y_{ij}$$ with common $$j$$ are not independent, because they share the common random effect $$u_j$$. But the independence assumption in this models are not about the observations, it is about the error terms $$\epsilon_{ij}$$. Both models contain this terms.
For the fixed effects ANOVA this details does not matter much, since in fact also the $$Y_{ij}$$ are independent. But when entering into random effects models, this detail matters!