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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.

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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!

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