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ANOVA estimates 3 sample variances: a total variance based on all the observation deviations from the grand mean, an error variance based on all the observation deviations from their appropriate treatment means, and a treatment variance. The treatment variance is based on the deviations of treatment means from the grand mean, the result being multiplied by the number of observations in each treatment to account for the difference between the variance of observations and the variance of means--wikipedia.org

In ANOVA I know the null hypothesis is

$\mu_1=\mu_2=...\mu_n$

This satisfies the definition of a nuisance parameter.

The analysis of variance can be presented in terms of a linear model, which makes the following assumptions about the probability distribution of the responses: ${ }^{[14][15][16][17]}$

  • Independence of observations - this is an assumption of the model that simplifies the statistical analysis.
  • Normality - the distributions of the residuals are normal.
  • Equality (or "homogeneity") of variances, called homoscedasticity-the variance of data in groups should be the same.

The separate assumptions of the textbook model imply that the errors are independently, identically, and normally distributed for fixed effects models, that is, that the errors $(\varepsilon)$ are independent and $$ \varepsilon \sim N\left(0, \sigma^{2}\right) $$

Do we know $\sigma^2$ or is that a parameter we have to estimate? We do not even hypothesis test on the error variance. Doesn't that mean the residual variance is also a nuisance parameter?

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  • $\begingroup$ So that we might know how to respond, and at what level of detail, could you indicate which parts of this question (and your subsequent one) are not answered by the definition of "nuisance parameter" on the first line of the Wikipedia article? $\endgroup$
    – whuber
    Aug 5 at 13:59

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