Intuition behind One-Way Random Effects ANOVA

I am trying to understand a single factor Random effects anova from a book.

In this case : -

$y_ij = \mu+ \tau_i + e_{ij}$, where $i = 1,...,a$ and $j= 1,...n$.

Assumptions:

• The treatment effects $\tau_i$ are $N(0,\sigma_\tau^2)$ and $e_{ij}$ are $N(0,\sigma^2)$
• $\tau_i$ and $e_{ij}$ are independent

Hence we have : $Variance(y_{ij}) = \sigma_\tau^2 + \sigma^2$

The observations in the random effects model are normally distributed because they are linear combinations of the two normally and independently distributed random variables $\tau_i$ and $e_{ij}$. However, unlike the fixed effects case in which all of the observations $y_{ij}$ are independent,in the random model the observations $y_{ij}$ are only independent if they come from different factor levels.

Note that the observations within a specific factor level all have the same covariance, because before the experiment is conducted, we expect the observations at that factor level to be similar because they all have the same random component. Once the experiment has been conducted, we can assume that all observations can be assumed to be independent, because the parameter $\tau_i$ has been determined and the observations in that treatment differ only because of random error.

I don't understand the above paragraph. Can some one please explain?

• I formatted your formulae using MathJax. Feel free to edit it if I missed something – Marquis de Carabas Jan 15 '17 at 6:59
• Thank you Marquis.Can you please fix the y_ij also on the 3rd line? – user2338823 Jan 15 '17 at 7:35
• go ahead and click "edit" to make the changes. I can't make edits if they are less than 10 characters (but if you are the author of the post, you can make as great or little change as you want). You would need to add curly brackets around $ij$, i.e. type y_{ij} – Marquis de Carabas Jan 15 '17 at 13:54