Consider a one-way layout in which $\displaystyle I $ groups being compared are regarded as a sample from some larger population. $\displaystyle J $ is the number of subsamples taken from each $\displaystyle I $. The random effects model is:

$\displaystyle Y_{ij} = \mu + A_i + \epsilon_{ij}$

$\displaystyle A_i$ are random and independent of each other with $\displaystyle E(A_i) = 0 $ and $\displaystyle Var(A_i) = \sigma_A^2 $

$\displaystyle \epsilon_{ij} $ are independent of $\displaystyle A_i $, independent of each other and have $\displaystyle E(\epsilon_{ij}) = 0 $ and $\displaystyle Var(\epsilon_{ij}) = \sigma_{\epsilon}^2 $

$\displaystyle Var(Y_{ij}) =\sigma_A^2 + \sigma_{\epsilon}^2 $

Show that for this model:

$\displaystyle E(MS_W) = \sigma_{\epsilon}^2 $

and

$\displaystyle E(MS_B) = \sigma_{\epsilon}^2 + J\sigma_A^2$

MS is the sum of squares divided by degrees of freedom. W means within subsamples and B means between groups.