# Thread: ANOVA and random effects model

1. ## ANOVA and random effects model

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.

2. Bump. I am stuck on this problem as well. To do this, I know I have to figure out Cov(Yij, Yi.) where Yi. is the mean of group i.

I am not quite sure how to find this covariance. Any suggestions?