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Thread: ANOVA and random effects model

  1. #1
    Newbie
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    Feb 2009
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    4

    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.
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  2. #2
    Junior Member
    Joined
    Feb 2007
    Posts
    39
    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?
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