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My goal is to drive $E(MSA)$ for this design(the 1-way ANOVA); $$y_{ij}=\mu +\alpha_i + \epsilon_{ij}$$for i=1,2,...,a (that is, there are a groups)and j=1,2,...,n. $\epsilon_{ij}$ and $\alpha_i$ are random as follows;$$E(\alpha_i)=0, E(\epsilon_{ij})=0$$ $$var(\alpha_i)=\sigma_{\alpha}^2, var(\epsilon_{ij})=\sigma_{\epsilon}^2 $$and all the covariances are zero.

The answer shoud be $E(MSA)=n\sigma_{\alpha}^2+\sigma_{\epsilon}^2$.Could I let mathematica derive the answer? I have tried inputting as below naively, but I couldn't get the answer.

    Subscript[y, i] = (*definition for average value yi for each group i*)
      Sum[Subscript[y, i, j], 
        {j, 1, n}]/n
    
    
    
     Subscript[y] =  (*definition for average of all yij*)
      Sum[Subscript[y, i, j], 
        {i, 1, a}, {j, 1, n}]/
       (a*n)
    
    
    
Expectation[(n/(a - 1))*
   Sum[(Subscript[y, i] - 
      Subscript[y])^2, 
    {i, 1, a}], 
  {Distributed[Subscript[y, 
     i], NormalDistribution[
     0, Subscript[\[Sigma], \[Alpha]]]], 
   Distributed[Subscript[y], 
    NormalDistribution[0, 
     Subscript[\[Sigma], \[Epsilon]]]]}]

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    $\begingroup$ Please include all other definitions in your code, e.g. $y_i$ etc. Also see: How to copy code from Mathematica so it looks good on this site. $\endgroup$
    – MarcoB
    Commented Jun 6, 2023 at 13:50
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    $\begingroup$ Either your definition of $MSA$ or $E(MSA)$ is wrong. As evidence for that consider $a=1$. Then Sum[(Subscript[y, i] - Subscript[y])^2, {i, 1, a}] is identically 0 and $n \sigma_\alpha^2+\sigma^2$ can't be zero. Also, you should use indexed variables rather than Subscript. See mathematica.stackexchange.com/questions/245319/…. $\endgroup$
    – JimB
    Commented Jun 6, 2023 at 17:13
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    $\begingroup$ I suspect you want to multiply Sum[(Subscript[y, i] - Subscript[y])^2, {i, 1, a}] by n/(a - 1). $\endgroup$
    – JimB
    Commented Jun 6, 2023 at 17:29

1 Answer 1

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Here are three approaches that work for specified values of a and n:

For the first two approaches perform the following:

a = 6;
n = 7;
y = Table[μ + α[i] + ϵ[i, j], {j, 1, n}, {i, 1, a}];
overallMean = Total[Total[y]]/(n a);
groupMean = Total[y]/n;
msa = (n/(a - 1)) Sum[(groupMean[[i]] - overallMean)^2, {i, 1, a}] // Expand;

Approach 1: Using TransformedDistribution

dist = TransformedDistribution[msa, Flatten[{
  Table[α[i] \[Distributed] NormalDistribution[0, σα], {i, 1, a}], 
  Table[ϵ[i, j] \[Distributed] NormalDistribution[0, σ], {i, 1, a}, {j, 1, n}]}]];
Mean[dist] // Expand
(* σ^2 + 7 σα^2 *)

Approach 2: Using substitution.

Emsa = msa /. {α[i_]^2 -> σα^2, α[i1_] α[i2_] -> 0, ϵ[i_, j_]^2 -> σ^2, 
  α[i1_] ϵ[i2_, j_] -> 0, ϵ[i1_, j1_] ϵ[i2_, j2_] -> 0}
(* σ^2 + 7 σα^2 *)

Approach 3: Knowledge of distribution of sample group means

If it is recognized that the sample group means all have a mean of $\mu$ and variance $\sigma_\alpha^2+\sigma^2/n$, then one can use the following:

msa = (n/(a - 1)) Sum[(ybar[i] - Sum[ybar[j], {j, 1, a}]/a)^2, {i, 1,  a}]
Emsa = (msa // Expand) /. ybar[i_]^2 -> σα^2 + σ^2/n + μ^2 /. ybar[i_] -> μ // Expand
(* σ^2 + n σα^2 *)

With this 3rd approach we get the general result when giving a specific value of $a$.

Maybe someone else knows how to do this in a more general way.

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    $\begingroup$ Thanks for the accept but you might want to wait a day or two before accepting an answer because someone might give the general answer (and I'd certainly like to see that, too). $\endgroup$
    – JimB
    Commented Jun 7, 2023 at 15:34

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