On page one of the handout entitled "Two-way Mixed Effects Models" we have the following values given for the expected values of the mean squares for the two-way fixed effects model:

$\begin{matrix} E ({MS}_{α}) & = & σ_{ε} + K n θ_{α}^{2} \end{matrix}$
$\begin{matrix} E ({MS}_{β}) & = & σ_{ε} + K n θ_{β}^{2} \end{matrix}$
$\begin{matrix} E ({MS}_{(α β)}) & = & σ_{ε} + n θ_{α, β}^{2} \end{matrix}$

On page three the expected values for the two-way mixed effects are given:

$\begin{matrix} E ({MS}_{α}) & = & σ_{ε} + n σ_{α, β}^{2} + K n θ_{α}^{2} \end{matrix}$
$\begin{matrix} E ({MS}_{β}) & = & σ_{ε} + K n σ_{β}^{2} \end{matrix}$
$\begin{matrix} E ({MS}_{(α β)}) & = & σ_{ε} + n σ_{α, β}^{2} \end{matrix}$

I understand the addition of $σ_{α, β}^{2}$ to $E ({MS}_{α})$ .

What I don't understand is why the additive term for $E ({MS}_{β})$ went from including $θ_{β}^{2}$ to instead using $σ_{β}^{2}$ . I can't see why randomly sampling subjects to take fixed level measurements from should change the expected value of those fixed levels.

So far as the null hypothesis it doesn't matter. So far as the F-ratio though it should matter because:

\begin{matrix} θ_{β}^{2} & = & \frac{\sum_{β_{i} \in β} β_{i}^{2}}{J - 1} & = & \frac{J}{J - 1} σ_{β}^{2} \end{matrix}

Those quantities to not grow at the same rate. It seems like the degrees of freedom for the F should be changed if it is $σ_{β}^{2}$ .