Deriving the expectation of SSE in randomized block design I am trying to derive the expectation of the $\operatorname{SSE}$ in a randomized block design, $\mathbb{E}(\operatorname{SSE})$, with
$$\operatorname{SSE} = \sum_{i=1}^b \sum_{j=1}^k(y_{ij} - \overline{y}_{i\bullet} - \overline{y}_{\bullet j} + \overline{y}_{\bullet\bullet})^2$$
with
\begin{align*}
\overline{y}_{i\bullet}&=\frac{1}{k}\sum_{j=1}^k y_{ij},\\
\overline{y}_{\bullet j}&=\frac{1}{b}\sum_{i=1}^b y_{ij},\;\text{and}\\
\overline{y}_{\bullet\bullet}&=\sum_{i=1}^b \sum_{j=1}^k y_{ij}.
\end{align*}
I know that the answer should be $(k-1)(b-1)\sigma^2$, but I am unsure how to prove it. I thought about first showing that
$$\operatorname{SSE} = \operatorname{SS}_{\text{total}} - \operatorname{SS}_{\text{treatment}} - \operatorname{SS}_{\text{block}}$$
which I know how to do, and then taking the expectation of both sides. But this leads me to the problem of not knowing how to show that $\mathbb{E}(\operatorname{SS}_{\text{total}})=(bk-1)\sigma^2$ (I know how to take the expectation of $\operatorname{SS}_{\text{treatment}}$ and $\operatorname{SS}_{\text{block}}$).
If anyone has any pointers on how to find  $\mathbb{E}(\operatorname{SS}_{\text{total}})$ or how to manipulate $\operatorname{SSE}$ in such a way that I do not need to do it this way it would be greatly appreciated.
If it is useful I assume that $\sum_{i=1}^b\alpha_i =\sum_{j=1}^k \tau_j=0 $ with $\alpha_i$ and $\tau_j$ representing the block effect and treatment effect respectively.
Thank you.
 A: In a two-way classified data with one observation per cell, the fixed effects model is of the form
$$y_{ij}=\mu+\alpha_i+\tau_j+\varepsilon_{ij}\,,\quad i=1,2,\ldots,b\,;\,j=1,2,\ldots,k \tag{$\star$}$$
Here $\mu$ is a general effect, and $\alpha_i, \tau_j$ are additional effects subject to
$$\sum_i \alpha_i=\sum_j \tau_j=0$$
The $\varepsilon_{ij}$'s are i.i.d random errors with mean $0$ and variance $\sigma^2$.
If you write the expressions for $y_{ij},\overline y_{i\bullet}, \overline y_{\bullet j}$ and $\overline y_{\bullet\bullet}$ from $(\star)$, you will end up with
\begin{align}
\text{SSE}&=\sum_{i,j}(y_{ij} - \overline y_{i\bullet} - \overline y_{\bullet j} + \overline y_{\bullet\bullet})^2
\\&=\sum_{i,j}(\varepsilon_{ij} - \overline \varepsilon_{i\bullet} - \overline \varepsilon_{\bullet j} + \overline \varepsilon_{\bullet\bullet})^2
\\&=\sum_j\sum_i\left\{(\varepsilon_{ij} - \overline \varepsilon_{i\bullet})-(\overline \varepsilon_{\bullet j} - \overline \varepsilon_{\bullet\bullet})\right\}^2
\\&=\sum_j \left\{\sum_i (\varepsilon_{ij} - \overline \varepsilon_{i\bullet})^2 - b(\overline \varepsilon_{\bullet j} - \overline \varepsilon_{\bullet\bullet})^2 \right\}
\\&=\sum_{i,j} (\varepsilon_{ij} - \overline \varepsilon_{i\bullet})^2 - b \sum_j (\overline \varepsilon_{\bullet j} - \overline \varepsilon_{\bullet\bullet})^2 
\\&=\sum_{i,j}\varepsilon_{ij}^2-k\sum_i \overline \varepsilon_{i\bullet}^2 - b\sum_j \overline \varepsilon_{\bullet j}^2 + bk \,\overline \varepsilon_{\bullet\bullet}^2
\end{align}
Taking expectation gives you the desired answer.
