Could anybody demonstrate or direct me to a readily available proof of the following:
For the cell means model:
$$ y_{ij} = \mu_{i} + \epsilon_{ij},\ \text{ for }\ i = 1, \ldots, r\ \text{ and }\ j = 1, \ldots, n_{i}.$$
Show that:
$$ \sum_{i=1}^{r}\sum_{j=1}^{n_{i}}(y_{ij}-\bar{y}_{\cdot\cdot})^{2} = \sum_{i=1}^{r}\sum_{j=1}^{n_{i}}(y_{ij}-\bar{y}_{i\cdot})^{2} + \sum_{i=1}^{r}n_{i}(\bar{y}_{i\cdot}-\bar{y}_{\cdot\cdot})^{2} $$
where the first term is $\text{SS}_{\text{TO}}$, the second $\text{SS}_{\text{E}}$ and the third $\text{SS}_{\text{TR}}$.
(I have an upcoming examination in linear models and this proof was required for one of the previous year's examinations, but, so far, I haven't had much success finding the proof.)
