Sums of Squares for Hierarchical Designs
We will explore the Expected Mean Squares for a standard Split Plot Design.
Suppose $$G$$ levels of a factor $$A$$ with $$K$$ families $${\F_text{gk},\text{k}=1\ldots K\}$$ nested within each group $$\{A_text{g},text{g}=1\ldots G\}$$, and within that family there are $$\mbox{n}_text{jk}$$ individuals on whom measurements are made.
Then pooling the K variances we have
$$\mbox{Pooled Variance V = } \frac{\sum_text{k}text{K} (n_text{k}-1) \mbox{V}_text{k}}{\sum_text{k}text{K} (n_text{k} -1)} $$
and we can use this pooled variance to obtain the standard error of the mean since
$$\mbox{Pooled Mean Standard Error = } \sqrt{ \frac{\mbox{V}}{\sum_text{k}^text{K} n_text{k}} } $$