An apparently classic mantra appearing in about a dozen of ANOVA textbooxs i consulted reads roughly as: given any choice of $T-1$ orthogonal contrasts $C_1, \cdots, C_{T-1}$ in a (not necessarily balanced) $T$-groups (one-way) ANOVA design, the treatment (or "between-groups") sum of squares $SS_{Treat}$ can be decomposed as the sum $\sum_{j=1}^{T-1}SS_{C_j}$ of contrasts $C_j$ sums of squares.
I think it would be helpful to have a proof of this fact or a reference to one, as most textbooks seem to lack.
I do not know if of any help, but here are some of the definitions of the statistics involved: let $\overline{Y}_i$ be the mean of the $n_i$ observations in the $i$-th group and $\overline{Y}$ the mean of all the observations; then
--a contrast is a linear combination $\sum_{i=1}^{T} \lambda_i\overline{Y}_i$ such that $\lambda_i\in \mathbb{R}$ and $\sum_i\lambda_i=0$;
--if two contrasts have coefficients $\lambda_i$ and $\nu_i$ such that $\sum_i\lambda_i\nu_i/n_i=0$, they are said to be orthogonal;
--sum of squares are defined as: $SS_{Treat}=\sum_{k=1}^Tn_k(\overline{Y}_k-\overline{Y})^2$ and $SS_C=\dfrac{C^2}{\sum_{i=1}^T\frac{\lambda_i^2}{n_i}}$
thanks a lot.