# Treatment subspaces in Bailey's Design of Comparative Experiments

A draft of Bailey's Design of Comparative Experiments may be found here; I am referring to Chapter 2, figure 2.1.

Consider $$t$$ treatments. We have $$N$$ plots labeled $$1, \dots, N$$. We apply treatment $$1$$ to plots $$1, \dots, r_1$$; $$2$$ to plots $$r_1 + 1, \dots, r_1 + r_2$$, and so forth.

Let $$\Omega = \{1, \dots, N\}$$ be the set of plots and let $$\mathcal{T}$$ be the set of treatments, for which $$|\mathcal{T}| = t$$. We define $$T: \Omega \to \mathcal{T}$$ as the treatment factor. The treatment subspace $$V_T$$ consists of the vectors in $$\mathbb{R}^{\Omega}$$ which are constant on each treatment.

Bailey's text refers to the following figure: What is the relevance of $$V_T$$? Usually when running an experiment, I would have the responses $$\mathbf{y}$$, the treatments $$T$$, and the covariates (of which there are none here), so I don't quite understand what $$V_T$$ is. The $$\mathbf{u}_i$$ seem to be nothing more than vectors of treatment indicators for $$A$$, $$B$$, and $$C$$.

(I believe you can ignore what the "typical $$\mathbf{v}$$" row is without any issues.)

• Chapter 3 gives some examples of where $V_T$ and its decomposition into subspaces is of interest Feb 2 at 1:30