Randomization

Definition

Let $T$ be a finite set of treatments, let $U$ be a set of $n$ experimental units, and let $r_t \ge 1$ be integers with $\sum_{t \in T} r_t = n$. Let $\mathcal{D}$ be the set of experimental designs $d \colon U \to T$ with $\abs{d^{-1}(t)} = r_t$ for every $t \in T$. Complete randomization is the choice of a design $D$ at random from $\mathcal{D}$ with the uniform distribution, that is, $$P(D = d) = \frac{1}{\abs{\mathcal{D}}} = \frac{\prod_{t \in T} r_t!}{n!} \quad \text{for every } d \in \mathcal{D}.$$ More generally, a randomized design is a design chosen at random from a set of designs by a known probability distribution. In a restricted randomization the set is smaller, for example the designs that give every treatment once in each block.