Randomized complete-block design

Fix integers $t\ge 2$ and $b\ge 1$. There are $b$ blocks, each containing the same $t$ fixed experimental units, indexed within block $j$ by $i\in \{1,\dots ,t\}$. The $t$ treatments form a fixed set $\mathcal{T}$ with $\mathrm{\mid }\mathcal{T}\mathrm{\mid }=t$. A complete block is a block in which each treatment is assigned to exactly one unit. The blocks and their units are fixed before randomization.

Let $S_t$ be the set of permutations from the unit labels to $\mathcal{T}$. The randomized complete-block design (RCBD) uses assignment space $\mathcal{Z}=(S_t{)}^b$ and the uniform probability law on this finite nonempty set. Thus each $z=({\pi }_1,\dots ,{\pi }_b)$ has probability $(t!{)}^{-b}$; the within-block treatment permutations are independent and individually uniform. This is a finite randomized-treatment assignment model (Finite randomized treatment-assignment models) whose units are the $tb$ cells and whose assignment is the only randomness supplied by the design. The design specifies no response model, Gaussian errors, additivity, or absence of interference. A pair is the special case $t=2,b=1$.