Blocks in experimental design

A block groups experimental units expected to have similar background responses. Comparing treatments within blocks can reduce the variance of an estimator. Block allocation must be specified before responses are used to evaluate treatment effects.

A block design assigns treatments to experimental units partitioned into blocks. Its incidence counts $n_{ij}$ record how often treatment $i$ occurs in block $j$. Balance and connectedness determine which treatment contrasts are estimable and how much adjustment for blocks costs.

A row-column design has two crossed partitions of experimental units into rows and columns. Equal treatment proportions in every row and every column make treatment contrasts orthogonal to both centered block spaces. A repeated three-treatment arrangement in a six-by-six array can give a balanced example.

A randomized complete block design places every treatment in every block, with a common replication pattern, and uses independent within-block randomization. The usual basic version has one occurrence of each treatment per block; equal repeated occurrences give a replicated complete-block version. Such equal-proportion designs are orthogonal block designs.

An orthogonal block design has centered treatment-indicator vectors orthogonal to centered block-indicator vectors on the experimental units. With treatment replication $r_i$, block size $k_j$ and total $n$, this is equivalent to $n_{ij}=r_i{k}_j/n$. It permits additive block adjustment without changing treatment estimates.