---
tag: VC8286D7TZ
kind: definition
name: Blocking
msc: [62K10]
defines: XCWKYS1DFA
uses: [2TE42ZWJ3E, 734Q6Q51X7]
sources:
  - url: https://en.wikipedia.org/wiki/Blocking_(statistics)
---

## Definition

Let $T$ be a finite set of treatments and $U$ a finite set of experimental units.
*Blocking* is a partition $U = B_1 \cup \cdots \cup B_b$ of the units into disjoint *blocks* of units that are expected to respond alike.
The partition is fixed before the treatments are assigned, and treatments are then assigned and compared within blocks.
A [[design|2TE42ZWJ3E]] $d \colon U \to T$ is a *randomized complete block design* if:

1. each block has exactly $\abs{T}$ units;
2. each treatment occurs exactly once in each block, that is, $d$ restricted to $B_j$ is a bijection onto $T$ for every $j$;
3. the bijections on the different blocks are chosen independently and uniformly at random, by [[randomization|734Q6Q51X7]] within blocks.

## Remarks

The usual model for a randomized complete block design is $y_{ij} = \mu + \tau_i + \beta_j + \varepsilon_{ij}$, with treatment effects $\tau_i$, block effects $\beta_j$ and independent errors.
Differences $y_{ij} - y_{i'j}$ within block $j$ do not contain $\beta_j$, so block effects drop out of treatment comparisons.

Example: three varieties $T = \set{A, B, C}$ on four fields, each cut into three plots.
The fields are the blocks, each of size $3$; in each field the three varieties are put on the three plots in an order chosen at random.

Non-example: if each field receives a single variety on all its plots, clause 2 fails.
The variety effect is then confounded with the field effect, and the two cannot be separated.

When blocks are smaller than the number of treatments, clause 1 fails and one uses incomplete block designs, such as balanced incomplete block designs.