Blocking

Builds on Design of experiments Randomization

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 $d \colon U \to T$ is a randomized complete block design if:

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

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Design of experiments concept
Randomization concept
Blocking definition

Tag	`VC8286D7TZ`
URI	`m2://VC8286D7TZ`
Kind	definition
Writing	W0 stub
MSC	62K10
Defines	Blocking
Uses	Design of experiments, Randomization

Blocking

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here