Definition:Balanced Incomplete Block Design

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Definition

A balanced incomplete block design or BIBD with parameters $v, b, r, k, \lambda$ is a block design such that:

$v$ is the number of treatments
$b$ is the number of blocks
$k$ is the size of each block
$r$ is the number of blocks any treatment can be in
$\lambda$ is the number of times any two treatments can occur in the same block

and has the following properties:

Each block is of size $k$
All of the $\dbinom v 2$ pairs occur together in exactly $\lambda$ blocks.

A BIBD with parameters $v, b, r, k, \lambda$ is commonly written several ways, for example:

$\map {\operatorname {BIBD} } {v, k, \lambda}$
$\tuple {v, k, \lambda}$-$\operatorname{BIBD}$


Treatments may be compared after eliminating block effects by an appropriate analysis of variance.


Examples

Arbitrary Example

Let there be $4$ treatments $A, B, C, D$.

Let there be $6$ blocks of $2$ units each.

Then we can arrange the $4$ treatments into the $6$ blocks as:

$AB, AC, AD, BC, BD, CD$


Also see

  • Results about balanced incomplete block designs can be found here.


Linguistic Note

The incomplete descriptor of a balanced incomplete block design arises from the fact that there are not enough units to receive all treatments: $k < v$.


Sources