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
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): balanced incomplete block design
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): balanced incomplete block design