Fisher's inequality

Fisher's inequality is a necessary condition for the existence of a balanced incomplete block design, that is, a system of subsets that satisfy certain prescribed conditions in combinatorial mathematics. Outlined by Ronald Fisher, a population geneticist and statistician, who was concerned with the design of experiments such as studying the differences among several different varieties of plants, under each of a number of different growing conditions, called blocks.

Let:

$v$ be the number of varieties of plants;
$b$ be the number of blocks.

To be a balanced incomplete block design it is required that:

$k$ different varieties are in each block, $1 \leq k < v$ ; no variety occurs twice in any one block;
any two varieties occur together in exactly $λ$ blocks;
each variety occurs in exactly $r$ blocks.

Fisher's inequality states simply that

b \geq v

.

Proof

Let the incidence matrix $M$ be a $v \times b$ matrix defined so that $M i,j$ is 1 if element $i$ is in block $j$ and 0 otherwise. Then $B = MM T$ is a $v \times v$ matrix such that $B i,i = r$ and $B i,j = λ$ for $i \neq j$ . Since $r \neq λ$ , $det(B) \neq 0$ , so $rank(B) = v$ ; on the other hand, $rank(B) \leq rank(M) \leq b$ , so $v \leq b$ .

Generalization

Fisher's inequality is valid for more general classes of designs. A pairwise balanced design (or PBD) is a set $X$ together with a family of non-empty subsets of $X$ (which need not have the same size and may contain repeats) such that every pair of distinct elements of $X$ is contained in exactly $λ$ (a positive integer) subsets. The set $X$ is allowed to be one of the subsets, and if all the subsets are copies of $X$ , the PBD is called "trivial". The size of $X$ is $v$ and the number of subsets in the family (counted with multiplicity) is $b$ .

Theorem: For any non-trivial PBD, $v \leq b$ .^[1]

This result also generalizes the Erdős–De Bruijn theorem:

For a PBD with $λ = 1$ having no blocks of size 1 or size $v$ , $v \leq b$ , with equality if and only if the PBD is a projective plane or a near-pencil (meaning that exactly $n - 1$ of the points are collinear).^[2]

In another direction, Ray-Chaudhuri and Wilson proved in 1975 that in a $2 s -(v, k, λ)$ design, the number of blocks is at least ${\binom {v}{s}}$ .^[3]

Notes

^ Stinson 2003, pg.193
^ Stinson 2003, pg.183
^ Ray-Chaudhuri, Dijen K.; Wilson, Richard M. (1975), "On t-designs", Osaka Journal of Mathematics, 12: 737–744, MR 0592624, Zbl 0342.05018

References

R. C. Bose, "A Note on Fisher's Inequality for Balanced Incomplete Block Designs", Annals of Mathematical Statistics, 1949, pages 619–620.
R. A. Fisher, "An examination of the different possible solutions of a problem in incomplete blocks", Annals of Eugenics, volume 10, 1940, pages 52–75.
Stinson, Douglas R. (2003), Combinatorial Designs: Constructions and Analysis, New York: Springer, ISBN 0-387-95487-2
Street, Anne Penfold; Street, Deborah J. (1987). Combinatorics of Experimental Design. Oxford U. P. [Clarendon]. ISBN 0-19-853256-3.