Fisher's inequality is a
necessary condition
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...
for the existence of a balanced incomplete
block design
In combinatorial mathematics, a block design is an incidence structure consisting of a set together with a family of subsets known as ''blocks'', chosen such that number of occurrences of each element satisfies certain conditions making the co ...
, that is, a system of subsets that satisfy certain prescribed conditions in
combinatorial
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
. Outlined by
Ronald Fisher
Sir Ronald Aylmer Fisher (17 February 1890 – 29 July 1962) was a British polymath who was active as a mathematician, statistician, biologist, geneticist, and academic. For his work in statistics, he has been described as "a genius who a ...
, a
population geneticist and
statistician
A statistician is a person who works with Theory, theoretical or applied statistics. The profession exists in both the private sector, private and public sectors.
It is common to combine statistical knowledge with expertise in other subjects, a ...
, who was concerned with the
design of experiments
The design of experiments (DOE), also known as experiment design or experimental design, is the design of any task that aims to describe and explain the variation of information under conditions that are hypothesized to reflect the variation. ...
such as studying the differences among several different
varieties of plants, under each of a number of different growing conditions, called ''blocks''.
Let:
* be the number of varieties of plants;
* be the number of blocks.
To be a balanced incomplete block design it is required that:
* different varieties are in each block, ; no variety occurs twice in any one block;
* any two varieties occur together in exactly blocks;
* each variety occurs in exactly blocks.
Fisher's inequality states simply that
:: .
Proof
Let the incidence matrix be a matrix defined so that is 1 if element is in block and 0 otherwise. Then is a matrix such that and for . Since , , so ; on the other hand, , so .
Generalization
Fisher's inequality is valid for more general classes of designs. A ''pairwise balanced design'' (or PBD) is a set together with a family of non-empty subsets of (which need not have the same size and may contain repeats) such that every pair of distinct elements of is contained in exactly (a positive integer) subsets. The set is allowed to be one of the subsets, and if all the subsets are copies of , the PBD is called "trivial". The size of is and the number of subsets in the family (counted with multiplicity) is .
Theorem: For any non-trivial PBD, .
This result also generalizes the
Erdős–De Bruijn theorem:
For a PBD with having no blocks of size 1 or size , , with equality if and only if the PBD is a
projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane (geometry), plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, paral ...
or a near-pencil (meaning that exactly of the points are
collinear
In geometry, collinearity of a set of Point (geometry), points is the property of their lying on a single Line (geometry), line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, t ...
).
In another direction,
Ray-Chaudhuri and
Wilson proved in 1975 that in a design, the number of blocks is at least
.
Notes
References
*
R. C. Bose, "A Note on Fisher's Inequality for Balanced Incomplete Block Designs", ''
Annals of Mathematical Statistics
The ''Annals of Mathematical Statistics'' was a peer-reviewed statistics journal published by the Institute of Mathematical Statistics from 1930 to 1972. It was superseded by the '' Annals of Statistics'' and the '' Annals of Probability''. In 1 ...
'', 1949, pages 619–620.
* R. A. Fisher, "An examination of the different possible solutions of a problem in incomplete blocks", ''
Annals of Eugenics
The ''Annals of Human Genetics'' is a bimonthly peer-reviewed scientific journal covering human genetics. It was established in 1925 by Karl Pearson as the ''Annals of Eugenics'', with as subtitle, Darwin's epigram "I have no Faith in anything sho ...
'', volume 10, 1940, pages 52–75.
*
*
{{Experimental design, state=expanded
Combinatorial design
Design of experiments
Families of sets
Statistical inequalities
Extremal combinatorics
Ronald Fisher