34
M. Slatkin
2.2
Tests of whether D = 0
The quantity D is, as will be seen below, useful for some purposes, but its value
alone does not tell us whether there is statistically significant LD between a pair
of loci, that is, whether the hypothesis that D = 0 can be rejected. Two tests are
commonly used. One is the standard χ 2 test of significance for a 2×2 contingency
table whose entries are the numbers of each of the four haplotypes. Closely related
to the χ 2 test is the r 2 statistic,
r
2
=
D 2
f A f a f B f b
(2.6)
which provides another way to quantify the extent of LD. r 2 is formally a correlation
coefficient. Although the upper bound of r 2 is 1, it does in general not take the value
1 even when D = 1. It is convenient to use r 2 because, when testing for significance
in the contingency table, χ 2 = nr 2 where n is the number of haplotypes sampled.
This result follows from the fact that D is the difference between the observed
haplotype frequency and the haplotype frequency expected if alleles at the two loci
are randomly associated. Such a difference arises naturally when doing a χ 2 test of
statistical significance in a 2×2 contingency table. When sample sizes are large and
neither allele is rare, the χ 2 test is powerful and easy to use. When sample sizes are
small or at least one of the haplotypes is rare (<5 in count), then Fisher’s exact test
is preferable (Weir 1996).
For most practical purposes, D and r 2 are equally useful, and in real data sets,
their values are highly correlated. One important feature of D is that when its value
is 1, at least one of the four haplotype frequencies is 0. That situation is particularly
important because, when a new mutation arises at a previously monomorphic locus,
D = 1 with all polymorphic loci on the same chromosome. The single copy of the
new mutant arises on only one genetic background. After this time, D between the
mutant and another locus becomes less than 1 only if there is recombination between
the two loci.
2.3
More than Two Alleles per Locus
Describing LD for a pair of biallelic loci is relatively simple because a single quantity D, combined with the allele frequencies, provides a complete characterization.
If there are more than two alleles at either or both loci, more coefficients of LD
are needed, one for the difference between the frequency of each haplotype and the
frequency expected under random association. If the alleles at one locus are A 1 , A 2 ,
A 3 . . . and those at the other are B 1 , B 2 , B 3 . . . , then
D ij = f A i B j − f A i f B j .
(2.7)
M. Slatkin
2.2
Tests of whether D = 0
The quantity D is, as will be seen below, useful for some purposes, but its value
alone does not tell us whether there is statistically significant LD between a pair
of loci, that is, whether the hypothesis that D = 0 can be rejected. Two tests are
commonly used. One is the standard χ 2 test of significance for a 2×2 contingency
table whose entries are the numbers of each of the four haplotypes. Closely related
to the χ 2 test is the r 2 statistic,
r
2
=
D 2
f A f a f B f b
(2.6)
which provides another way to quantify the extent of LD. r 2 is formally a correlation
coefficient. Although the upper bound of r 2 is 1, it does in general not take the value
1 even when D = 1. It is convenient to use r 2 because, when testing for significance
in the contingency table, χ 2 = nr 2 where n is the number of haplotypes sampled.
This result follows from the fact that D is the difference between the observed
haplotype frequency and the haplotype frequency expected if alleles at the two loci
are randomly associated. Such a difference arises naturally when doing a χ 2 test of
statistical significance in a 2×2 contingency table. When sample sizes are large and
neither allele is rare, the χ 2 test is powerful and easy to use. When sample sizes are
small or at least one of the haplotypes is rare (<5 in count), then Fisher’s exact test
is preferable (Weir 1996).
For most practical purposes, D and r 2 are equally useful, and in real data sets,
their values are highly correlated. One important feature of D is that when its value
is 1, at least one of the four haplotype frequencies is 0. That situation is particularly
important because, when a new mutation arises at a previously monomorphic locus,
D = 1 with all polymorphic loci on the same chromosome. The single copy of the
new mutant arises on only one genetic background. After this time, D between the
mutant and another locus becomes less than 1 only if there is recombination between
the two loci.
2.3
More than Two Alleles per Locus
Describing LD for a pair of biallelic loci is relatively simple because a single quantity D, combined with the allele frequencies, provides a complete characterization.
If there are more than two alleles at either or both loci, more coefficients of LD
are needed, one for the difference between the frequency of each haplotype and the
frequency expected under random association. If the alleles at one locus are A 1 , A 2 ,
A 3 . . . and those at the other are B 1 , B 2 , B 3 . . . , then
D ij = f A i B j − f A i f B j .
(2.7)