Page 97 - Applied Probability

5
Genetic Identity Coefficients
5.1 Introduction

Genetic identity coefficients are powerful theoretical tools for genetic analy-
sis. Geneticists have devised these indices to measure the degree of inbreed-
ing of a single individual and the degree of relatedness of a pair of relatives.
Since the degree of inbreeding of a single individual can be summarized by
the relationship between his or her parents, we will focus on identity coef-
ficients for relative pairs. These coefficients pertain to a generic autosomal
locus and depend only on the relevant pedigree connecting two relatives
and not on any phenotypes observed in the pedigree. In Chapter 6 we will
investigate the applications of identity coefficients. Readers desiring moti-
vation for the combinatorial problems attacked here may want to glance at
Chapter 6 first.

5.2 Kinship and Inbreeding Coefficients

Two genes G 1 and G 2 are identical by descent (i.b.d.) if one is a physical
copy of the other or if they are both physical copies of the same ancestral
gene. Two genes are identical by state if they represent the same allele.
Identity by descent implies identity by state, but not conversely. The sim-
plest measure of relationship between two relatives i and j is their kinship
coefficient Φ ij . Mal´ecot [12] defined this index to be the probability that a
gene selected randomly from i and a gene selected randomly from the same
autosomal locus of j are i.b.d. The kinship coefficient takes into account
the common ancestry of i and j but not their observed phenotypes at any
particular locus. When i and j are the same person, the same gene can
be drawn twice because kinship sampling is done with replacement. The
inbreeding coefficient f i of an individual i is the probability that his
or her two genes at any autosomal locus are i.b.d.; inbreeding sampling is
1
done without replacement. Since Φ ii = (1+f i ) and f i =Φ kl , where k and
2
l are the parents of i, an inbreeding coefficient entails no new information.
Note that f i = 0 unless i’s parents k and l are related. If f i > 0, then i is
said to be inbred.
The last column of Table 5.1 lists kinship coefficients for several com-
mon types of relative pairs. The table also contains probabilities for other
identity coefficients. Before defining these additional indices of relationship,
let us focus on a simple algorithm for computing kinship coefficients. This