Page 117 - Applied Probability

6. Applications of Identity Coefficients
us do this calculation under the simplifying assumption that neither i nor
j is inbred. Conditioning on the various identity states and using the facts
δ
α k p k =0,

p k =1,

k
k
E(X i X j )
l kl p l = 0, and α k = l µ kl p l , we deduce 101
2
=∆ 7ij (α k + α l + δ kl ) p k p l
k l

+∆ 8ij (α k + α l + δ kl )(α k + α m + δ km )p k p l p m
k l m

+∆ 9ij (α k + α l + δ kl )(α m + α n + δ mn )p k p l p mp n
k l m n

2 2 2
=∆ 7ij 2 α p k +
k δ p k p l +∆ 8ij α p k
kl
k
k k l k

1 1 2 2
=2 ∆ 7ij + ∆ 8ij 2 α p k +∆ 7ij δ p k p l
k
kl
2 4
k k l
2
2
=2Φ ij σ +∆ 7ij σ ,
a d
2
2
2
2
δ p
where σ =2 α p k and σ = l kl k p l are explicit expressions for
a k k d k
the additive and dominance genetic variances. Since E(X i )= E(X j )= 0,
the desired covariance Cov(X i ,X j )= E(X i X j ). When i and j represent
2
2
the same person, Var(X i )= σ + σ is the total genetic variance. If i
a
d
σ .If i and j are siblings, then
is a parent of j, then Cov(X i ,X j )= 1 2 a 2
2
1
2
1
Cov(X i ,X j )= σ + σ .
2 a
4 d
The above arguments generalize to allow some environmental determi-
nation of the trait. Suppose that W i and W j are the random genotypes of
two non-inbred relatives i and j.If X i and X j are independent given W i
and W j , then the expression for Cov(X i ,X j ) continues to hold provided
we define µ kl =E(X | W = a k /a l ) for the trait value X and genotype W
of a random person. Indeed, in view of our convention that E(X)=0, we
find that
Cov(X i ,X j )=E(X i X j )
=E[E(X i X j | W i ,W j )]
=E[E(X i | W i )E(X j | W j )].
However, the total trait variance of any person is inflated because
Var(X)=Var[E(X | W)] + E[Var(X | W)]
2
2
= σ + σ + E[Var(X | W)].
a
d
These simple variance and covariance expressions extend straightfor-
wardly to polygenic traits, where many genes of small effect act additively
to determine a quantitative trait. Many interesting statistical problems
arise in this classical biometrical genetics setting.