Matvec Users’ Guide

Chapter 12
Segregation and Linkage Analyses
12.1 Genotype Probability Computation
It is very difficult, in general, to compute the genotype probabilities for each member of a pedigree with
loops . Van Arendonk et al. (1889), Janss et al. (1992), Fernando et al. (1993), and Wang et al. (1995)
discussed and proposed an iterative algorithm to compute the genotype probabilities for large and complex
livestock pedigrees. The proposed iterative algorithms are based on primary works by Murphy and Mutalik
(1969), Elston and Stewart (1971), and Heuch and Li (1972).
The conditional probability that pedigree member i has genotype u i given all the phenotypes y 1 , . . . , y n
can be computed as
Pr(u i | y 1 , . . . , y n ) = a i(u i )g(y i | u i ) ∏ jɛS i
{p ij (u i )}
∑
u i
a i (u i )g(y i | u i ) ∏ (12.1)
jɛS i
{p ij (u i )}
where a i (u i ) is the joint probability of phenotypes of members anterior to i and of genotype u i for i,
g(y i | u i ) is the penetrance function, and Π jɛSi {p ij (u i )} is the conditional probability of phenotypes of
members posterior to i, given i has genotype u i .
The anterior probability, a i (u i ), and posterior probability, p ij (u i ), can be computed as
a i (u i ) = ∑ u m
{a m (u m )g(y m | u m )
× ∑ u f
{a f (u f )g(y f | u f )
∏
jɛS m,j≠f
∏
jɛS f ,j≠m
p mj (u m )
p fj (u f ) (12.2)
×tr(u i | u m , u f )
⎡
⎤
× ∏ ⎣ ∑ tr(u j | u m , u f )g(y j | u j ) ∏ p jk (u j ) ⎦}}
u j kɛS j
jɛC mf ,j≠i
and
p ij (u i ) = ∑ ∏
{a j (u j )g(y j | u j ) p jk (u j ) (12.3)
u j
× ∏
kɛS j,k≠i
kɛC ij
[ ∑
u k
tr(u k | u i , u j )g(y k | u k ) ∏ lɛS k
p kl (u k )
]
}
where C ij (C mf ) is the set of offspring of parents i and j (m and f).
The step-by-step iterative algorithm is given below:
1. For each member i:
87