Page 206 - Applied Probability

9. Descent Graph Methods
m

s r (j r ). Hence,
t i (j)=
r=1
1
m

ˆ t i (k)=
j 1 =0
1
m 1 ··· j m =0 r=1 (−1) k r j r s r (j r ) 191

= (−1) k r j r s r (j r )
r=1 j r =0
m ,
1 k r =0
=
1 − 2θ i k r =1
r=1
m
1 r=1 k r =0
,
= m m .
(1 − 2θ i ) r=1 k r k r > 0
r=1
If we posit unequal female θ xi and male θ yi recombination fractions sepa-
rating loci i and i+1, then a straightforward adaptation of these arguments
yields
ˆ t i (k)=(1 − 2θ xi ) p(k) (1 − 2θ yi) q(k) ,
where p(k) is the number of components k r = 1 with r a female meiosis
and q(k) is the number of components k r = 1 with r a male meiosis.
9.12 Genotyping Errors
One or two unfortunately placed genotyping errors can profoundly influ-
ence the magnitude of lod and location scores. Although many errors can
be detected by a careful rereading of gels or other phenotypic tests, it is
often more powerful to use the contextual evidence provided by relatives.
Overt violations of Mendel’s laws are the easiest to detect. The less ob-
vious errors such as unlikely double recombinants do more damage. For
these more subtle errors, the best approach is to construct a genotyping
error model and compute posterior error probabilities. The single-locus ver-
sion of this tactic is advocated in the papers [5, 26, 30, 36]. In practice, any
realistic error model forces one to consider many alternative genotypes. At
a highly polymorphic marker locus, this complication creates a major com-
putational bottleneck for the Elston-Stewart method. Computing posterior
error probabilities with multiple linked markers just exacerbates the prob-
lem. The deterministic and stochastic descent graph methods discussed in
this chapter are capable of handling posterior error computations involving
linked markers. Only in this setting can unlikely double recombinants be
detected.
There are several plausible models for mistyping error. All invoke inde-
pendence of typing errors from person to person and locus to locus. The