We first explain the multilocus or multipoint calculations in the context of incomplete IBD information on sibs discussed in section 9 of week 6. We consider the case where only the sibs are observed. For example, the sibs multilocus genotypes could be:
The sibs unordered multilocus genotypes contain information on the haplotypes they inherited from their parents. Here for example, the most likely haplotypes as reconstructed by MAPMAKER/SIBS are:
Computing haplotype probabilities conditional on marker data is not
however the final step of the analysis. The probability of interest is
where 
is the IBD status at locus x
(cf. week 6, section 9) and is derived from the haplotype probabilities.
In what follows, the notation from week 6 is used unless otherwise
specified. One of the elements of the HMM is the probability of the
observed marker genotypes 
given the inheritance vector 
at
marker locus 
. To compute it, we must condition
on the ordered parental genotypes at the marker (
).
We can write the last expression because the 
are independent
of the inheritance vectors 
under Assumption G2. Note that
since 
and 
completely
determine the sibs genotypes at the marker (
).
is a product of allele frequencies under
the random mating and HW assumptions. 
is 0 or 1
depending on whether the marker genotype is consistent with 
or
not. For a marker with k alleles, there are 
terms in the
summation. This number is usually small enough that the computation is
done quickly, but we will see that in general pedigrees complexity
increases and it becomes advantageous to restrict the summation to
terms where 
is 1.
The next elements we need are the transition matrices between adjacent marker loci. They have the same form as the transition matrix between a disease and a marker locus described in section 5 of week 6, namely:
where 
indexes the intervals between markers.
We now have all the ingredients to set up a forward and backward
recurrence relation to compute 
, the inheritance
distribution at marker locus l conditional on the multilocus marker
data m. This is similar to what we did in section 5.1 of week 3.
Similarly,
We can see that
The inheritance distribution at locus l is given by the 
-variables
which are functions of the forward and backward variables.
From there, the conditional inheritance distribution at a locus x on
which we have no data 
is obtained as a function of
and recombination fractions 
between
the point x and the marker l. We get the IBD probabilities at locus
x (or a marker locus) 
by summing
over 
s corresponding to IBD = 0,1 or 2. The
use of 
in linkage
analysis on sib pairs is described in section 9.6 of week 6.
The assumption of linkage equilibrium between the marker
loci ( Assumption G6) required to establish the recurrence
relations is a strong assumption on top of the random mating and
HW assumptions made to compute 
. It is the price we pay
for not observing the parent's genotypes. However, if these
assumptions seem reasonable, genotyping only the sibs and not their
parents at a number of markers (not too widely spaced) is a big saving
in genotyping work with almost no loss of information. Researchers can
then employ their resources to type more sib pairs.
