The multilocus approach that we just described at
section 3.2 extends to pedigrees of arbitrary structure.
The program GENEHUNTER (Kruglyak et al. [1]) has been
created to perform multilocus linkage analysis in this setting.
We consider an arbitrary pedigree with f founders and n
non-founders. (A sib-pair pedigree is a special case with 2 founders
and 2 non-founders). The length of the inheritance vector is now 2n,
and the number of possible vectors is 
. The multilocus genotypes
of some number of pedigree members (including possibly founders)
have been collected and constitute the marker data m.
The formulation of the computation of 
is modified. We
give distinct labels 
to the alleles carried by the founders
at a marker locus, like we did in section 1.1 of week 6, and we call
them ``genes'', in the broad sense of an heritable DNA segment, to
make a distinction between them and the observable allele types.
An allelic assignment is then defined as a mapping from the founder genes to the allele types. For example, the assignment could be:
When the unordered genotypes of the founders are known, the actual
assignment is known up to a permutation of the labels within a
founder. When the genotypes of some or all founders are missing, we
have to sum over the allelic assignments for the missing founders,
just as we summed over ordered parental genotypes in
section 3.2 above.
Denote by 
the vector of alleles
assigned to the founders at locus l.
where 
is the genotype of the non-founder j at locus l and
may be either observed or missing.
The probability of the vector of assigned alleles 
is
computed as a product of allele frequencies by assuming independent
sampling of the founder genes from the gene pool of the population
(random mating). 
is 1 if 
is compatible
with 
and 
and 0 otherwise.
With k allele types at a marker locus, there are 
different
allelic assignments, a number growing exponentially with the number of
founders in the pedigree. Most of these assignments are however
incompatible with the genotype data. Summing over all of them is a
waste of time. An efficient algorithm as been developed that restricts
the summation to the non-zero terms only (Sobel and Lange [6],
Kruglyak et al. [1]). The algorithm is efficient because the
number of operations it involves grows only linearly with the number
of founders. A description of the algorithm can be found in
appendix A.
The transition matrix between adjacent marker loci generalizes the sib-pair transition matrix.
The 
s and 
s of the forward-backward algorithm can now
be computed using equations (1) and (2) of
section 3.2. Difficulties arise because we have to
repeatedly multiply 
transition matrices
by 
vectors 
, a computation that
appear to require 
operations. It is possible to take
advantage of the fact that 
is a Kronecker product to
perform the computation in 
operations. Kruglyak et
al. [3] describe an algorithm to achieve that
improvement. Their algorithm achieves the same order of simplification
as the Yates' algorithm presented in appendix B.
The conditional inheritance distribution 
at any point x along a chromosome is obtained in the same way
as in the sib-pair analysis of section 3.2. This
probability is used differently with arbitrary pedigrees than with sib
pairs. The most common statistical procedure is to compute a location
score. Before explaining how GENEHUNTER calculates a location score,
we discuss what it is in the next section.