Page 136 - Applied Probability

7. Computation of Mendelian Likelihoods
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plications, this action will not change the likelihood of the pedigree, pro-
vided the lumped allele is assigned the appropriate summed population
frequency. For instance, suppose only the first three alleles from the set
{A i :1 ≤ i ≤ 6} of six possible alleles are seen among the typed people of
a pedigree. Then one can consolidate alleles A 4 , A 5 , and A 6 into a single
artificial allele A 7 with frequency p 7 = p 4 + p 5 + p 6 in obvious notation.
If allele consolidation is carried out on a pedigree-by-pedigree basis, then
substantial computational savings can be realized. Even more dramatic
savings can occur when allele consolidation is carried out locally within a
pedigree. O’Connell and Weeks’ [26] explanation of local consolidation is
well worth reading but a little too lengthy to recite here. Finally, note that
there are some problems such as allele frequency estimation from pedigree
data [2] where allele consolidation is disastrous. A little common sense
should be an adequate safeguard against these abuses.
7.4 Array Transformations and Iterated Sums

To elaborate on some of the comments made earlier about iterated sums
and arrays, we now strip away the genetics overlay and concentrate on
issues of numerical analysis. As an example [18], consider the problem of
computing the sum of products

A(G 1 ,G 2 )B(G 2 )C(G 2 ,G 3 ), (7.2)
G 1 ∈S 1 G 2 ∈S 2 G 3 ∈S 3
where S i is the finite range of summation for the index G i , and where A,
B, and C are arrays of real numbers. Let S i have m i elements. Computing
(7.2) as a joint sum requires 2m 1 m 2 m 3 multiplications and m 1 m 2 m 3 − 1
additions. If we compute (7.2) as an iterated sum in the sequence (3, 2, 1)
specified, we first compute an array

D(G 2 ) = C(G 2 ,G 3 )
G 3 ∈S 3
in m 2 (m 3 − 1) additions. Note that the arrays A and B do not depend on
the index G 3 so it is uneconomical to involve them in the sum on G 3 . Next
we compute an array

E(G 1 )= A(G 1 ,G 2 )B(G 2 )D(G 2 ) (7.3)
G 2 ∈S 2
in 2m 1m 2 multiplications and m 1 (m 2 −1) additions. Last of all, we compute
the sum E(G 1 )in m 1 −1 additions. The total arithmetic operations
G 1 ∈S 1
needed for the joint sum is 3m 1m 2 m 3 − 1; for the iterated sum the total
is m 2 (3m 1 + m 3 − 1) − 1. It is clear that the iterated sum requires the