Page 131 - Applied Probability

7
Computation of Mendelian
Likelihoods

7.1 Introduction

Rigorous analysis of human pedigree data is a vital concern in genetic epi-
demiology, human gene mapping, and genetic counseling. In this chapter
we investigate efficient algorithms for likelihood computation on pedigree
data, placing particular stress on the pioneering algorithm of Elston and
Stewart [8]. It is no accident that their research coincided with the in-
troduction of modern computing. To analyze human pedigree data is te-
dious, if not impossible, without computers. Pedigrees lack symmetry, and
all simple closed-form solutions in mathematics depend on symmetry. The
achievement of Elston and Stewart [8] was to recognize that closed-form so-
lutions are less relevant than good algorithms. However, the Elston-Stewart
algorithm is not the end of the story. Evaluation of pedigree likelihoods re-
mains a subject sorely in need of further theoretical improvement. Linkage
calculations alone are among the most demanding computational tasks in
modern biology.

7.2 Mendelian Models

Besides the raw materials of pedigree structure and observed phenotypes,
a genetic model is a prerequisite for likelihood calculation. At its most
elementary level, a model postulates the number of loci necessary to explain
the phenotypes. Mendelian models, as opposed to polygenic models, involve
only a finite number of loci. For purposes of discussion, it is convenient to
use the term “genotype” when discussing the multilocus, ordered genotypes
of an underlying model. Because ordered genotypes preserve phase, they
are preferable to unordered genotypes for theoretical and computational
purposes. Of course, observed genotypes are always unordered.
Any Mendelian model revolves around the three crucial notions of pri-
ors, penetrances, and transmission probabilities [8]. Prior probabil-
ities pertain only to founders. If G is a possible genotype for a founder,
then in the absence of other knowledge, Prior(G) is the probability that
the founder carries genotype G. Almost all models postulate that prior
probabilities conform to Hardy-Weinberg and linkage equilibrium.
Penetrance functions specify the likelihood of an observed phenotype X