Page 140 - Applied Probability

7. Computation of Mendelian Likelihoods
124
can be factored uniformly into two terms involving a given locus and loci to
the right and left of it, respectively, only if the contributing parent is an ob-
ligate heterozygote at the locus. A combination of inspection and genotype
elimination quickly identifies all obligate heterozygous loci in parents.
In this reformulated model, the summations in the likelihood representa-
tion (7.1) are replaced by analogous summations over person–locus combi-
nations G ij . Although this substitution increases the complexity of finding
a good summation sequence, most other features of likelihood evaluation re-
main unchanged. For instance, genotype elimination is already carried out
one locus at a time. Array creation and annihilation are handled similarly
in both likelihood formulations, except that more numerous but smaller
arrays are encountered in the person–locus mode of calculation.
7.6 Examples of Pedigree Analysis

Example 7.6.1 Paternity Testing
Paternity testing confirms or eliminates a putative father as the actual
father of a child. Phenotyping of the mother, child, and putative father
is done at a number of different marker loci. If a genetic inconsistency is
found, then the putative father is absolved. On the other hand, if the trio
is consistent at all loci typed, then either a rare event has occurred or the
putative father is the actual father. There are two ways of quantifying the
rarity of this event. The Bayesian approach is to compute a likelihood ratio
of the trio with the putative father as real father versus the trio with the
real father as a random male. This likelihood ratio or paternity index
can be transformed into a posterior probability if a prior probability of
paternity is supplied.
A strictly frequentist approach to the problem is to compute the prob-
ability that a random male would be excluded by at least one of the tests
based on the phenotypes of the mother and child. This exclusion proba-
bility relieves a judge or jury from the necessity of quantifying their prior
probabilities of paternity. Both posterior and exclusion probabilities can
be computed for each locus separately and then cumulated over all loci
jointly. The locus-by-locus statistics are useful in determining which loci
are critically important in confirming paternity. The cumulative statistics
are the ones quoted in court.
To compute the paternity index, imagine two pedigrees. The first pedi-
gree, Ped 1 , contains the mother and child and the putative father as actual
father. The second pedigree, Ped 2 , substitutes a random male with all phe-
notypes unknown for the actual father. The putative father is present as
an isolated individual unrelated to the child in Ped 2 . Suppose the vec-
tor X j denotes the observed phenotypes for the trio of mother, child,
and putative father at the jth locus of a set of marker loci in Hardy-