#General pedigrees

Kinship handles a pair. A missing person identified through two siblings and a niece, or an alleged father known only through his parents, needs the likelihood of the whole pedigree.

#Peeling

The likelihood at one locus sums over every genotype of every untyped member, which is exponential if done literally. Elston-Stewart peeling makes it linear in the number of people: each branch of the individual/family tree is reduced to a vector over the genotypes of the person joining it to the rest, and these vectors are passed along the tree's edges. The recursion is in Formulae and sources.

• Simple pedigrees only: the individual/family graph must be a forest. Loops throw (see Loops).
• Alleles no typed member shows are lumped into one class with their combined frequency. Untyped members' alleles are exchangeable, so this is exact, and it keeps a twenty-allele locus to a few genotype classes.
• θ = 0.

#Building a pedigree

def ped as pedigree.Pedigree init pedigree.pedigree("case 42");
$ped = pedigree.withFounder($ped, "GF", "m");
$ped = pedigree.withFounder($ped, "GM", "f");
$ped = pedigree.withChild($ped, "FA", "m", "GF", "GM");

A founder's genotype prior is the population frequency. Everyone else names both parents: a member with one parent recorded throws, because the missing parent still contributes alleles. Parents may be added after the child.

sex is "m", "f" or "". It is recorded and not used: the autosomal calculation treats both parents alike.

pedigree.members($ped);            # ids, insertion order
pedigree.isFounder($ped, "GF");    # bool
pedigree.childrenOf($ped, "GF");   # ids
pedigree.personOf($ped, "FA");     # the Person

#Validation

pedigree.validate($ped);      # throws, naming the problem
pedigree.isSimple($ped);      # bool

validate checks that every named parent is a member, that nobody is their own ancestor, and that the individual/family graph is a forest. Every likelihood function calls it first.

#Loops

# Two full siblings have a child together.
$p = pedigree.withChild($p, "A", "m", "GF", "GM");
$p = pedigree.withChild($p, "B", "f", "GF", "GM");
$p = pedigree.withChild($p, "C", "m", "A", "B");    # loop

pedigree 'loop' contains a loop (a consanguineous mating or a repeated
marriage); Elston-Stewart peeling needs a simple pedigree

Loops need cutset conditioning, which is not implemented. A half-sibling pedigree (one father, two mothers) is not a loop.

#Likelihoods

def l as float init pedigree.locusLikelihood($ped, $genotypes, $db, "TH01");

def loci as list of string init pedigree.pedigreeLoci($ped, $profiles, $db);
def lp as float init pedigree.pedigreeLikelihood($ped, $profiles, $db);
def logL as float init pedigree.log10PedigreeLikelihood($ped, $profiles, $db);

$genotypes maps member id to Genotype at one locus; $profiles maps member id to Profile. Members without an entry are untyped and summed out; ids that are not members are ignored. A locus is used if the table covers it and at least one member is typed there. pedigreeLikelihood underflows on a long panel; use the log form or a ratio.

#The likelihood ratio

def lr as float init pedigree.pedigreeLR($h1, $h2, $profiles, $db);
pedigree.typedMembers($h1, $profiles);   # who contributes data

The ratio is taken locus by locus, so it stays in range where either likelihood alone would underflow. Both pedigrees must type the same members, or the call throws; untyped members may differ, which is how a hypothesis adds an unknown alternative father. A denominator likelihood of zero at any locus throws.

#Worked example: deficient paternity

The alleged father is unavailable. The grandparents, the mother and the child are typed.

  H1                              H2
    GF x GM                         GF   GM   (unrelated bystanders)
       |                                 UN x MO
      FA x MO                                |
           |                                CH
          CH

The pedigrees are built as in the Quick start. examples/pedigree.j prints:

locus       L(H1)          L(H2)          LR
D3S1358      0.0000210529 0.0000111749 1.8839
vWA          0.0000030558 0.0000026145 1.1688
FGA          0.0000002152 0.0000000707 3.0451
D8S1179      0.0000001059 0.0000000174 6.0827
D21S11       0.0000000211 0.0000000187 1.1302
D18S51       0.0000026876 0.0000008977 2.9940
TH01         0.0000013055 0.0000004047 3.2258
D16S539      0.0000003625 0.0000003200 1.1328

combined LR = 504.29

for contrast, the pairwise grandparent LR (GF vs CH alone) = 53.67

The pairwise question (is CH a grandchild of GF?) ignores GM and the constraint that both grandparents must jointly explain the child's paternal allele. Using both is worth about ten times as much here.

#Correctness

The test overlay checks peeling against kinship.j's closed forms, which share no code with it: parent/child, full siblings, half siblings, grandparent, avuncular and the trio paternity index, to a relative tolerance of 1e-9. See Testing and contributing.