#Paternity

With mother, child and alleged father all typed, conditioning on the mother tells you which of the child's alleles the father had to supply. That is more informative than a pairwise father/child LR, so paternity has its own trio calculation.

#The paternity index

At one locus:

X  = P(child | mother, alleged father)
Y  = P(child | mother, random man from the reference population)
PI = X / Y

X uses the alleged father's transmission probabilities, with mutation; Y uses population allele frequencies. The full formula is in Formulae and sources.

kinship.paternityIndexAt($db, $motherGt, $childGt, $fatherGt, kinship.NO_MUTATION);

def cpi as float init kinship.paternityIndex($mother, $child, $allegedFather, $db);
kinship.trioLoci($db, $mother, $child, $allegedFather);   # the loci used

The combined paternity index (CPI) is the product over the loci all three profiles and the table share. No shared locus throws.

#Probability of paternity

def w as float init kinship.probabilityOfPaternity($cpi);              # prior 0.5
def w01 as float init kinship.probabilityOfPaternityPrior($cpi, 0.1);  # prior 0.1

W = CPI / (CPI + 1) at prior 0.5, or W = CPI·π / (CPI·π + (1-π)) for a prior π in (0, 1). W restates the CPI and adds no evidence; a 99.8% rests on assuming the man was as likely as not the father before any DNA was examined. Report both numbers and the prior.

#Maternal exclusion throws

If the child carries neither of the mother's alleles at a locus, Y is zero and no index exists. That points to a sample or pedigree error (a mislabelled tube, the wrong mother), so the call throws:

locus 'D8S1179': the mother cannot have transmitted either of the child's
alleles - this is a maternal exclusion, not a paternity index

#Mutation

STR loci mutate at roughly one or two per thousand meioses, almost always by one repeat. Over a fifteen-locus panel a true father will occasionally fail to explain one locus. Without a mutation model that locus gives X = 0 and a false exclusion.

def bad as list of string init kinship.inconsistentLoci($mother, $child,
    $allegedFather, $db);

lists the Mendelian-inconsistent loci. One is the classic single inconsistency; several mean exclusion, and no mutation model should rescue that.

The stepwise model (formula in Formulae and sources) gives 1 - rate for no change and rate · (1 - decay) · decay^(k-1) / 2 for a k-step change in one direction. A microvariant difference below one repeat (15 to 15.2) is one step. It applies to the alleged father's transmission only; with NO_MUTATION the index is the strict Mendelian one.

kinship.mutationProbability(kinship.DEFAULT_MUTATION, "D8S1179", "16", "15");

A rescued trio's CPI is proportional to the rate assumed at the inconsistent locus, and observed rates vary about tenfold between loci. Set published per-locus rates:

def model as kinship.MutationModel init kinship.DEFAULT_MUTATION;
$model = kinship.withLocusRate($model, "D8S1179", 0.0016);
$model = kinship.withLocusRate($model, "FGA", 0.0028);

def cpi as float init kinship.paternityIndexWith($mother, $child,
    $allegedFather, $db, $model);

#Worked example

examples/paternity.j. The alleged father is 16/16 at D8S1179 and the child needs a 15:

locus        mother  child    father     PI strict  PI mutation
D3S1358      15/16    15/17    17/18      3.7679 3.7638
vWA          16/17    17/19    19/19      4.6751 4.6657
FGA          21/22    22/24    24/25      6.0901 6.0834
D8S1179      12/13    13/15    16/16      0.0000 0.0219
D21S11       29/30    30/31.2  31.2/32.2  2.2604 2.2579
D18S51       14/15    15/17    17/18      5.9880 5.9814
TH01         6/7      7/9.3    9.3/10     6.4516 6.4445
D16S539      11/12    12/13    13/14      2.2655 2.2630

Mendelian-inconsistent loci: ["D8S1179"]

strict Mendelian CPI  = 0.000000
  -> excluded: no allowance is made for mutation

CPI with mutation     = 460.77
W (prior 0.5)         = 0.99783441
W (prior 0.1)         = 0.98084163

Each consistent locus loses about 0.1% to the (1 - rate) factor. The inconsistent locus contributes 0.0219 instead of 0. The prior moves W from 99.78% to 98.08% on the same evidence.

When the alleged father cannot be typed, the case goes through his relatives: see General pedigrees.