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Formulas for sibship
Paternity and mutation
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Likelihood ratios for paternity situations

(excluding mutation. For mutation, see link at right.)

Each row of the table lists a possible pattern of gene sharing among child, [mother], and alleged father (on the left) and paternity index formulas (on the right). The paternity index formulas were derived automatically by the Symbolic Kinship Program.
The letters o, p, q, r, s in the "Child, Mother, Alleged Father" columns represent alleles.
p is often in the child, and in the mother.
q is always in the child and usually in the alleged father.
r or s are alleles not found in the child.
o means there is no allele or none detected in one chromosome, such as primer dropout.
The letters p, q, o in the likelihood ratio columns represent the allele frequency of the corresponding allele.

If the possibility of an undetected allele changes the formula, the more general form is listed in the silent allele column, using abbreviations q'=q+o, q"=q+2o, r"=r+2o, where o is the frequency of the silent allele.

"Silent allele" or null allele of course includes the possibility of RFLP bands that are not visible for whatever reason, and of PCR primer dropout. The only caveat is that one must somehow estimate the frequency o with which alleles are unobserved.

Child Mother Alleged father Likelihood ratio for paternity vs. non-paternity
Co-dominant system With silent allele o Notes
q pq q 1/q 1 / q'
q p q impossible (q'/q") / q (1)
pq p or pr q 1/q (q'/q") / q (1)
q q q 1/q 1 / (q'-o3/q'(q+3o)) (1*)
pq p or pr or ps qr (or pq) 1/2q
q p qr impossible 1 / 2q (2)
q pq qr (or pq) 1/2q 1 / 2q' (2)
q q qr 1/2q 1 / 2(q'-o²/q") (2*)
pq pq pq 1/(p+q)
pq pq q 1/(p+q) (q'/q") / (p+q)
pq pq qr 1/(2p+2q)
q pq r 0 o / q'r" (3)
q q r 0 o / (q'r" + oqr"/q') (3*)
q p r impossible 0
q Mother not tested q 1/q (1 + o(1-o/q")/q) / q" (4*)
pq Mother not tested q 1/2q (q'/q") / 2q
q Mother not tested qr 1/2q (q'/q") / 2q
pq Mother not tested pq (p+q) / 4pq
pq Mother not tested qr 1/4q
q Mother not tested r 0 o / q"r" (3*)

Notes re implementation of silent allele formulas in DNA•VIEW

Notes 1*, 2*, 3*: The complicated formulas are not implemented. Instead, these situations are approximated using the similar formulas 1, 2, 3.
Note 4*: In the motherless case with identically homozygous man and child (both q), DNA•VIEW ignores the silent allele possibility and uses the approximate formula 1/q.