Sometimes a paternity test is needed based on a mixture of mother and child. That is, the embryo died or was aborted, and the only DNA available is the mother's type, the alleged father's, and a type from the abortus that is a combination of the mother and the embroyo.
In such a situation it's easy enough to work out the proper likelihood ratios from first principles. Before doing that ...
Mother | PR | |
Mixture | PQR | |
Alleged father | QS |
Definitions of probability symbols: |
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The paternity index PI=X/Y where
X = Pr(such types | paternity by Alleged father), and Y = Pr(such types | man unrelated). Define M=Pr(mother's type)
|
X = M×F×Pr(Mixture | parents as alleged) = M×F×1/2;
Y = M×F×Pr(Mixture | Mother & random sperm) =
M×F×Pr(Q).
PI = 1 /
2Pr(Q)
Note that this is the same formula as that for the normal paternity pattern
Mother | PR | |
Child | PQ | |
Alleged father | QS |
If the placental genetic data suggests more than one embroyo, a possible explanation is non-identical twins from a single father. Computing likelihoods isn’t much harder than in the one-embryo problem.
Mother | PR | |
Mixture | PQRS | |
Alleged father | QS | |
Define the probabilities X, Y, M, F as above. |
X = M×F×Pr(Mixture | parents as alleged) = M×F×1/2
— same formula though this time 1/2 is Pr(F contributes different
alleles to each embryo).
Y = M×F×Pr(Mixture | Mother & random pair of sperm) =
M×F×Pr(QS) =
M×F×2Pr(Q)Pr(S).
PI = 1 /
4Pr(Q)Pr(S)