1. paternity with placental mixture
  2. paternity with placental twins

  1. Paternity calculation with placental mixtures
  2. (initially posted November 2020)

    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 ...


    It turns out that it comes down to a simple rule that works for calculating any pattern of the placental mixture situation. Just relate it to a standard paternity pattern as follows:


    Let's check the rule for the pattern
    Alleged fatherQS
    Definitions of probability symbols:
    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)
    and F=Pr(Alleged father'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

    Alleged fatherQS

  3. Paternity with placental twins
  4. (added August 2023)

    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.


    Suppose the pattern at a locus is
    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)

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