## Paternity calculation with placental mixtures

(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 ...

### Shortcut

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:
• If the mixture has one or two alleles, pretend that the child has that type.
• If the mixture has three alleles, pretend that the child has the non-maternal allele plus either of the others.
• Then compute as a normal paternity calculation.

### Example

Let's check the rule for the pattern
 Mother PR Mixture PQR Alleged father QS
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).

Then

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

## Paternity with placental twins

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.

### Example

Suppose the pattern at a locus is
 Mother PR Mixture PQRS Alleged father QS Define the probabilities X, Y, M, F as above.
Then

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)