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# Paternity Calculations

In paternity calculations the likelihood ratio is the same as the paternity index. The three tables below show the calculations of the likelihood ratio for different possible combinations of genotypes assuming that the mother, alleged father and father are unrelated.

In the tables

• GC is the child's genotype
• GM is the mother's genotype
• GAF is the alleged father's genotype
• Hp is the mother and alleged father are the true parents
• Hd is the mother and an unrelated male are the true parents
• LR numerator is P(GC|GM, GAF, Hp)
• LR denominator is P(GC|GM, GAF, Hd).01

Likelihood Ratio and Paternity Calculation

GC

GM

GAF

Num

Denominator

LR

Ai Ai

Ai Ai

Ai Ai

1

pi

1/ pi

Ai Aj, j≠ i

1/2

pi

1/2pi

Aj Ak, k≠i, j

0

pi

0

Ai Ai

Ai Aj, i ≠ j

Ai Ai

1/2

pi /2

1/ pi

Ai Aj, j≠i

1/4

pi /2

1/2 pi

Aj Ak, k≠i, j

0

pi /2

0

Ai Aj, i≠j

Ai Ai

Aj Aj

1

pj

1/pj

Aj Ak, k≠j

1/2

pi

1/2 pj

Ak Al, k, l≠j

0

pi

0

Ai Aj, i≠j

Ai Aj, i≠j

Ai Ai

1/2

(pi + pj)/2

1/(pi + pj)

Ai Aj

1/2

(pi + pj)/2

1/( pi + pj)

Aj Ak, k≠j

1/4

(pi + pj)/2

1/(2(pi + pj))

Ak Al, k, l≠i, j

0

(pi + pj)/2

0

Ai Aj, i≠j

Ai Ak, k≠i, j

Aj Aj

1/2

pj /2

1/ pj

Aj Al,l ≠j

1/4

pj /2

1/2 pj

Ak Al, k, l≠j

0

pj /2

0

Example:

 Locus Mother Child Alleged Father D3S1358 16 16 15, 16 vWA 17, 18 17, 18 18, 20 FGA 25, 27 23, 25 21, 23 D18S51 14, 18 18 18, 20

The following frequencies were used:

 Locus Allele Frequency Allele Frequency D3S1358 16 0.2315 vWA 17 0.2627 18 0.2219 FGA 23 0.1581 D18S51 18 0.0918

• D3S1358 — LR = 1/2p16 = 1/(2 x 0.2315) = 2.1598
• vWA — LR = 1/[2(p17 + p18)] = 1/[2(0.2627 + 0.2219)] = 1.0317
• FGA — LR = 1/2p23 = 1/(2 x 0.1581) = 3.1625
• D18S51 — LR = 1/2p18 = 1/(2 x 0.0918) = 5.4466

Therefore, LR = 2.1598 x 1.0317 x 3.1625 x 5.4466 = 38.3815

In other words, it is 38 times more likely that the child's DNA profile would be observed if the alleged father is the true father rather than an unrelated individual selected at random from the Caucasian population.