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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

GCGMGAFNumDenominatorLR
Ai AiAi AiAi Ai1pi1/ pi
Ai Aj, j≠ i1/2pi1/2pi
Aj Ak, k≠i, j0pi0
Ai AiAi Aj, i ≠ jAi Ai1/2pi /21/ pi
Ai Aj, j≠i1/4pi /21/2 pi
Aj Ak, k≠i, j0pi /20
Ai Aj, i≠jAi AiAj Aj1pj1/pj
Aj Ak, k≠j1/2pi1/2 pj
Ak Al, k, l≠j0pi0
Ai Aj, i≠jAi Aj, i≠jAi Ai1/2(pi + pj)/21/(pi + pj)
Ai Aj1/2(pi + pj)/21/( pi + pj)
Aj Ak, k≠j1/4(pi + pj)/21/(2(pi + pj))
Ak Al, k, l≠i, j0(pi + pj)/20
Ai Aj, i≠jAi Ak, k≠i, jAj Aj1/2pj /21/ pj
Aj Al,l ≠j1/4pj /21/2 pj
Ak Al, k, l≠j0pj /20

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.