Calculation of the Paternity Index for the Alleged Father Related to the Child’s Mother

doi:10.12116/j.issn.1004-5619.2023.530209

Abstract

Abstract:

Objective To derive the paternity index （PI） calculation formula of the alleged father （AF） when the AF is a relative （parent/child， siblings， grandparent/grandchild， uncle/nephew， first cousins） of the child’s biological mother. Methods For the case when the AF is related to the child’s biological mother， the existence of the relationship in the numerator and denominator hypothesis of PI was considered. The genotype frequency of the AF was calculated by using the frequency formula in which the mother’s genotype was considered， while the random male in the denominator was substituted as another relative of the mother’s same rank. The PI calculation formula was derived to eliminate the effect of the relationship between AF and the child’s biological mother. Results When the AF and the biological mother have first， second and tertiary kinship， a more conservative PI was obtained from the PI calculation formula derived in this study compared with the PI calculation method which did not consider kinship. Conclusion The calculation method provided in this study can eliminate the effect of the relation of the AF and mother on the PI in incest cases， to obtain more accurate and conservative identification conclusions.

Key words: forensic genetics, paternity testing, incest, paternity index, likelihood ratio

CLC Number:

Yu-ting WANG, Qiang ZHU, Yu-han HU, Yi-fan WEI, Ting-yun HOU, Ji ZHANG. Calculation of the Paternity Index for the Alleged Father Related to the Child’s Mother[J]. Journal of Forensic Medicine, 2023, 39(3): 271-275.

Figures/Tables 3

Tab. 1 Possible genotype combinations of the AF who is mother’s relatives

母亲的基因型	与母亲具有亲子关系的亲属可能的基因型	母亲的全同胞、二级亲属、三级亲属可能的基因型
AA	AA、AQ、AU^1）	AA、AQ、QQ、AU、QU、UU^2）
AQ	AA、AQ、QQ、AU、QU	AA、AQ、QQ、AU、QU、UU^2）

Tab. 2 Conditional probabilities for the genotype of an unknown individual， given the genotype of the relative

基因型		条件概率计算公式
已知个体	未知个体	亲子	全同胞	一级表亲	叔侄、爷孙、半同胞
AA	AA	$p A$	$1 + p A 2 4$	$p A 1 + 3 p A 4$	$p A 1 + p A 2$
AA	QQ	0	$p Q 2 4$	$3 p Q 2 4$	$p Q 2 2$
AA	AQ	$p Q$	$p Q 1 + p A 2$	$p Q 1 + 6 p A 2$	$p Q 1 + 2 p A 2$
AA	QR	0	$p Q p R 2$	$3 p Q p R 2$	$p Q p R$
AQ	AA	$p A 2$	$p A 1 + p A 4$	$p A 1 + 6 p A 8$	$p A 1 + 2 p A 4$
AQ	AQ	$p A + p Q 2$	$1 + p A + p Q + 2 p A p Q 4$	$p A + p Q + 12 p A p Q 8$	$p A + p Q + 4 p A p Q 4$
AQ	AR	$p R 2$	$p R 1 + 2 p A 4$	$p R 1 + 12 p A 8$	$p R 1 + 4 p A 4$
AQ	RR	0	$p R 2 4$	$3 p R 2 2$	$p R 2 2$
AQ	RS	0	$p R p S 2$	$3 p R p S 2$	$p R p S$

Tab. 2 Conditional probabilities for the genotype of an unknown individual， given the genotype of the relative

基因型		条件概率计算公式
已知个体	未知个体	亲子	全同胞	一级表亲	叔侄、爷孙、半同胞
AA	AA	$p A$	$1 + p A 2 4$	$p A 1 + 3 p A 4$	$p A 1 + p A 2$
AA	QQ	0	$p Q 2 4$	$3 p Q 2 4$	$p Q 2 2$
AA	AQ	$p Q$	$p Q 1 + p A 2$	$p Q 1 + 6 p A 2$	$p Q 1 + 2 p A 2$
AA	QR	0	$p Q p R 2$	$3 p Q p R 2$	$p Q p R$
AQ	AA	$p A 2$	$p A 1 + p A 4$	$p A 1 + 6 p A 8$	$p A 1 + 2 p A 4$
AQ	AQ	$p A + p Q 2$	$1 + p A + p Q + 2 p A p Q 4$	$p A + p Q + 12 p A p Q 8$	$p A + p Q + 4 p A p Q 4$
AQ	AR	$p R 2$	$p R 1 + 2 p A 4$	$p R 1 + 12 p A 8$	$p R 1 + 4 p A 4$
AQ	RR	0	$p R 2 4$	$3 p R 2 2$	$p R 2 2$
AQ	RS	0	$p R p S 2$	$3 p R p S 2$	$p R p S$

Tab. 3 Formula for calculating the likelihood ratio under different genetic relationships in incest cases

基因型			AF和母亲存在亲子关系	AF和母亲存在全同胞关系	AF和母亲存在一级表亲关系	AF和母亲存在叔侄、爷孙、半同胞关系
孩子	母亲	可疑父	AF和母亲存在亲子关系	AF和母亲存在全同胞关系	AF和母亲存在一级表亲关系	AF和母亲存在叔侄、爷孙、半同胞关系
AP	AQ	AA	$4 2 p A + 1$	$4 2 p A + 1$	$16 14 p A + 1$	$8 6 p A + 1$
AP	AQ	AQ	$2 2 p A + 1$	$2 2 p A + 1$	$8 14 p A + 1$	$4 6 p A + 1$
AP	AQ	AU	$2 2 p A + 1$	$2 2 p A + 1$	$8 14 p A + 1$	$4 6 p A + 1$
AP	AA	AA	$2 p A + 1$	$2 p A + 1$	$8 7 p A + 1$	$4 3 p A + 1$
AP	AA	AQ	$1 p A + 1$	$2 p A + 1$	$4 7 p A + 1$	$2 3 p A + 1$
AP	AA	AU	$1 p A + 1$	$2 p A + 1$	$4 7 p A + 1$	$2 3 p A + 1$
AQ	AA	AQ	$1 p Q$	$1 p Q$	$4 7 p Q$	$2 3 p Q$
AQ	AA	QQ	-	$2 p Q$	$8 7 p Q ?$	$4 3 p Q$
AQ	AA	QQ	-	$2 p Q$	$8 7 p Q ?$	$4 3 p Q$
AQ	AA	QU	-	$1 p Q$	$4 7 p Q$	$2 3 p Q$
AQ	AQ	AA	$2 1 + p A + p Q$	$2 1 + p A + p Q$	$8 1 + 7 p A + 7 p Q$	$4 1 + 3 p A + 3 p Q$
AQ	AQ	AQ	$2 1 + p A + p Q$	$2 1 + p A + p Q$	$8 1 + 7 p A + 7 p Q$	$4 1 + 3 p A + 3 p Q$
AQ	AQ	AU	$1 1 + p A + p Q$	$1 1 + p A + p Q$	$4 1 + 7 p A + 7 p Q$	$2 1 + 3 p A + 3 p Q$
AQ	AU	AQ	$1 p Q$	$1 p Q$	$4 p Q 7 + 6 p Q$	$2 3 p Q$
AQ	AU	QQ	-	$2 p Q$	$8 p Q 7 + 6 p Q$	$4 3 p Q$
AQ	AU	QU	$1 p Q$	$1 p Q$	$4 p Q 7 + 6 p Q$	$2 3 p Q$

Tab. 3 Formula for calculating the likelihood ratio under different genetic relationships in incest cases

基因型			AF和母亲存在亲子关系	AF和母亲存在全同胞关系	AF和母亲存在一级表亲关系	AF和母亲存在叔侄、爷孙、半同胞关系
孩子	母亲	可疑父	AF和母亲存在亲子关系	AF和母亲存在全同胞关系	AF和母亲存在一级表亲关系	AF和母亲存在叔侄、爷孙、半同胞关系
AP	AQ	AA	$4 2 p A + 1$	$4 2 p A + 1$	$16 14 p A + 1$	$8 6 p A + 1$
AP	AQ	AQ	$2 2 p A + 1$	$2 2 p A + 1$	$8 14 p A + 1$	$4 6 p A + 1$
AP	AQ	AU	$2 2 p A + 1$	$2 2 p A + 1$	$8 14 p A + 1$	$4 6 p A + 1$
AP	AA	AA	$2 p A + 1$	$2 p A + 1$	$8 7 p A + 1$	$4 3 p A + 1$
AP	AA	AQ	$1 p A + 1$	$2 p A + 1$	$4 7 p A + 1$	$2 3 p A + 1$
AP	AA	AU	$1 p A + 1$	$2 p A + 1$	$4 7 p A + 1$	$2 3 p A + 1$
AQ	AA	AQ	$1 p Q$	$1 p Q$	$4 7 p Q$	$2 3 p Q$
AQ	AA	QQ	-	$2 p Q$	$8 7 p Q ?$	$4 3 p Q$
AQ	AA	QQ	-	$2 p Q$	$8 7 p Q ?$	$4 3 p Q$
AQ	AA	QU	-	$1 p Q$	$4 7 p Q$	$2 3 p Q$
AQ	AQ	AA	$2 1 + p A + p Q$	$2 1 + p A + p Q$	$8 1 + 7 p A + 7 p Q$	$4 1 + 3 p A + 3 p Q$
AQ	AQ	AQ	$2 1 + p A + p Q$	$2 1 + p A + p Q$	$8 1 + 7 p A + 7 p Q$	$4 1 + 3 p A + 3 p Q$
AQ	AQ	AU	$1 1 + p A + p Q$	$1 1 + p A + p Q$	$4 1 + 7 p A + 7 p Q$	$2 1 + 3 p A + 3 p Q$
AQ	AU	AQ	$1 p Q$	$1 p Q$	$4 p Q 7 + 6 p Q$	$2 3 p Q$
AQ	AU	QQ	-	$2 p Q$	$8 p Q 7 + 6 p Q$	$4 3 p Q$
AQ	AU	QU	$1 p Q$	$1 p Q$	$4 p Q 7 + 6 p Q$	$2 3 p Q$

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