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Pedigree likelihood ratio for lineage markers

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Abstract

Lineage-based haplotype markers (e.g., Y chromosome STRs and mitochondrial DNA sequences) are important adjunct tools to the autosomal markers for kinship analysis and for specialized kinship applications such as database searching. Traditionally, the prosecution or kinship hypothesis considers the haplotypes in the same lineage and the probability of genotype data given the lineage hypothesis is simply set at 1 if the number of mismatched loci or nucleotides between the questioned person and the references is less than a predefined threshold. In this study, a kinship hypothesis based on a fixed relationship of the questioned person in the reference family is introduced. A graphical model is proposed to calculate the probability of the genotype data given the kinship hypothesis, which is the product of haplotype frequency of the founder in the pedigree and the transmission probability from the founder to all descendants. Proper mutation models are suggested for Y chromosome STRs and mitochondrial DNA sequence variants (i.e., SNPs) to calculate the transmission probability. The methods to infer the genotypes of the untyped individuals in the pedigree and the computational complexity of handling these untyped individuals are also addressed. Lastly, numerical examples of the applications are given to demonstrate the kinship hypothesis and the algorithms.

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Acknowledgements

This work was partially supported by USA National Institute of Justice (2009-DN-BX-K188) and National Natural Science Foundation of China (No. 81072511).

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Authors and Affiliations

  1. Department of Forensic and Investigative Genetics, University of North Texas Health Science Center, Ft Worth, TX, 76107, USA

    Jianye Ge, Arthur Eisenberg, Ranajit Chakraborty & Bruce Budowle

  2. Institute of Investigative Genetics, University of North Texas Health Science Center, Ft Worth, TX, 76107, USA

    Jianye Ge, Arthur Eisenberg, Ranajit Chakraborty & Bruce Budowle

  3. Beijing Institute of Genomics, China Academy of Science, Beijing, China

    Jiangwei Yan

Authors
  1. Jianye Ge
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  4. Ranajit Chakraborty
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Corresponding author

Correspondence to Jianye Ge.

Appendix—numerical examples

Appendix—numerical examples

Case 1: Y-STR

Figure 1 is used as an example to explain the Y-STR haplotype-based likelihood ratio calculation. Let H k be the hypothesis that d is the brother of e and nephew of b, and H nk be the hypothesis that d is unrelated to the family of b and e. The likelihood of the family given H k is the sum of the following four likelihoods, each of which is the likelihood of a subpedigree with untyped data determined.

  1. 1.

    a = {11,20} and c = {11,20}

    L1 = Pr({11,20}) × Pr(11 à 11)4 × Pr(20 à 20)3 × Pr(20 à 21)

  2. 2.

    a = {11,20} and c = {11,21}

    L2 = Pr({11,20}) × Pr(11 à 11)4 × Pr(20à 20) × Pr(20 à 21) × Pr(21à 20) × Pr(21 à 21)

  3. 3.

    a = {11,21} and c = {11,20}

    L3 = Pr({11,21}) × Pr(11 à 11)4 × Pr(20 à 20) × Pr(20 à 21) × Pr(21 à 20)2

  4. 4.

    a = {11,21} and c = {11,21}

    L4 = Pr({11,21}) × Pr(11 à 11)4 × Pr(21 à 20)2 × Pr(21 à 21)2

The likelihood of the family given H nk is the product of the haplotype frequency of d, Pr({11,20}), and the likelihood of the family only containing b and e, which is the sum of the following four subpedigrees with inferred genotypes for a and c.

  1. 1.

    a = {11,20} and c = {11,20}

    L1 = Pr({11,20}) × Pr({11,20}) × Pr(11 à 11)4 × Pr(20 à 20)2 × Pr(20 à 21)

  2. 2.

    a = {11,20} and c = {11,21}

    L2 = Pr({11,20}) × Pr({11,20}) × Pr(11 à 11)4 × Pr(20 à 20) × Pr(20 à 21) × Pr(21 à 21)

  3. 3.

    a = {11,21} and c = {11,20}

    L3 = Pr({11,20}) × Pr({11,21}) × Pr(11 à 11)4 × Pr(20 à 21) × Pr(21 à 20)2

  4. 4.

    a = {11,21} and c = {11,21}

    L4 = Pr({11,20}) × Pr({11,21}) × Pr(11 à 11)4 × Pr(21 à 20) × Pr(21 à 21)2

Let the mutation rate μ = 0.002, one-step mutation proportion be 0.95, haplotype frequency of {11,20} be 0.2, and haplotype frequency of {11,21} be 0.3. Pr(11 à 11) = Pr(20 à 20) = Pr(21 à 21) = (1 − μ) and Pr(21 à 20) = Pr(20 à 21) = 1/2μα in terms of Eq. (3). The likelihood ratio is 1.43. To reduce computational time, L 2, L 3, and L 4 in H k and L 3 in H nk, which have higher number of mutation steps than others, can be ignored, and the likelihood is still about 1.43.

Case 2: mtDNA

Figure 2 is used as an example to explain the mtDNA-based likelihood ratio calculation. Let H k be the hypothesis that c is the sister of b, and H nk be the hypothesis that c is unrelated to the family of b. The likelihood of the family given H k is the sum of the following four likelihoods, each of which is the likelihood of a subpedigree with untyped data determined.

  1. 1.

    a = {A, T} and b = {A,T}

    L1 = Pr({A,T}) × Pr(A àA)2 × Pr(T à T)2

  2. 2.

    a = {A, T} and b = {A,C}

    L2 = Pr({A,T}) × Pr(A à A)2 × Pr(T à T) × Pr(T à C)

  3. 3.

    a = {A, C} and b = {A,T}

    L3 = Pr({A,C}) × Pr(A à A)2 × Pr(Cà T)2

  4. 4.

    a = {A, C} and b = {A,C}

    L4 = Pr({A,C}) × Pr(A à A)2 × Pr(C à T) × Pr(C à C)

The likelihood of the family given H nk is the product of the haplotype frequency of c, Pr({A,T}), and the likelihood of the family only containing b, which is the sum of frequencies of two haplotypes, Pr({A,T}) and Pr({A,C}). Let the frequency of {A,T} and {A,C} be 0.2 and 0.3, respectively. Using the mutation rates in Table 1, the likelihood ratio is about 2.0.

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Ge, J., Eisenberg, A., Yan, J. et al. Pedigree likelihood ratio for lineage markers. Int J Legal Med 125, 519–525 (2011). https://doi.org/10.1007/s00414-010-0514-9

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