Evaluation of DNA Mixtures Accounting for Sampling Variability - - PowerPoint PPT Presentation

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Evaluation of DNA Mixtures Accounting for Sampling Variability

Yuk-Ka Chung

• Dept. of Statistics and Actuarial Science

University of Hong Kong yukchung@hku.hk

Joint work: Y. Q. Hu, D. G. Zhu, W. K. Fung

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Outline

Background Bayesian model Example Conclusion and future works

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DNA Profile

Set of numbers representing the genetic

characteristics of an individual

D19S433 vWA TPOX 13 / 14 14 / 17 11 / 11 D18S51 D5S818 FGA 13 / 16 10 / 11 23.2 / 25

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Two-person Mixture

Crime stain M Reference samples K

= (V, S)

Victim V, Suspect S

Evidentiary value

( )

( )

d p

H K M P H K M P LR | , | , =

( )

( )

d p

H K M P H K M P , | , | =

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Two-person Mixture

Locus Mixture Victim Suspect Frequency D5S818 7 7 0.035 10 10 0.252 13 13 0.165

Rape case in HK

Hp : the victim and the suspect were contributors

( )

1 , | =

p

H K M P

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Two-person Mixture

Locus Mixture Victim Suspect Frequency D5S818 7 7 0.035 10 10 0.252 13 13 0.165

Rape case in HK

Hd : the victim and one unknown were contributors

( )

0306 . 2 2 , |

13 7 10 7 2 7

= + + = p p p p p H K M P

d

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Two-person Mixture

Locus Mixture Victim Suspect Frequency D5S818 7 7 0.035 10 10 0.252 13 13 0.165

Rape case in HK

Hd : the victim and one unknown were contributors Hp : the victim and the suspect were contributors

7 . 32 0306 . 1 = = LR

Estimated by database D

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Two-person Mixture

Locus Mixture Victim Suspect Frequency D5S818 7 7 0.050 10 10 0.252 13 13 0.165

Rape case in HK

Hd : the victim and one unknown were contributors Hp : the victim and the suspect were contributors

62 . 22 0442 . 1 = = LR

22.6

31% reduction

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Sampling Variability

Underestimate the allele frequencies “Overstate the strength of the evidence

against the defendant” (Balding, 1995)

Bayesian approaches on identification cases

Balding (1995) Balding & Donnelly (1995) Foreman et al. (1997) Curran et al. (2002) Corradi et al. (2003)

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Bayesian Model

Database D with n individuals Reference samples

(K, D)

Evidential value

( )

( )

d p

H D K M P H D K M P LR | , , | , , =

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Bayesian Model

Allele frequencies Dirichlet prior

( )

l

x x x x

, 2 , 1 ,

..., , ,

θ θ θ θ =

( )

α

θ |

x Dir

( ) ( ) ( )

∫

=

θ

χ θ θ θ

α dx x Dir H x D K M P H D K M P | , | , , | , ,

( ) ( ) ( )

∫

=

θ

χ θ θ θ θ

α dx x Dir x D K P H K x M P | | , , , |

Probability by “plug-in” approach Product multinomial

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Bayesian Model

Locus Mixture Victim Suspect Frequency in D D5S818 7 7 n7 10 10 n10 13 13 n13

Rape case in HK Hd : the victim and one unknown were contributors Hp : the victim and the suspect were contributors

( )( ) ( )( )

13 10 7 7

2 2 12 3 2 13 2 12 n n n n n n LR + + + + + + =

( )'

1 ... 1 1 = α

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Bayesian Model

Locus Mixture Victim Suspect Frequency in D D5S818 7 7 20 10 10 143 13 13 94

Rape case in HK

Hd : the victim and one unknown were contributors Hp : the victim and the suspect were contributors

96 . 28 = LR

n = 284

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Bayesian Model

Locus Mixture Victim Suspect Frequency in D D5S818 7 7 70 10 10 504 13 13 331

Rape case in HK

Hd : the victim and one unknown were contributors Hp : the victim and the suspect were contributors

58 . 31 = LR

n = 1000

32.7

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Q-function

“plug-in” probability P(M | xθ , K, H) as a linear

combination of the Q-function (Fung & Hu, 2008)

Probability of j random alleles that all belong

to M and explain all alleles in B

( ) ( )

∑ ∑

⊂ ⊂ ∈

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − =

M C B M j C i i C M

x x B j Q

\ , \

1 | ,

θ θ

( )

M X B P ⊂ ⊂

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Q-function

M = { 7, 10, 13 }, V = { 10, 13 }, S = { 7, 7 } Hd : V and one unknown were contributors Hd’ : V and one relative of S were contributors

( ) ( )

} 7 { , 2 , , | Q H K x M P

d = θ

( ) ( ) ( ) ( )

φ φ

θ

, , 1 2 } 7 { , 2 , , |

2 1 '

Q k Q k Q k H K x M P

d

+ + =

ki : Kinship coefficients

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Q-function

Suspect is typed but prosecution involves a

close relative

Suspect may be unavailable for typing and a

close relative is typed

H : the victim, one relative R of a typed person T and other x – 1 unknowns were contributors

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Q-function

U set of alleles in M but absent in known contributors declared in H

H : the victim, one relative R of T = t1 t2 and

• ther x – 1 unknowns were contributors

( ) ( ) ( ) ( ) ( ) { } ( ) ( ) ( ) { } ( )

, \ , 2 2 \ , 1 2 } { \ , 1 2 , 2 , , |

2 1 2 1 2 2 2 1 1 1 1

t t U x Q t I t I k t U x Q t I k t U x Q t I k U x Q k H K x M P

M M M M

− + − + − + =

θ

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Modified Q-function

Replacing Q(.,.|xθ) in P(M | xθ , K, H) by

Q*(.,.) gives P(M ,D, K | H)

( ) ( ) ( ) ( )

∫

=

θ

χ θ θ θ θ

α dx x Dir x D K P H K x M P H D K M P | | , , , | | , ,

( ) ( ) ( ) ( )

∫

=

θ

χ θ θ θ θ

α dx x Dir x D K P x B j Q B j Q | | , | , , *

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Modified Q-function

Closed-form formula Easy to be implemented by computer program No simulation or approximation is need Does not increase computational complexity

( ) ( ) ( ) ( )

( ) ( )

∑

⊂ ⊂ ∪ ∪

• +

Γ + + Γ − + + + Γ + + Γ ∝

M C B M D K C C D K C C C M

n j n j n k n k B j Q

\ , , \

1 2 2 2 2 , * α α α α ( ) ( ) ( ) ( )

1 1 − + + = Γ + Γ j r r r r j r L

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Numerical Example

Locus Mixture Victim Suspect Frequency FGA 18 18 0.025 19 19 0.065 24 24 0.166 26 26 0.048 D5S818 7 7 0.035 10 10 0.252 13 13 0.165 D8S1179 12 12 0.118 16 16 0.098

Rape case in HK

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Numerical Example

Illustration: hypothetical database D with

fixed allele frequencies and different sample sizes

Hp : the victim and the suspect were contributors Hd’ : the victim and a relative of the suspect were contributors Hd : the victim and one unknown were contributors

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LR of Hp : the victim and the suspect are contributors vs Hd : the victim and one unknown are contributors

56% 10%

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LR of Hp : the victim and the suspect are contributors vs Hd : the victim and the cousin of the suspect are contributors

35% 5%

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Summary

Bayesian approach provides conservative

evaluation of DNA mixtures

Can be implemented efficiently by modifying

existing plug-in formulae

Incorporate subpopulation models to handle

cases involving different ethnic groups

Consideration of allele drop-out

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Thank You!