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 1/26
2/26 � Conclusion and future works � Bayesian model � Background � Example Outline
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 3/26
Two-person Mixture � Crime stain M � Reference samples K = ( V , S ) � Victim V , Suspect S � Evidentiary value ( ) ( ) | , P M K H , | P M K H = = p p LR ( ) ( ) , | P M K H | , P M K H d d 4/26
Two-person Mixture Rape case in HK Locus Mixture Victim Suspect Frequency D5S818 7 7 0.035 10 10 0.252 13 13 0.165 H p : the victim and the suspect were contributors ( ) = | , 1 P M K H p 5/26
Two-person Mixture Rape case in HK Locus Mixture Victim Suspect Frequency D5S818 7 7 0.035 10 10 0.252 13 13 0.165 H d : the victim and one unknown were contributors ( ) = + + = 2 | , 2 2 0 . 0306 P M K H p p p p p 7 7 10 7 13 d 6/26
Two-person Mixture Rape case in HK Locus Mixture Victim Suspect Frequency D5S818 7 7 0.035 Estimated 10 10 0.252 by database D 13 13 0.165 H p : the victim and the suspect were contributors H d : the victim and one unknown were contributors 1 = = 32 . 7 LR 0 . 0306 7/26
Two-person Mixture Rape case in HK Locus Mixture Victim Suspect Frequency D5S818 7 7 0.050 10 10 0.252 13 13 0.165 H p : the victim and the suspect were contributors H d : the victim and one unknown were contributors 1 = = 22.6 22 . 62 LR 31% reduction 0 . 0442 8/26
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) 9/26
Bayesian Model � Database D with n individuals � Reference samples ( K , D ) � Evidential value ( ) , , | P M K D H = p LR ( ) , , | P M K D H d 10/26
Bayesian Model ( ) θ = � Allele frequencies , , ..., x x x x θ θ θ , 1 , 2 , l ( ) α θ | Dir x � Dirichlet prior ( ) ( ) ( ) ∫ = α dx , , | , , | , | P M K D H P M K D x H Dir x θ θ θ χ θ ( ) ( ) ( ) ∫ = α dx | , , , | | P M x K H P K D x Dir x θ θ θ θ χ θ Probability by “plug-in” approach Product multinomial 11/26
Bayesian Model Rape case in HK Locus Mixture Victim Suspect Frequency in D D5S818 7 7 n 7 10 10 n 10 13 13 n 13 H p : the victim and the suspect were contributors H d : the victim and one unknown were contributors ( ) ' α = 1 1 ... 1 ( )( ) + + 12 2 13 2 n n = LR ( )( ) + + + + 3 12 2 2 n n n n 7 7 10 13 12/26
Bayesian Model Rape case in HK Locus Mixture Victim Suspect Frequency in D D5S818 7 7 20 n = 284 10 10 143 13 13 94 H p : the victim and the suspect were contributors H d : the victim and one unknown were contributors = 28 . 96 LR 13/26
Bayesian Model Rape case in HK Locus Mixture Victim Suspect Frequency in D D5S818 7 7 70 n = 1000 10 10 504 13 13 331 H p : the victim and the suspect were contributors H d : the victim and one unknown were contributors = 32.7 31 . 58 LR 14/26
Q-function � “plug-in” probability P ( M | x θ , K , H ) as a linear combination of the Q-function (Fung & Hu, 2008) j ⎛ ⎞ ( ) ( ) ∑ ∑ = − \ M C ⎜ ⎟ , | 1 Q j B x x θ θ , i ⎝ ⎠ ⊂ ⊂ ∈ \ M B C M i C � Probability of j random alleles that all belong to M and explain all alleles in B ( ) ⊂ ⊂ P B X M 15/26
Q-function � M = { 7, 10, 13 }, V = { 10, 13 }, S = { 7, 7 } � H d : V and one unknown were contributors ( ) ( ) d = | , , 2 , { 7 } P M x K H Q θ � H d’ : V and one relative of S were contributors ( ) ( ) ( ) ( ) = + φ + φ | , , 2 , { 7 } 2 1 , 0 , P M x K H k Q k Q k Q θ ' 0 1 2 d k i : Kinship coefficients 16/26
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 17/26
Q-function H : the victim, one relative R of T = t 1 t 2 and other x – 1 unknowns were contributors U set of alleles in M but absent in known contributors declared in H ( ) ( ) ( ) ( ) = + − | , , 2 , 2 1 , \ { } P M x K H k Q x U k I t Q x U t θ 0 1 1 1 M ( ) ( { } ) + − 2 1 , \ k I t Q x U t 1 2 2 M ( ) ( ) ( { } ) + − 2 2 , \ , k I t I t Q x U t t 2 1 2 1 2 M M 18/26
Modified Q-function ( ) ( ) ( ) ( ) ∫ = α dx , , | | , , , | | P M K D H P M x K H P K D x Dir x θ θ θ θ χ θ ( ) ( ) ( ) ( ) ∫ = α dx * , , | , | | Q j B Q j B x P K D x Dir x θ θ θ θ χ θ � Replacing Q (.,.| x θ ) in P ( M | x θ , K , H ) by Q *(.,.) gives P ( M ,D, K | H ) 19/26
Modified Q-function ( ) ( ) Γ α + + Γ α + + 2 2 n j ( ) ( ) k n ∑ ∪ ∝ − • \ M C , C C K D * , 1 ( ) Q j B ( ) Γ α + + + Γ α + 2 2 k n j n ⊂ ⊂ • \ ∪ M B C M , C C K D ( ) Γ + r j ( ) ( ) = + + − L 1 1 r r r j ( ) Γ r � Closed-form formula � Easy to be implemented by computer program � No simulation or approximation is need � Does not increase computational complexity 20/26
Numerical Example Rape case in HK 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 21/26
Numerical Example H p : the victim and the suspect were contributors H d : the victim and one unknown were contributors H d’ : the victim and a relative of the suspect were contributors � Illustration: hypothetical database D with fixed allele frequencies and different sample sizes 22/26
LR of H p : the victim and the suspect are contributors vs H d : the victim and one unknown are contributors 10% 56% 23/26
LR of H p : the victim and the suspect are contributors vs H d : the victim and the cousin of the suspect are contributors 5% 35% 24/26
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 25/26
26/26 Thank You!
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