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Statistics and genetics: A criminal perspective

Statistics and genetics: A criminal perspective. Thore.Egeland National University Hospital & Section of Medical Statistics, Oslo. Contents. Motivation. Genetical background, data. Principles for evaluation of evidence.

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Statistics and genetics: A criminal perspective

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  1. Statistics and genetics:A criminal perspective Thore.Egeland National University Hospital & Section of Medical Statistics, Oslo Thore.Egeland@basalmed.uio.no

  2. Contents • Motivation. Genetical background, data. • Principles for evaluation of evidence. • Mixtures evidence (e.g., OJ-case)-Multiple hypotheses.-How many contributors?-Incomplete data, related contributors, eg incest. • References.

  3. Forensic genetics:Two perspectives • Civil cases: Paternity-cases, identification (disasters), family reunion, (immigration). ... • Criminal cases

  4. Criminal cases involving DNA, Norway

  5. Biological material Tissue Bone Teeth ... Blood Hair Salvia Semen Urine Nails Vaginal swab

  6. Allele frequencies for HUMVWA Througout allele frequencies are considered fixed.

  7. Stain Victim Suspect 16 17 18 20 Suspect cleared

  8. Stain Victim Suspect A1 A2 A3 A4 Suspect not cleared

  9. Three principles of interpretation • To evaluate the uncertainty of any given proposition it is necessary to consider at least one alternative proposition. • Interpretation is based on asking “What is the probability of the evidence given the proposition?” • Interpretation is conditioned on competing propositions and context.Evett and Weir (1998).

  10. Applying the principles • Principle 1: H1: “Stain from defendant and victim”H2: “Stain from unknown and victim” • Principle 2:Calculate L(data|H1) and L(data|H2) and form the likelihood ratio

  11. Example • We calculate the likelihoods given the data:L(stain|H1)=1 L(stain|H2)=2pA1 pA2 Hardy-Weinberg Likelihood ratio=1/( 2pA1 pA2) =50 frequences 1/10.

  12. Several loci • Assume for simplicity four independent loci give same numerical result, then the overall likelihood ratio becomes.LR=50^4 • Throughout we consider one locus for simplicity.

  13. Principle 3 • How is other evidence included? • More than two conceivable propositions? -For instance, more than two contributors? • What if contributors are related?

  14. Bayes Theorem on odds form • Posterior odds = LR * prior odds • Principle 2: Expert should report LR. This can be converted to statements regarding guilt using Bayes theorem.

  15. Multiple hypotheses • Bayesian framework: Possible hypotheses H1, ..., HNPriors P(H1) ,..., P(HN) Compute P(data|Hi) and finally P(data|Hi) • Problematic in legal settings? • Numbers of previous example are reproduced for uniform prior, not interpretation.

  16. General formula. Weir (1998). x = no. of U-contributors, E = evidence (stain), U = alleles contributed by unknown persons, Bo = sum of allele frequencies, B-j,1 = sum of allele frequencies except allele j in U.

  17. Previous example H2: “Stain from unknown and victim” P1({1,2,3,4}|{1,2}) = (p1+ p2+ p3+ p4)2- ( p2+ p3+ p4)2 -(p1+ p3+ p4)2 +(p3+ p4)2 = 2 p1 p2

  18. Unknown no. x of contributors • Give several alternative calculationsOJ-case. Weir et al. (1995, 1997). • Give inequalities, e.g., most favorable alternative for defendant.Brenner et al. (1996), Buckleton et al. (1998). • Maximize likelihood wrt to xStockmarr (2000).

  19. Alternative approach • Put a probability distribution on X. • Convenient choices (x=0,1,2,...):

  20. Example Stain: a and b, both with frequencies p. H: x unknown contributors

  21. E(no contributors)=1+lambda

  22. Top curve: p=0.4. Uncertain number of contributors. Reasonable! Bottom: p=0.05 E(no contributors)=1+lambda

  23. OJ case H1: Stain from OJ and NB H2: Stain not from OJ or NB Stain OJ NB During court proceedings additional calculations were required allowing for varying no. of unknown contributors. Buckleton et al (1998) argue thatLR is between 7.2 and 27.6 if x<=10.

  24. OJ case H1: Stain from OJ and NB and x unknown (geometrically dist.) H2: Stain not from OJ or NB and x unknown (geometrically dist.)

  25. Summary • Important biostatistical application. • Possible to model and estimate no of contributors. • Challenge: model raw data realistically.

  26. References • Brenner et al. (1996). Int J Legal Med 109, pp 218-219 • Buckleton et al (1998). Science and Justice 38, pp. 23-26. • Dalen (2001). Diploma Thesis, NTNU, Trondheim, Norway • Evett and Weir (1998). “Interpreting DNA evidence”. Sinauer, MA, USA. • Weir et al (1997). J Forensic Sci 42, pp. 213-222. • Weir (1995). DNA statistics in the Simpson matter. Nat. Gen, 11, 366-368 • Stockmarr (2000). In “Stat Science in the courtroom”, ed Gastwirth, Springer. • http://www.uio.no/~thoree

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