1 Statistical genetics and genetical statistics Thore Egeland, Rikshospitalet and Section of Medical Statistics Joint work with P. Mostad, NR, B. Olaisen, B. Mevåg, M. Stenersen, Inst of Forensic Medicine. Grimstad 6/6/ Grimstad 6/6/2000

2 Contents What did we learn in school and what have we read in the papers? Erik Essen-Möller Identification problems: - origin of wine grapes (Science, 3/10/99), - wolves and dogs (Villmarksliv 3, 2000), - disasters, (Nature gen. 15/4/97), - paternity, e.g., Jefferson (Nature. 5/11/98).

3 Peas!

4 Nature Genetics, OJ

5 Dispute laid to rest

6 Tre slides på Essen-Møller

7 On the theory and practice of Essen-Möller's W value and Gurtler's paternity index (PI). Hummel K Forensic Sci Int 1984 May;25(1):1-17

8 H 1 : M1 father H 2 : Random man father P(data| H 1 )= P(data| H 2 )=p B Paternity index=LR=1/ p B Five independent loci, p B =0.05: LR=(1/p B ) 5 = Paternity index (PI). LR A,AB,B A,B M1 F1 M2

9 Bayes Theorem on odds form Posterior odds = LR * prior odds Essen-Möller’s W=P(H 1 |data) assuming prior odds=1

10 Bayes theorem: Framework for merging independent data Nuclear DNA. Several independent loci mitochondrial DNA: maternally inherited All these mitochondrial DNAs stem from one woman who is postulated to have lived about 200,000 years ago, probably in Africa. Cann, Nature, 1987 Y-chromosome. Paternal

11 Dual origins of finns

12 Ambitions We would like to: - determine most likely family among many, - include non-DNA data (prior), e.g. age, - m odel mutations, - model kinship (departures from Hardy-Weinberg), - model measurement uncertainty.

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14 Bayesian solution Find a set of “possible” pedigrees Set up prior probabilities based on non-DNA information. Compute for each pedigree Make inferences from the posterior distribution:

15 Example of use: The Romanov family 9 bodies found, presumed to be Tsar Nicolay II, Tsarina and his three daughters, three servants, and a doctor. Age and sex information for the bodies narrow down possible pedigrees to Our method picked among these the accepted pedigree. Mitochondrial DNA link with Prince Philip, Duke of Edinburgh.

16 Prior distribution

17 Modelling mutations Mutation rate varies with –Sex of parent and locus. Alleles tend to mutate to close alleles:

18 database

19 Kinship and uncertainty in allele frequencies Vector of allele frequencies p Dirichlet by evolutionary argument data|p ~ Multinomial Then p|data ~ Dirichlet Basis for simulation

20 Paper challenge

21 Alternatives to consider One extra woman and man introduced gives 1074 possible families Flat prior Three examples:

22 w2 childwom childwom man m2 man Full sibs Incestuous Unrelated childwom man

23 w2 childwom childwom man m2 man most probable among 1074 I II III LR (I/II) =2.1 LR(I/III) = 1.6*10^18 childwom man

24 Further results Number reduced from 1074 to 193 disregarding incestuous pedigrees. Same result; now LR=165. Full sib alternative most likely also when allowing for larger pedigrees. Non-flat prior not needed, even so...

25 F F F b G = 2 b I = 3 b P = 3 b G = 1 b I = 0 b P = 0 Example Prior ratio A/B= A: B: childwommanchildwomman

26 Non-flat prior All M-parameters 0.1: same result.

27 Literature Evett og Weir. "Interpreting DNA evidence". Sinauer, MA, USA, Egeland, Mostad, Mevåg og Stenersen. "Beyond traditional paternity and identification cases. Selecting the most probable pedigree". Forensic Science International, 110(1), 2000