1 statistical genetics and genetical statistics thore egeland, rikshospitalet and section of medical...
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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 6/6/2000
www.uio.no/~thoree
Grimstad 6/6/2000
www.uio.no/~thoree
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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).
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Peas!
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Nature Genetics, OJ
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Dispute laid to rest
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Tre slides på Essen-Møller
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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
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H1: M1 fatherH2: Random man father
P(data| H1)=
P(data| H2)=pB
Paternity index=LR=1/ pB
Five independent loci, pB=0.05:
LR=(1/pB)5 = 3 200 000
Paternity index (PI). LR
A,A B,B
A,B
M1F1
M2
22BA pp
22BA pp
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Bayes Theorem on odds form
• Posterior odds = LR * prior odds
)(
)(
)|(
)|(
)|(
)|(
2
1
2
1
2
1
HP
HP
HdataP
HdataP
dataHP
dataHP
Essen-Möller’s W=P(H1 |data) assuming prior odds=1
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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
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Dual origins of finns
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Ambitions
• We would like to:
- determine most likely family among many,- include non-DNA data (prior), e.g. age,- model mutations,- model kinship (departures from Hardy-Weinberg), - model measurement uncertainty.
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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:
NPP ,...,1
)(),...,( 1 NPP
)| dataDNA( iP iP
N
jjj
iii
PP
PPP
1
)()|dataDNA(
)()|dataDNA()dataDNA|(
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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 4536.
• Our method picked among these the accepted pedigree.
• Mitochondrial DNA link with Prince Philip, Duke of Edinburgh.
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Prior distribution
yPromiscuit: ;Inbreeding : ;Generation:
specifieduser :parameters
pedigree from calculated parameters
c
},,{)(pedigrees space sample 1
PIG
M
b
MMMonst.prior
PPPIG b
PbI
bG
n
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Modelling mutations
• Mutation rate varies with – Sex of parent and locus.
Alleles tend to mutate to close alleles:
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database
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Kinship and uncertainty in allele frequencies
• Vector of allele frequencies pDirichlet by evolutionary argument
• data|p ~ Multinomial
• Then p|data ~ Dirichlet
• Basis for simulation
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Paper challenge
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Alternatives to consider
• One extra woman and man introduced gives 1074 possible families
• Flat prior
• Three examples:
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w2
childwom
childwom man
manm2
manman
Full sibs
Incestuous
Unrelated
childwom man
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w2
childwom
childwom man
manm2
manman
most probable among 1074I
II
III
LR (I/II) =2.1
LR(I/III) = 1.6*10^18
childwom man
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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 ...
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F
F F
bG= 2
bI = 3bP = 3
bG= 1
bI = 0bP = 0
Example
Prior ratio A/B=
A:
B:
331PIG MMM
childwom man childwom man
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Non-flat prior
• All M-parameters 0.1: same result.
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Literature
• Evett og Weir. "Interpreting DNA evidence". Sinauer, MA, USA, 1998.
• http://www.nr.no/familias• Egeland, Mostad, Mevåg og Stenersen.
"Beyond traditional paternity and identification cases. Selecting the most probable pedigree". Forensic Science International, 110(1), 2000