1 Statistical genetics and genetical statistics Thore Egeland, Rikshospitalet and Section of Medical Statistics Joint work with P. Mostad, NR, B. Olaisen,

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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