Basic forensic geneticsWeight of evidence. Likelihood Ratio (LR)

Combining information. Bayes theorem

Workshop: Kinship analysisFirst lecture:

Basics: Genetics, weight of evidence. I.1

Thore Egeland(1),(2), Klaas Slooten(3),(4)

(1) Norwegian University of Life Sciences, (2) NIPH, (3) Netherlands ForensicInstitute, (4) VU University Amsterdam

Seoul, ISFG workshop, Aug 29 2017

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Basic forensic geneticsWeight of evidence. Likelihood Ratio (LR)

Combining information. Bayes theorem

Short bio

I Education:

Mathematics, statistics, computer science,mostly from University of Oslo.PhD in statistics, University of Oslo, 1989.

I Work:

Previously: Research, consulting.From 2011: Professor of statistics,Norwegian University of Life Sciences.20% position, Section of Forensics, Oslo University Hospital.

I Research interests:

Statistical, mathematical perspective on (forensic) genetics.Software: Familias (Windows and R).

I Norwegian: living in Oslo.

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Basic forensic geneticsWeight of evidence. Likelihood Ratio (LR)

Combining information. Bayes theorem

Contents of workshop

09:00-09:45 I.1 Basics: Genetics, weight of evidence (LR, Bayes). Thore.

09:45-10:30 I.2 Pairwise comparisons, IBD, mutation, theta,... Klaas.

10:30-11:30 Exercises. Coffee 11:00.

11:30-12:30 II LR properties. Interpretation. Klaas.

12:30-13:00 Exercises.

13:00-14:00 Lunch.

14:00-15:00 III.1 Familial searching, DVI. Klaas.

15:00-15:30 III.2 Mixtures and relatives. relMix demo. Thore.

15:30-16:00 Exercises.

16:00-16:30 IV.1,IV.2 Software: R. paramlink. Thore

16:30-17:45 Exercises.

17:45-18:00 Summary.

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Basic forensic geneticsWeight of evidence. Likelihood Ratio (LR)

Combining information. Bayes theorem

Contents 09:00–09:45

I Basic forensic genetics very briefly:

Mendelian inheritanceMarkers: autosomal, X, Y, mtDNA, STR-s.

I Weight of evidence. Likelihood Ratio (LR). Assumptions:

Hardy Weinberg Equilibrium (HWE).Linkage.Linkage disequilibrium (LD).

I Combining information. Bayes theorem:

Standard version.On odds form.On log form.

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Basic forensic geneticsWeight of evidence. Likelihood Ratio (LR)

Combining information. Bayes theorem

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Basic forensic geneticsWeight of evidence. Likelihood Ratio (LR)

Combining information. Bayes theorem

Pedigree

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Basic forensic geneticsWeight of evidence. Likelihood Ratio (LR)

Combining information. Bayes theorem

Genetic markers I

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Basic forensic geneticsWeight of evidence. Likelihood Ratio (LR)

Combining information. Bayes theorem

Genetic markers II

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Basic forensic geneticsWeight of evidence. Likelihood Ratio (LR)

Combining information. Bayes theorem

. Genetic markers III. Example: Fusion 6C

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Basic forensic geneticsWeight of evidence. Likelihood Ratio (LR)

Combining information. Bayes theorem

Mendelian inheritance

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Basic forensic geneticsWeight of evidence. Likelihood Ratio (LR)

Combining information. Bayes theorem

X linked inheritance

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Basic forensic geneticsWeight of evidence. Likelihood Ratio (LR)

Combining information. Bayes theorem

Y linked inheritance

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Basic forensic geneticsWeight of evidence. Likelihood Ratio (LR)

Combining information. Bayes theorem

Mitochondrial (mtDNA) inheritance

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Basic forensic geneticsWeight of evidence. Likelihood Ratio (LR)

Combining information. Bayes theorem

Hypotheses

AF17/18

8/8

MO−/−−/−

CH17/17

8/8

I H1: AF biological father of CH.

I H2: AF and CH unrelated.

I Notation. Sometimes:

I H1 = HP :“prosecution hypothesis”,

I H2 = HD :“defence hypothesis”.

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Basic forensic geneticsWeight of evidence. Likelihood Ratio (LR)

Combining information. Bayes theorem

Likelihood ratio. Definition

Forensic framework

LR = LRH1,H2(E ) =P(E | H1)

P(E | H2)

is the likelihood ratio for evidence E with respect to the twohypotheses H1 and H2. The LR measures how much better H1

explains the evidence E than H2.

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Basic forensic geneticsWeight of evidence. Likelihood Ratio (LR)

Combining information. Bayes theorem

Likelihood Ratio. Example

AF17/18

8/8

MO−/−−/−

CH17/17

8/8

LR =P(E | H1)

P(E | H2)= · · · =

P(gCH | gAF )

P(gCH)

LR1 =12p17

p217=

1

2× 0.204= 2.45

LR2 =p8p28

=1

0.554= 1.81

LR = LR1 × LR2 = 2.45× 1.81 = 4.4.

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Basic forensic geneticsWeight of evidence. Likelihood Ratio (LR)

Combining information. Bayes theorem

Likelihood Ratio. Interpretation and assumptions

AF17/18

8/8

MO−/−−/−

CH17/17

8/8

I Interpretation LR=4.4: Thedata is 4.4 times more likelyif AF is assumed to be thefather compared to theunrelated alternative.

I Assumptions

Hardy–WeinbergEquilibrium (HWE).Independent markers.No artefacts:(no mutation, no silentalleles, no drop–out/in,no error).

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Basic forensic geneticsWeight of evidence. Likelihood Ratio (LR)

Combining information. Bayes theorem

Realistic number of markers

Marker CH AF LR LR(mut)

D3S1358 17/17 17/18 2.45 2.45TPOX 8/8 8/8 1.81 1.80

D6S474 16/17 14/15 0.000 0.001. . . . . . . . . . . . . . .

D19S433 12/15 12/14 3.34 3.34

Total 0 25070642

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Basic forensic geneticsWeight of evidence. Likelihood Ratio (LR)

Combining information. Bayes theorem

W = Posterior probability of paternity

I Assume prior probabilities P(HP) = P(HD) = 0.5(reasonable?)

I Prior odds P(HP)P(HD)

= 1.

Then

W = P(HP | E ) =LR

LR + 1=

25070642

25070642 + 1

= 0.99999996 = ”Probability of HP given evidence”

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Basic forensic geneticsWeight of evidence. Likelihood Ratio (LR)

Combining information. Bayes theorem

Bayes theorem on odds form

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Basic forensic geneticsWeight of evidence. Likelihood Ratio (LR)

Combining information. Bayes theorem

Blackstone ratio

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Basic forensic geneticsWeight of evidence. Likelihood Ratio (LR)

Combining information. Bayes theorem

Optimal decision rule

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Basic forensic geneticsWeight of evidence. Likelihood Ratio (LR)

Combining information. Bayes theorem

Adding evidence I

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Basic forensic geneticsWeight of evidence. Likelihood Ratio (LR)

Combining information. Bayes theorem

Adding evidence II

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Basic forensic geneticsWeight of evidence. Likelihood Ratio (LR)

Combining information. Bayes theorem

T Egeland, D Kling, and P Mostad.Relationship Inference with Familias and R: StatisticalMethods in Forensic Genetics.Academic Press, 2015.

A Tillmar and P Mostad.Choosing supplementary markers in forensic casework.Forensic Science International: Genetics, 13:128–133, 2014.

IJ Wood.Weight of evidence: A brief survey.Bayesian Statistics, 1985.

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