Hinda Haned, Guro Dorum, Thore Egeland, Peter Gill And EUROFORGEN-NoE

On the meaning of the likelihood ratio: is a large number always an indication of

strength of evidence?

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Outline

We discuss the use of a LR-based model for evaluating the weight of DNA evidence

The presented concepts can be applied to other models

We illustrate using the LRmix tool from the Forensim package.

Consider contributors in the following epg. We could regard this as a typical LTDNA profile

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C

Epithelial swab from female victim (V)

Sexual assault with two suspects under Hp (S1, S2)

Casework example

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Pre-case assessment

Crime-stain alleles

Marker Allele1 Allele2 Allele3 Allele4 S1 S1 S2 S2 Unique alleles

AMEL X Y X Y X Y 2

D3S1358 14 16 17 16 17 15 17 4

VWA 16 17 18 19 16 18 18 19 4

D16S539 11 12 13 15 12 13 12 12 4

D2S1338 17 19 20 (24) 19 20 17 18 4

D8S1179 9 10 13 14 9 13 13 13 4

D21S11 29 31 32 28 32 30 30 5

D18S51 12 16 12 15 12 20 4

D19S433 12 14 15.2 16 12 16 12 15 5

TH01 6 9.3 6 9.3 6 9.3 2

FGA 19 24 26 19 21 20 21 5

 

(15) 

 

(15)   

   

 

14 alleles found in the crime stain that match the victim

alleles shared with Victim under Hp

() alleles below the detection threshold but appear to be distinct

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Pre-case assessment

Crime-stain alleles

Marker Allele1 Allele2 Allele3 Allele4 S1 S1 S2 S2 Unique alleles

AMEL X Y X Y X Y 2

D3S1358 14 16 17 16 17 15 17 4

VWA 16 17 18 19 16 18 18 19 4

D16S539 11 12 13 15 12 13 12 12 4

D2S1338 17 19 20 (24) 19 20 17 18 4

D8S1179 9 10 13 14 9 13 13 13 4

D21S11 29 31 32 28 32 30 30 5

D18S51 12 16 12 15 12 20 4

D19S433 12 14 15.2 16 12 16 12 15 5

TH01 6 9.3 6 9.3 6 9.3 2

FGA 19 24 26 19 21 20 21 5

 

(15) 

 

(15)   

   

 

Trace is a two- or three-person mixture The mixture is low level and dropout is expected. It is reasonable to condition on the victim under Hp and Hd

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Evaluate the first scenario based on epg and case circumstances

The proposition under Hp is S1,S2,V

The proposition under Hd is U1,U2,V

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Hp: S1+S2+Vic , Hd: Vic + U1+ U2

We estimate log10(LR) 5.3

But can Hp rely on this LR to prosecute both suspects or is this a naïve approach?

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Non-contributor tests

The process is exploratory Propositions are not obvious So what will happen if we replace a suspect with a random man? We would expect the LR to be very low (an exclusion!!) because we would expect a model to distinguish random man from a true perpetrator Therefore, the non-contributor test is a measure of robustness and we consider this to be an important part of model validation This idea must apply to all models - not just LRmix

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Non-contributor tests

S1 replaced by 1000 random men S2 replaced by 1000 random men

This means that the model is insensitive to S2 because the same result can be achieved with random man

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What does this mean?

Beware complex propositions – the relative weightings of the S1,S2 ‘contributions’ are not reflected in the likelihood ratio Therefore complex propositions must be simplified and qualified before they can be reported The non-contributor test is a useful adjunct to verify the likelihood ratio (define limitations of the model) and also provides an additional way to think about the results (court-friendly)

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Simplify the propositions

So far we don’t have evidence for S2 under Hp Therefore we need to think about different propositions in order to re-evaluate the evidence There seems to be good evidence under Hp for S1

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Evaluate the results and decide if new propositions are required

Evidence for S2 under Hp is exclusionary Very strong evidence for S1 under Hp, regardless of propositions tested

three person mixture Robustness estimation

Hp Hd log10(LR) LR distribution Random man substituted

S1,V,U V,U,U 7.3 (-10,-5,-0.9) S1

S2,V,U V,U,U -3 (-10,-5,-0.9) S2

S1,S2,V V,U,U 5.3 (-23,-16,-8) S1

S1,S2,V V,U,U 5.3 (+0.1,+3.7,+7.9) S2

two person mixture Robustness estimation

Hp Hd log10(LR) LR distribution Random man substituted

S1,V V,U 7.9 (-45,-30,-15) S1

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Summary Not a black box – this is: Exploratory

Data Analysis

Evaluation of S1,V,U under Hp gives Reported LR=7.3(-10,-5,-0.9) Evaluation of S2,V,U under Hp gives Reported LR=-3.5 (-10,-5,-0.9) Recall S1,S2,V under Hp gave LR= 5.3 (this is our naïve estimate)

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Further analyses: Exact p-values (Guro Dorum)

Percentiles are a bit crude – can we do better? • To calculate an exact probability we need the combined probabilities

of all genotypes that give an LR >= the observed LR But for 16 markers, 10 alleles each, there are 7x1027 genotype permutations

A recursive method has been developed to do the calculation

What is a p-value?

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• We ask: What is the probability of observing a LR at least as large as the one observed IF Hd is true?

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• The observed LR=7.3 (-10, -5, -0.9) • But we are primarily interested in the p-value, which is the chance that A random man substitution of S1 will give an LR >7.3 and this is Pr=9.68E-10 (so the propositions and the resultant LR appear to be probative)

As an example, consider the non-contributor test for S1 using hypotheses: Hp=S1,V,U; Hd=U,U,V Is this reasonable ?

three person mixture Robustness estimation

Hp Hd log10(LR) LR distribution Random man substituted

S1,V,U V,U,U 7.3 (-10,-5,-0.9) S1

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Final remarks what if the perpetrator is a relative?

• Hd does not necessarily have to specify “a random man from the population” as alternative

Example: Hp: Victim + Suspect

Hd: Victim + Unknown half sib of suspect

Suspect can be replaced with possible profiles for half sib to get distribution of LR under Hd

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Summary

Both S1 and S2 are suspects of sexual assault and a sample is taken from the victim. We condition on the victim under Hd No evidence for S2 in the crime stain [even though a three person evaluation with S1,S2 under Hp gives a high LR= log10(5)] Advice: Simplify propositions if there are two suspects always evaluate them separately, replacing the other with an unknown under Hp and Hd

We provide tools to calculate the LR and also ask whether the LR is meaningful in the context of the propositions (hypotheses) that are formulated. We simultaneously provide a novel, but simple method to evaluate any level of relatedness

http://forensim.r-forge.r-project.org/

https://sites.google.com/site/forensicdnastatistics/