1. 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?

2. 2
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

3. 3
C
 Epithelial swab from female victim (V)
 Sexual assault with two suspects under Hp (S1, S2)
Casework example

4. 4
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

5. 5
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

6. 6
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

7. 7
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?

8. 8
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

9. 9
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

10. 10
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)

11. 11
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

12. 12
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

13. 13
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)

14. 14
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

15. What is a p-value?
15
• We ask: What is the probability of observing a
LR at least as large as the one observed IF Hd
is true?

16. 16
• 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

17. 17
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

18. 18
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

19. http://forensim.r-forge.r-project.org/
https://sites.google.com/site/forensicdnastatistics/