Statistics in Practice Forensic Science Dr. David Lucy - - PowerPoint PPT Presentation

Statistics in Practice Forensic Science

• Dr. David Lucy

d.lucy@lancaster.ac.uk

Lancaster University

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Forensic Science

Criminal evidence becoming increasingly “scientific”.

• Greater use of trace evidence (paint/glass/fibres).
• DNA revolution.

The rise of DNA was coincident with a greater awareness

• n the part of courts that observations are subject to

uncertainty.

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Forensic Science

Greater realisation that uncertainty is important has lead to:

• Trace evidence (glass/paint/fibres) being treated

statistically.

• More evidence types:
• common observations - shoe types - facial features
• all being treated with some form of statistical

method.

• observation of co-incidence of treatment in cases

where carers are suspected of harming their charges.

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Forensic Consultancies

Typically:

• Police.
• Customs and Excise.
• Criminal Defence lawyers.

Start with an approach by one of the above.

• Usually concludes with the submission of a statistical

report.

• Rarely concludes with a court appearance.

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Case work - Tinley

The facts were:

• Evening of 9th April 2004 Andrew Tinley was assaulted

by his partner Sally Rose.

• He picked up a champagne bottle and struck her on

the head.

• Rose died at the scene of the incident.

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Case work - Tinley

Under interrogation:

• Tinley said he had struck Rose twice with the bottle.
• The pathologist could find no evidence for two strikes.

A double blow is more incriminating than a single -

• bviously of interest to the court

The question was: what is the probability of administering two blows with a champagne bottle and leaving only a single wound?

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Case work - Tinley

1 2

r3 r r

If:

• 1. r1 is the radius head.
• 2. r2 is the radius of the area of the wound.
• 3. r3 is the radius of the area of the implement.

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Case work - Tinley

Pr = 2π(r2 − r3)2 2π(r1 − r3)2 Given the values for r in the pathologists report gives a probability of about 8%.

• Only a guide for the court.
• Could spend a lot of time working out a more exact

value.

• Limits of knowledge given by Tinley’s account.

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Firearm rifling patterns

Two firearms offences committed:

• The first incident was the shooting of a man in an

Ulster town in June 2000.

• The second was a shooting of a man in the same town

in March 2003. Both incidents featured a 0.32 calibre revolver with a 5-right rifling pattern. What is the evidential value of the “match”?

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5-right 0.32 calibre

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Firearm rifling patterns

A suspect had been located.

• That suspect was been found to possess a firearm with

a five-right rifling pattern. Start with two propositions

• 1. Hp is that two firearms offences employed the same

weapon, and that weapon is that found in the possession of the suspect.

• 2. Hd is that the firearms used in the two offences were

different weapons to that of the suspect.

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Likelihood ratios

A standard measure of evidential value is a likelihood ratio Let:

• E ≡ the firearm used was of calibre 0.32 and rifling

pattern 5 right. Then: LR = Pr(E|Hp, I) Pr(E|Hd, I) where I ≡ a firearm has been used in the commission of the offences.

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Numerator

The numerator is: Pr(E|Hp, I).

• What is the probability of observing 5-right, 0.32

calibre, were the firearm used that of the suspect.

• The suspect has only one firearm.
• Some other individual may have used the suspect’s

firearm - should be from defence case if so. The probability is quite high ≈ 1

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Denominator

The denominator is: Pr(E|Hd, I).

• What is the probability of observing 5-right, 0.32

calibre, were the firearm used some other firearm

• ther than that of the suspect.
• This is proportional to the frequency of 5-right, 0.32

calibre, firearms from the population of illegally held firearms. Need for data - supplied by investigators.

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Denominator

• From data supplied of observations from a suitable

sample of weapons recovered

• There are 716 illegal firearms known to the firearms

intelligence branch

• 4 were revolvers with right handed rifling of 5 grooves,

and were of 0.32 calibre. The likelihood ratio is 716/4 = 179

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Likelihood ratio

The likelihood ratio is 716/4 = 179 can be interpreted: The observation of 0.32 calibre, and 5-right rifling, is 179 more likely were the weapon used in these offences that of the suspect rather than any other weapon from the population of illicit firearms. This does not take into account the fact that there were two scenes - should it be 1792. Should I have done so? - some disquiet about using the higher figure.

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How incriminating

Evett et al. (2000) give the following table: likelihood ratio verbal equivalent 1 < LR ≤ 10 limited support for Hp 10 < LR ≤ 100 moderate support for Hp 100 < LR ≤ 1000 moderately strong support for Hp 1000 < LR ≤ 10000 strong support for Hp 10000 < LR very strong support for Hp 179 is in the middle of this range - thus implying moderately strong support for Hp.

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Case work

The “comparison problem” is the archetypal forensic problem:

• where a fragment of an item found to be associated

with a suspect, compared to an item known to be associated with an offence,

• found to “match” in some sense.

What is the evidential value of that “match”.

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Glass

Surprisingly common material of forensic interest

• Criminal activity often includes glass breakage.
• Shards scattered over area, including offenders.

To what extent do the observations from the fragments of glass found upon a suspect suggest that some, or all, of those fragments came from the crimescene glass?

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Glass

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Glass

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Glass

With glass the observables tend to be:

• Refractive index.
• Major, minor and trace element measurements.
• Isotopic measurements - not attempted yet - should be

possible.

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Continuous variables

For continuous variables there are a few problems:

• Within source variation.
• Leads to some conceptual confusion - no such thing as

an absolute match. Repeated observations of the same object are unlikely to give exactly the same observations - similarity not as simple as for shoe types.

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Continuous variables

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Traditional evaluation

Traditional evidence evaluation methods include:

• 1. nσ rules - where if the means between the fragments

for all elements are within nσ of each other then a “match” is declared - frighteningly bad.

• 2. multiple t testing - variables are t tested between

fragments - poor.

• 3. eyeball - scatterplots of pairwise elements are

examined by eye. The really hardened scientist examines principle component plots. Each and every one of these is unfounded, and none answers the question.

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Proximity based measures

These only take into account the proximity of one object to another.

• Proximity is not the same, and cannot be easily

equated to identity.

• Does not calculate a “weight of evidence” within a

propositional framework. It can:

• be used to generate the best guess from limited

number of entities,

• more tenuously be used to “exclude”.

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The fallacy of proximity

An easily conceived illustration of this:

• A friend collects coloured marbles - stores them in

bags.

• each bag has marbles of only one colour.
• there may be more than one bag containing

marbles of each colour.

• A marble has dropped out of a bag - the marble is red -

you select a bag and sample a marble - that marble is red. To what extent does the observation that both marbles are red support the notion that the marble came from that particular bag?

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The fallacy of proximity

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The fallacy of proximity

?

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The fallacy of proximity

Similarity of observation on its own:

• Gives no idea as to identity.

Dissimilarity of observation:

• Can be used to reject in cases where observation is

unambiguous.

• Does not apply in any logical manner where
• bservations are continuous.

Need knowledge of population to make any legitimate probabilistic inference about identity of source.

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Lindley’s formulation

• X sample of m measurements from crimescene,
• Y sample of n measurements of suspect properties.
• X ∼ N(θ1, σ2/m), Y ∼ N(θ2, σ2/n).

Under Under Hp : θ1 = θ2 = θ, under Hd : θ1 = θ2. LR =

• Pr(X|θ) Pr(Y |θ) Pr(θ)dθ
• Pr(X|θ1) Pr(θ1)dθ1
• Pr(Y |θ2) Pr(θ2)dθ2

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Lindley’s formulation

LR = σ1σ2 aσσ3 exp

• − (X − Y )2τ 2

a2σ2(σ2

1 + σ2 2)

• exp
• − (W − µ)2)

2σ2

3

+ (Z − µ)2(σ2

1 + σ2 2)

2σ2

1σ2 2

• proximity

rarity where: X mean of observations from object 1 σ2

1 = τ 2 + σ2/m

Y mean of observations from object 2 σ2

2 = τ 2 + σ2/n

m number of observations from object 1 σ2

3 = τ 2 + σ2/(m + n)

n number of observations from object 2 Z = (σ2

2X + σ2 1Y )/(σ2 1 + σ2 2)

σ2 within object variance W = (mX + nY )/(m + n) µ population mean a2 = 1/m + 1/n τ 2 population variance

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Generalisations

Lindley’s formulation of the comparison problem has now:

• extended to the multivariate case,
• for high multivariate spaces DAGs used,
• non-normal between item means using
• 1. kernel density estimates
• 2. log-concave estimation
• with structural zeros

This is, at the moment, in constant development

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Future work

There are a number of outstanding problems:

• 1. isotopic systems tend to be compositional - what about

multiple isotopic systems?

• 2. dimensionality - curse of - lots of dimensions - few
• bservations.

Comparison problems, at the moment, are developing quickly

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Professional forensic statistics

Intellectually forensic statistics is of great interest:

• Most have distinct features.
• Many revolve around “comparison” problems.

Academia one of the best occupations for those involved in this sort of work:

• Quite often your conclusions favour the side who has

not employed you.

• You can become quite unpopular with both sides.

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Professional forensic statistics

A career in forensic statistics:

• has variety of challenging problems.
• Is in a growing field, unfortunately the main employer,

FSS, is being shut down.

• Commercial forensic science providers are unlikely to

employ statisticians. But:

• Is difficult to operate in, particularly independently.
• Really need institutional support.

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