Statistics in Practice Forensic Science

Transcription

1 Statistics in Practice Forensic Science Dr. David Lucy Lancaster University Statistics in Practice p.1/36

2 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 on the part of courts that observations are subject to uncertainty. Statistics in Practice p.2/36

3 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. Statistics in Practice p.3/36

4 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. Statistics in Practice p.4/36

5 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. Statistics in Practice p.5/36

6 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 - obviously 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? Statistics in Practice p.6/36

7 Case work - Tinley r 1 r 3 r 2 If: 1. r 1 is the radius head. 2. r 2 is the radius of the area of the wound. 3. r 3 is the radius of the area of the implement. Statistics in Practice p.7/36

8 Case work - Tinley Pr = 2π(r 2 r 3 ) 2 2π(r 1 r 3 ) 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. Statistics in Practice p.8/36

9 Firearm rifling patterns Two firearms offences committed: The first incident was the shooting of a man in an Ulster town in June The second was a shooting of a man in the same town in March Both incidents featured a 0.32 calibre revolver with a 5-right rifling pattern. What is the evidential value of the match? Statistics in Practice p.9/36

10 5-right 0.32 calibre Statistics in Practice p.10/36

11 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. H p is that two firearms offences employed the same weapon, and that weapon is that found in the possession of the suspect. 2. H d is that the firearms used in the two offences were different weapons to that of the suspect. Statistics in Practice p.11/36

12 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 H p,i) Pr(E H d,i) where I a firearm has been used in the commission of the offences. Statistics in Practice p.12/36

13 Numerator The numerator is: Pr(E H p,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 Statistics in Practice p.13/36

14 Denominator The denominator is: Pr(E H d,i). What is the probability of observing 5-right, 0.32 calibre, were the firearm used some other firearm other 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. Statistics in Practice p.14/36

15 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 Statistics in Practice p.15/36

16 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 Should I have done so? - some disquiet about using the higher figure. Statistics in Practice p.16/36

17 How incriminating Evett et al. (2000) give the following table: likelihood ratio verbal equivalent 1 < LR 10 limited support for H p 10 < LR 100 moderate support for H p 100 < LR 1000 moderately strong support for H p 1000 < LR strong support for H p < LR very strong support for H p 179 is in the middle of this range - thus implying moderately strong support for H p. Statistics in Practice p.17/36

18 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. Statistics in Practice p.18/36

19 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? Statistics in Practice p.19/36

20 Glass Statistics in Practice p.20/36

21 Glass Statistics in Practice p.21/36

22 Glass With glass the observables tend to be: Refractive index. Major, minor and trace element measurements. Isotopic measurements - not attempted yet - should be possible. Statistics in Practice p.22/36

23 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. Statistics in Practice p.23/36

24 Continuous variables Statistics in Practice p.24/36

25 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. Statistics in Practice p.25/36

26 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. Statistics in Practice p.26/36

27 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? Statistics in Practice p.27/36

28 The fallacy of proximity Statistics in Practice p.28/36

29 The fallacy of proximity? Statistics in Practice p.29/36

30 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 observations are continuous. Need knowledge of population to make any legitimate probabilistic inference about identity of source. Statistics in Practice p.30/36

31 Lindley s formulation X sample of m measurements from crimescene, Y sample of n measurements of suspect properties. X N(θ 1,σ 2 /m), Y N(θ 2,σ 2 /n). Under Under H p : θ 1 = θ 2 = θ, under H d : θ 1 θ 2. LR = Pr(X θ)pr(y θ)pr(θ)dθ Pr(X θ1 )Pr(θ 1 )dθ 1 Pr(Y θ2 )Pr(θ 2 )dθ 2 Statistics in Practice p.31/36

32 Lindley s formulation LR = σ 1σ 2 aσσ 3 exp { (X Y)2 τ 2 } a 2 σ 2 (σ1 2 +σ2 2 }{{ ) } exp { (W µ)2 ) 2σ (Z µ)2 (σ1 2 +σ2 2 ) } 2σ 2 1 σ2 2 }{{} proximity rarity where: X mean of observations from object 1 σ1 2 = τ2 +σ 2 /m Y mean of observations from object 2 σ2 2 = τ2 +σ 2 /n m number of observations from object 1 σ3 2 = τ2 +σ 2 /(m+n) n number of observations from object 2 Z = (σ2 2X +σ2 1 Y)/(σ2 1 +σ2 2 ) σ 2 within object variance W = (mx +ny)/(m+n) µ population mean a 2 = 1/m+1/n τ 2 population variance Statistics in Practice p.32/36

33 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 Statistics in Practice p.33/36

34 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 observations. Comparison problems, at the moment, are developing quickly Statistics in Practice p.34/36

35 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. Statistics in Practice p.35/36

36 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. Statistics in Practice p.36/36