### Background:

- In 1969, Jane Mixer, a law student, was murdered. The case went cold.
- The case was reopened 33 years later when crime-scene evidence was submitted to DNA analysis.
- The DNA yielded two matches; both matches were from samples that were analyzed in the same lab and at the same time of the crime-scene DNA analysis. All three samples were analyzed in late 2001 and early 2002.
- One match was to John Ruelas. Mr. Ruelas was 4 in 1969 and was excluded as a suspect.
- The other match was to Gary Leiterman. Mr. Leiterman was 26 at the time. He was convicted in 2005 and is serving life without parole. His appeal was denied in 2007.
- There is no doubt that Mr. Leiterman's DNA was deposited on the crime scene sample. The match is 176-trillion-to-1.
- The question is whether the DNA was deposited at the crime scene in 1969 or if there was a cross-contamination event in the lab in 2002.

### A Very Easy and Helpful PowerPoint:

- This case comes from John Wixted, a psychologist at UCSD
- He has made a detailed and convincing presentation. Click here for The Power Point from John's website.
- John has helped to persuade the Innocence Clinic at the University of Michigan to investigate the Leiterman case.
- John and I are convinced this is a an injustice. We are working pro bono.

### Our Job:

- Our job is to make an educated assessment of Mr. Leiterman's guilt or innocence. It would greatly help the Innocence Clinic to assess whether there is sufficient evidence to appeal.
- The jury heard that DNA is a trillion-to-1 accurate and there was only a very tine chance of cross contamination. Yet, we know these are the wrong conditional probabilities to compute.
- Consider the two hypotheses above that Leiterman's DNA was deposited at the crime scene or, alternatively, that it was deposited in the lab through cross contamination. Conditional on the match, compute posterior probabilities.

### My Analysis:

I have done my own analyses and typeset them. But reasoning is tricky, and I would like some backup. It is just too important to mess up. Can you try your own analysis? Then we can decide what is best.

You will need more information. I used the following specifications. Write me if you want more:

- John and I assumed 2.5M people are possible suspects in 1969. It is a good guess based on population estimate of Detroit metro area.
- The lab processes 12,000 samples a year. The time period the DNA overlapped can be assumed to be 6 months, that is 6,000 other samples could be cross contaminated with Mixer or Leiterman.
- The known rate of DNA cross-contamination is 1-in-1500. That is, each time they do a mouth swab from one person, they end up with two or more DNA profiles with probability of 1/1500. We assume this rate holds for unknowable cross-contamination such as that in processing a crime scene.
- The probability of getting usable DNA from a 33-year-old sample is 1/2.
- Need other facts? Just ask in the comments.

Thank you,

Jeff Rouder

John Wixted

## 22 comments:

Thank you, Jeff. I tried this analysis myself and discovered it was harder than I thought. I really appreciate your help. I hope others can help as well.

Wow, the evidence underlying this conviction is about the flimsiest I have ever seen. The DNA of a 4-yr old showing up in the same place inexplicably? huh? really?

Astonishing case. Really a hard one, I couldnt figure out how to start. I tried to used Bayes for P(Guilty | Positive Test) for a random Inhabitant of Detroit and tried modifying P(Positive Test | Not Guilty) for the chance of crosscontamination. I got a Number with five zeros as endresult, so its probably far from being right...really interested how you solve it

Hi Jeff and John,

I'm struggling with this one, but one thing that I am wondering about it is the probability of cross-contamination involving more than one spuriously introduced DNA profiles. E.g., if we know that a particular sample includes at least one DNA profile introduced by contamination, what is the probability that there will be one or more *additional* profiles present in the sample that were introduced by contamination?

Matt, that is key, isn't it. I assumed it was the same cross-contamination event that captured each. But who knows. I did it with and without Ruelas, but when I did, I used this assumption.

My answer at https://github.com/rouderj/leiterman, see the pdf.

The following is not a statistical suggestion but nevertheless is relevant: I am far from sure that the method of analysis used in DNA matching is infallible. It all comes down to the number of markers used for comparison, which markers, how good AND relevant for a criminal investigation/comparison are they and what method was used for processing the sample.

May be this from Andrew Gelman's blog of use: http://andrewgelman.com/2007/05/18/the_prosecutors/

I cant remember when but Andrew recently (within last year) wrote an invited paper for a criminology journal which decided not to publish it, so he wrote about it in his blog. That would be useful for you. Sorry for the memory hole there.

Good luck with the case. I am curious now.

> I assumed it was the same cross-contamination event that captured each.

I'm not sure I would make that assumption. I think you have to assume two separate cross-contamination events, since that's the assumption that gives the lower probability of cross-contamination being the reason for the match. Or at least, if you're going to assume one cross-contamination event, I think you need to justify that assumption based on knowledge of how DNA labs work and how likely it is that a single cross-contamination event would include two DNA profiles.

> I think you have to assume two separate cross-contamination events

That assumption, by my calculations, when it is figured into the Fourth Pass calculation (including the fluids and the second sample), gives odds of guilt of 0.595 to 1, or a probability of guilt of 0.373.

(The calculation is the same as your Second Pass calculation, but dividing the numerator by 3 to account for the fluids and dividing the denominator by 1500 to account for the second cross-contamination event.)

Was John Ruelas present at the crime scene or was his identification a guarantee that cross-contamination happened in the lab at least once?

The prosecution suffers from a deeply unprofessional lack of curiosity to find out how Ruelas got his DNA on the victim or the crime scene.

Ockham's knife points the finger at the lab.

Sloppiness is a hallmark of forensic labs as has been documented countless times.

Credulous juries enable prosecutorial fantasies.

You say the DNA tests matched? How much of a match? How many alleles matched?

Ahh, the DNA expert testimony shows that the DNA matches were partial matches (2 to 7 alleles out of 13). What this would seem to indicate is that the person who matched the DNA is contained in a group that may be a blood relation of Leiterman.

Peter, you write, "I'm not sure I would make that assumption." I agree. It is the one I am quite unhappy with. It is not so much that I think there were two x-contamination events, but that we need to compute the probability of a single x-contamination event with two separate contaminating profiles (Ruelas and Leiterman). That is a challenge.

Here's a simple analysis:

If the same lab asked all 2.5M people in Detroit to come in to be swabbed, how many matches would there be against the evidence in this case? You can estimate this using the number of samples the lab took in the few months around the time of the analysis. Suppose they acquired 10k samples in that time; then the odds of a match are at least 1 in 10k (because of the 1 false match, Ruelas, out of the 10k the lab acquired in this time).

So if the lab swabbed 2.5M people in Detroit, they would find that 250 people matched (1 in 10k). Of these, at most 1 is guilty. So the probability of guilt given a match is less than 1 in 250.

> It is not so much that I think there were two x-contamination events

I think you need to consider both possibilities: the possibility that one cross-contamination event occurred and resulted in both DNA traces being put into the sample, and the possibility that there were two separate cross-contamination events, each of which put one DNA trace into the sample.

> we need to compute the probability of a single x-contamination event with two separate contaminating profiles (Ruelas and Leiterman)

You already estimated that: it's 1.85x10^-11 (1 in 1500*6000^2).

If you're thinking there should be an extra factor in there, i.e., that the probability of that single cross-contamination event containing both Ruelas' and Leiterman's DNA samples is less than 1 in 6000^2, I don't think that's significant unless you think that extra factor should be on the order of 1 in 1500 or smaller. The reason is simple: the probability of two independent cross-contamination events, one containing Ruelas' DNA and one containing Leiterman's DNA, is 1.23x10^-14 (1 in 1500^2*6000^2). So the likelihood ratio for that case will dominate the probability of guilt unless there is a small enough extra factor in the probability of a single cross-contamination event to bring it down to the same order of magnitude.

I do work involving DNA analysis for research purposes, but it is sequencing-based. I have no specific knowledge about CODIS but I have considerable difficulty believing that the 13 (now 20) loci in it are really completely "independently assorted" as claimed. This assumption is used to claim that the probability of a false positive is the product of the population frequencies of each allele.

If we were talking about a SNP chip or DNA sequencing, I would agree that the FPR is truly negligible because there are so many loci being measured. With CODIS, the FBI acknowledges that "near matches" can be obtained by relatives, but even that is possibly understating the case. After a moderate amount of Googling and PubMed searching, I could not find evidence for the assertion that the CODIS alleles are completely independent. It is puzzling to me that this would be taken as a given since, because of how genetics works, few alleles really are independent. What is certain is that using a "product rule" to calculate the FPR is the most optimistic possible way to do it, and the unfortunate result would be prejudicial to defendants if in fact the FPR is considerably higher.

I am sure that whatever the FPR of CODIS is, it is small, certainly far below 1%. But with the large prior probabilities involved, even a small FPR may become quite important. So that is another possibility to consider. And of course if it is true that, as one commenter suggests, that the match against Leiterman was only partial, then the FPR potential skyrockets.

> the likelihood ratio for that case will dominate the probability of guilt

Actually, on thinking it over, this is not correct. The correct method is to compute the likelihood ratio for the single hypothesis of cross contamination with two terms in the denominator: one for the case of a single cross-contamination event, and one for the case of two cross-contamination events. With the numbers as we have them, the sum of those two terms is basically equal to the first, since its probability is three orders of magnitude higher. But information about how cross-contamination events happen could lower that difference and therefore possibly raise the probability of guilt. However, I think it's highly unlikely that information about cross-contamination events could raise the probability of guilt anywhere close to the value of 0.373 that I computed just for the case of two cross-contamination events. (And even that probability is still less than 50-50, which should equate to reasonable doubt at the very least.)

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