# DNA exoneration: what are the chances?

As with any biometric, both the False Acceptance Rate (i.e. what is the chance that two samples of different individuals match) as well as the False Rejection Rate (i.e. what is the chance that two samples of the same individual do not match) are important.

However, in DNA studies, it seems most attention goes to the FAR. The FAR is important indeed: in court it is unacceptable that an innocent person would have a large chance to be convicted based on a DNA sample. However, the FRR is also important: it is unacceptable that a criminal walks because the DNA test falsely rejected the match.

It seems that "DNA has an extremely low false acceptance rate, but an uncertain false rejection rate." (source: Cyber Crime: Concepts, Methodologies, Tools and Applications)

Now, my question is: are there any studies that have investigated this FRR for DNA? What are the statistics on this? The 'Innocence project' mentions that some criminals have been exonerated based on a DNA test after their conviction. Are we sure that those exonerated criminials were indeed innocent, or were they just falsely rejected?

• Good question. If you haven't read it already, I'd highly recommend Kaiser Fung's book Numbers Rule Your World. It's full of neat examples of this kind of practical applied stats – shadowtalker Feb 4 '15 at 12:57
• Chimerism brings an interesting spin to this whole story, as Lydia Fairchild can attest to. – Marc Claesen Feb 12 '15 at 18:46

Are there any studies that have investigated this FRR for DNA?

Yes, what you are refering to is also called a type II error or a false negative. People have investigated this and also for lab work in general:

A very broad range of error values, with Koehler reporting a ridiculous 12 out of 1000.

What are the statistics on this?

If the test would have been a perfect test it would still have an error rate due to the presence of homozygotic twins in the population and the small chance of two people having identical DNA profiles, called a coincidental match ($\approx 1\cdot 10^{-9}\%$). There are approximatly 0.3% homozygotic twins in the population which gives an error rate of 0.15% in the case of a perfect test.

but...

Since the test is not completely perfect and humans aren't either, there is a larger error introduced. Like every other test there are type I errors which might be due to equipment, hetrogeneous DNA mixture, human error or some unknown external source.

Whenever you design a test you also test the test on its error rate by testing some samples of which you know the outcome: $$False\;negative\;rate= \frac{False\;negative}{False\;negative+True\;positives}$$

This would yield the value of the error rate and is different for different laboratories (due to different people, equipment, etc.). If you are into bayes theorem and would like to know more about error statistics in forensic DNA test: Thompson et al.

It is unacceptable that a criminal walks because the DNA test falsely rejected the match.

Indeed, but a good judge would not make a decision solely on a DNA test and bears in mind there are errors in testing. However when a DNA test is presented, the reported error rate is the chance of getting a coincidental match!

• Interesting papers that you link to. You mention that I refer to type I errors, but I think I am referring to type II errors, right? Also, the fact that homozygotic twins exist and that there is a small change of two people having identical DNA profiles, isn't that an issue for the FAR, instead of the FRR? – Michael Feb 16 '15 at 16:39
• @Michael In both cases you are right, I quickly checked FRR on google and came across this site which also made the mistake! The overall error rate of a perfect forensic DNA test would be .15% which is almost exclusively contributed by the FAR (FRR part is in the coincidental match). – Mehdi Nellen Feb 16 '15 at 17:47
• I went through Kloosterman's paper, and in that paper it is reported that 14 type II errors occurred. However, it is also mentioned that those errors were all due to human error. That still doesn't give me an idea on the real FRR of the DNA algorithms that are currently used (supposing no human errors occur), if you understand what I mean. Koehler is talking about the FAR, and the Lapworth paper I have no access to, but it seems that this paper focuses on real blunders, and not FRRs of the algorithm. – Michael Feb 18 '15 at 15:47
• what do you mean by algorithm? Do you mean the biochemical DNA test? They indeed focus on the errors of humans because the test should not gives errors if it was perfectly performed only the coincidental match which is due to people having the same outcome in the test. The error is therefor almost exclusively contributed by external factors like humans and bad DNA mixtures. – Mehdi Nellen Feb 19 '15 at 8:43
• Well, I want to know the FRR of the biochemical DNA test itself indeed. Due to the finite nature of such a test (not the complete DNA chain is measured, but only certain loci) and due to measuring contaminations, the FRR may be larger or smaller. For example, arm length would not be a good measure, because there are probably a lot of people have the same arm length. DNA is unique, but the test cannot capture the complete DNA profile, so the test itself is not always unique. I don't want to consider post-analysis mistakes like sample switches etc. – Michael Feb 19 '15 at 9:24