Sunday, 1 January 2017

The problem with the likelihood ratio for DNA mixture profiles


We have written many times before (see the links below) about use of the Likelihood Ratio (LR) in legal and forensic analysis.

To recap: the LR is a very good and simple method for determining the extent to which some evidence (such as DNA found at the crime scene matching the defendant) supports one hypothesis (such as "defendant is the source of the DNA") over an alternative hypothesis (such as "defendant is not the source of the DNA"). The previous articles discussed the various problems and misinterpretations surrounding the use of the LR. Many of these arise when the hypotheses are not mutually exclusive and exhaustive. This problem is especially pertinent in the case of 'DNA mixture' evidence, i.e. when some DNA sample relevant to a case comes from more than one person. With modern DNA testing techniques it is common to find DNA samples with multiple (but unknown number of) contributors. In such cases there is no obvious 'pair' of hypotheses that are mutually exclusive and exhaustive, since we have individual hypotheses such as:
  • H1: suspect + one unknown
  • H2: suspect + one known other 
  • H3: two unknowns
  • H4: suspect + two unknowns 
  • H5: suspect + one known other + one unknown
  • H6: suspect + two known others
  • H7: three unknowns 
  • H8: one known other + two unknowns
  • H9: two known others + one unknown
  • H10: three known others
  • H11:  suspect + three unknowns 
  • etc.
It is typical in such situations to focus on the 'most likely' number of contributors (say n) and then compare the hypothesis "suspect + (n-1) unknowns" with the hypothesis "n unknowns". For example, if there are likely to be 3 contributors then typically the following hypotheses are compared:
  • H1: suspect + two unknowns
  • H2: three unknowns
Now, to compute the LR we have two compute the likelihood of the particular DNA trace evidence E under each of the hypotheses. Generally both of these are extremely small numbers, i.e. both the probability values P(E | H1) and P( E | H2) are very small numbers. For example, we might get something like
  • P(E | H1) = 0.00000000000000000001  (10 to the minus 20)
  • P(E | H2) = 0.00000000000000000000000001  (10 to the minus 26)
For a statistician, the size of these numbers does not matter – we are only interested in the ratio (that is precisely what the LR is) and in the above example the LR is very large (one million) meaning that the evidence is a million times more likely to have been observed if H1 is true compared to H2. This seems to be overwhelming evidence that the suspect was a contributor. Case closed?

Apart from the communication problem in court of getting across what this all means (defence lawyers can and do exploit the very low probability of E given H1) and how it is computed, there is an underlying statistical problem with small likelihoods for non-exhaustive hypotheses and I will highlight the problem with two scenarios involving a simple urn example. Superficially, the scenarios seem identical. The first scenario causes no problem but the second one does. The concern is that it is not at all obvious that the DNA mixture problem always corresponds more closely to the first scenario than the second.

In both scenarios we assume the following:

There is an urn with 1000 balls – some of which are white. Suppose W is the (unknown) number of white balls. We have 2 hypotheses:
  • H1: W=100
  • H2:  W=90
We can draw a ball as many times as we like, note its colour and replace it (i.e. sample with replacement). We wish to use the evidence of 10,000 such samples.

Scenario 1: We draw 1001 white balls. In this case using standard statistical assumptions we calculate P(E | H1) = 0.013, P(E|H2) = 0.0000036. Both values are small but the LR is large, 3611, strongly favouring H1 over H2.

Scenario 2: We draw 1100 white balls. In this case P(E | H1) = 0.000057, P(E|H2) < 0.00000001. Again both values are very small but the LR is very large, strongly favouring of H1 over H2.

(note: in both cases we could have chosen a much larger sample and got truly tiny likelihoods but these values are sufficient to make the point).

So in what sense are these two scenarios fundamentally different and why is there a problem?

In scenario 1 not only does the conclusion favouring H1 make sense, but the actual number of balls drawn is very close to the expected number we would get if H1 were true (in fact, W=100 is the 'maximum likelihood estimate' for number of balls). So not only does the evidence point to H1 over H2, but also to H1 over any other hypothesis (and there are 1000 different hypotheses W=0, W=1, W=2 etc.).

In scenario 2 the evidence is actually even much more supportive of H1 over H2 than in scenario 1. But it is essentially meaningless because it is virtually certain that BOTH hypotheses are false.

So, returning to the DNA mixture example, it is certainly not sufficient to compare just two hypotheses. The LR of one million in favour of H1 over H2 may be hiding the fact that neither of these hypotheses is true. It is far better to identify as exhaustive a set of hypotheses as is realistically possible and then determine the individual likelihood value of each hypothesis. We can then identify the hypothesis with the highest likelihood value and consider its LR compared to each of the other hypotheses.

3 comments:

  1. BRING BACK THE GOOD TIMES, THINK ABOUT THE HAPPIEST TIME IN YOUR RELATIONSHIP, DR OGUDU CAN MAKE IT HAPPEN AGAIN, YOU DESERVE TO BE HAPPY, TODAY BRING BACK LOST LOVE INSTANTLY USING HIS EFFECTIVE SPELLS

    Are you looking for ultimate spells that works effectively same time on your problems, ranges from Love spells, marriage spells, and carrier related problems, Gay and lesbian extreme love / attraction spells, women of the night Norton accruing powders and oil for extreme ecstasy and protection with in working ordeals. How to get your ex back permanently? Is this what you are asking yourself? Have you been dumped? Can't you just move on? Did he/she chose someone else over you? Your lover shows no interest in you anymore? You might have been asking yourself how to get your ex lover back permanently and you have been looking for unanswered questions about it, Well today is the day for your worries to be eliminated for good. Dr Ogudu have assisted me in getting my ex lover back,Are you going through emotional torture contact him now so he can use his birth gift of casting strong love spells and his super natural magical powers that mean no harm but motivated by peace, wellness and love. The fact that you ask how to get your lover back permanently is the reason why you should no longer face sorrow on your own, and that is because you are in a perfect condition to win over your ex lover and make him or her fall in love with you all over again. It is a great pain to be in love with someone who does not show the same affection back to you as you hope for. However, it is never too late to have all that you wish for and deserve in a lover. Love spells by (DR OGUDU) are pure nature of magic, they are inspired by true mystery of the beauty of the planet earth and they will do well for you as well. CONTACT HIM NOW AND BE HAPPY AGAIN""""


    EMAIL--[OGUDUSPELLTEMPLE@YAHOO.COM]
    WHATSAPP NUMBER +2348106058254
    Website...http://oguduspelltemple.wixsite.com/drogudu

    ReplyDelete
  2. Hi everyone
    This is the most wonderful thing i have ever experience and i need to share this great testimony. About how i get my ex back after a breakup. I never believed it, because i never heard nor learn anything about it before. Hey guys my wife and I have been married for 15 years. For over five years I have been dealing with jealousy issues because of flirting and her having emotional type affairs with other men. I felt like my masculinity was in question and if I said the way she was acting bothered me, it seemed to make matters worse or I was accused of being controling. During this period I did state I wanted a divorce if the behavior was not going to end: texting men a night, leaving for the weekend without letting me know where she was going or not responding to messages. We do have a son and basically it's been him and I for the last year on the weekends. She disconnected completely from being a good wife and mother. In April she said she wanted a separation and I said no we need to get into marriage counseling. We did try that, but she was not very responsive and didn't give any effort. Afterward she was adamant about separation and divorce. I continued to say no and that we needed to save our family. She presented a separation agreement and I had to hire a lawyer. Because of the above behavior my lawyer suggested a private investigator. The investigator discovered my wife was having an affair with a close friend of the family who also is married with children. We know the extended families. I feel like I should tell this man's wife about what happened. This adultery has devastated me emotionally, I feel betrayed and I'm physically drained. I know my wife is passionate in terms of her sexuality, and I can't get the thought of them out of my head. It makes me question my own manhood, and I feel very inferior or that he must be a better lover or what ever. The problem is my wife pursued him. She would go to him and she lured him into this adultery. I felt this was coming for some time and could not stop it. She was not only lying to me but also to our son about what she was doing and where she was going. My family is important, my son loves her and as crazy as it sounds so do I. Can you respond with a course of action on how to proceed? I was still have a very huge place in my heart for her. so i searched for help online and I came across a website that suggested that Dr Ahmed can help solve marital problems, restore broken relationships and so on. So I felt I should give him a try. I contacted him and he told me what to do and i did it then he did a spell for me. 28 hours later, my wife came to me and apolo[filtered]ed for the wrongs she did and promise never to do it again. Ever since then, everything has returned back to normal. I, my son and my wife are living together happily again.. All thanks to Dr Ahmed. as it is a place to resolve marriage/relationship issues, do you want to be sure if your spouse is being faithful to you or Do you want your Ex to come back to you Contact.: E-mail: Ahmedutimate@gmail.com or call/Whats-app: +2348160153829 save your crumbling home and change of grades its 100% safe. I suggest you contact him. He will not disappoint you. David L. Ollis, 43yrs, UK

    ReplyDelete
  3. Real And Urgent Spell Caster Reviews To Solve Your Relationship Problem Dr Goodluck..
    My mind, for some reason, wont let go of my memory's of My husband after i found out my Husband of 9years had been living a double life and we separated months ago because of him cheating I was totally frustrated and did not know how to bring him back home because i love him so much and have tried all my possible best all to no avail, as i was browsing on the internet i came across a post on YouTube about Dr Goodluck who help people with their relationship and marriage problem by the powers of love spells, at first I doubted if such thing ever exists but decided to give it a try, when I contacted him i told him everything and he told me that everything will be fine that my husband is going to come back to me, And he told me all i will need to provide him with and i did and he told me that he is going to cast a returning spell so that my husband will come back, he helped me cast a spell and guarantee me that my husband will come back to me and within 17hours, to my greatest surprise, my husband came back to me and started apologizing, now he has stopped going out with women and he is with me for good and for real. if you have any problem contact Dr Goodluck he is real and always ready to solve your problem, email him on goodluck05spellcaster@gmail.com and you can also whatapp him or call with his mobile +2349059610309 and i promise you that he is going to help you with any problem OKay…

    +2348138654465
    goodluck05spellcaster@gmail.com
    http://goodluck05spellcaster.yolasite.com

    ReplyDelete