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Saturday, 15 November 2014
Imagine if you asked people to roll eight dice to see if they can 'hit the jackpot' by rolling 8 out of 8 sixes. The chances are less than 1 in 1.5 million. So if you saw somebody - let's call him Fred - who has a history of 'trouble with authority' getting a jackpot then you might be convinced that Fred is somehow cheating or the dice are loaded. It would be easy to make a convincing case against Fred just on the basis of the unlikeliness of him getting the jackpot by chance and his problematic history.
But now imagine Fred was just one of the 60 million people in the UK who all had a go at rolling the dice. It would actually be extremely unlikely if less than 25 of them hit the jackpot with fair dice (and without cheating) - the expected number is about 35. In any set of 25 people it is also extremely unlikely that there will not be at least one person who has a history of 'trouble with authority'. In fact you are likely to find something worse, since about 10 million people in the UK have criminal convictions, meaning that in a random set of 25 people there are likely to be about 5 with some criminal conviction.
So the fact that you find a character like Fred rolling 8 out of 8 sixes purely by chance is actually almost inevitable. There is nothing to see here and nothing to investigate. As we showed in Section 4.6.3 of our book (or in the examples here) many events which people think of as 'almost impossible'/'unbelievable' are in fact routine and inevitable.
Now, instead of thinking about 'clusters' of sixes rolled from dice, think about clusters of patient deaths in hospitals. Just as Fred got his cluster of sixes, if you look hard enough it is inevitable you will find some nurses associated with abnormally high numbers of patient deaths. In Holland a nurse called Lucia deBerk was wrongly convicted of multiple murders as a result of investigators initially reading too much into such statistics (and then also getting the relevant probability calculations wrong). There have been other similar cases, and as my colleague Richard Gill explains so well it seems that Ben Geen may also have been the victim of such misunderstandings.
See also: Justice for Ben Geen
Update: See Richard Gill's excellent comments below
Update 16 Feb 2015: Guardian article talks about my statement made to the Criminal Cases Review Board.
From Richard Gill:
What happens in both Lucia and Ben's case is not only the surprising coincidence but the magnification of the coincidence, after it has been observed.
Doctors look back at past cases and start reclassifying them. So the dice analogy is not quite correct: it is more like you see someone rolling 5 out of 8 sixes (and it's someone you think is a bit odd in some way), and then you turn over the three non-sixes and make them into sixes too. The you go to the police: "8 out of 8".
This is exactly Lucia: 9 out of 9 - but actually three or four of those dice outcomes had been altered.
A further subtlety is that you never take the trouble to look at a further 20 dice-throws which had also been done and where, surprise surprise, there are only two or three sixes. The dice throws which are investigated are the ones which you remember. Part of the reason you remember them is exactly because that striking nurse about whom people have been gossiping was there.
Interestingly, just recently new cases started up in Italy and in Germany. They might be similar, they might not be. What is common is that the media immediately start spreading all kinds of extremely lurid tales, which, in the case of Lucia and of Ben, certainly turned out to be largely false, and even if there was a tiny snippet of truth in them, they were completely misleading. Here is what the UK media make of the Italian case: http://www.dailymail.co.uk/news/article-2798167/italian-nurse-murdered-38-patients-pleasure-enjoyed-challenge.html. Here is current German case. http://news.newsdirectory1.com/prosecution-nurses-should-have-murdered-100-times/ Sure, maybe these two really are serial killers. But maybe not.
The dice throw analogy is very accurate. A typical full time nurse works roughly half of the days of the whole year (take account of holidays, training courses, absence due to illness, "weekends"), and then just one of the three hospital shifts on a day on which she works. 1 in 6 shifts. So if something odd happens on one of the shifts, there is 1 in 6 chance it happens on his/her shifts. However ... more incidents happen in weekends and some nurses have more than average weekend shifts. Then there is the question: which shift did some event actually happen in? There's a lot of leeway in attributing some developing medical situation to one particular shift. Then there is the question, which shift did the nurse have? There is overlap between shifts, and anyway, sometimes a nurse arrives earlier or leaves earlier. This gives hospital authorities a great deal of flexibility in compiling statistics of shifts with incidents, and shifts with a suspicious nurse. Both in the Ben Geen and in the Lucia de Berk case, a great deal of use was made of this "flexibility" in order to inflate what might well have been chance fluctuations into such powerful numbers that a statistical analysis becomes superfluous: anyone can see "this can't be chance". Indeed. It was not chance. The statistics were fabricated using a prior conviction on the part of investigators (medical doctors at the same hospital, not police investigators) that they have found a serial killer. After that, no-one doubts them.
|Douglas Hubbard (left) and Norman Fenton in London 15 Nov 2014.|
Anyone who has read our Bayesian Networks book or the latest (3rd edition) of my Software Metrics book (the one I gave Douglas in the above picture!) will know how much his work has influenced us recently.
Although we have previously communicated about technical issues by email, today I had the pleasure of meeting Douglas for the first time when we were able to meet for lunch in London.We discussed numerous topics of mutual interest (including the problems with classical hypothesis testing - and how Bayes provides a better alternative, and evolving work on the 'value of information' which enables you to identify where to focus your measurement to optimise your decision-making).